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Collected Brief Summaries of
Graham Priest
An Introduction to NonClassical Logic:
From If to Is
Propositional Logic
Mathematical Prolegomenon
Settheoretic Notation
The following are definitions for basic settheoretical notions that will appear throughout this book. [The following is quotation from pp.xxviixxix.]
set
A set, X, is a collection of objects. If the set comprises the objects a_{1}, ... , a_{n}, this may be written as {a_{1}, ... , a_{n}}. If it is the set of objects satisfying some condition, A(x), then it may be written as {x :A(x)}.
membership
a ∈ X means that a is a member of the set X, that is, a is one of the objects in X. a ∉ X means that a is not a member of X.
singleton
for any a, there is a set whose only member is a, written {a}. {a} is called a singleton (and is not to be confused with a itself).
empty set
There is also a set which has no members, the empty set; this is written as φ.
subset
A set, X, is a subset of a set, Y, if and only if every member of X is a member of Y. This is written as X ⊆ Y. The empty set is a subset of every set (including itself).
proper subset
X ⊂ Y means that X is a proper subset of Y; that is, everything in X is in Y, but there are some things in Y that are not in X. X and Y are identical sets, X = Y, if they have the same members, i.e., if X ⊆ Y and Y ⊆ X. Hence, if X and Y are not identical, X ≠ Y, either there are some members of X that are not in Y, or vice versa (or both).
union
The union of two sets, X, Y, is the set containing just those things that are in X or Y (or both). This is written as X ∪ Y. So a ∈ X ∪ Y if and only if a ∈ X or a ∈ Y.
intersection
The intersection of two sets, X, Y, is the set containing just those things that are in both X and Y. It is written X ∩ Y. So a ∈ X ∩ Y if and only if a ∈ X and a ∈ Y.
relative complement
The relative complement of one set, X, with respect to another, Y, is the set of all things in Y but not in X. It is written Y − X. Thus, a ∈ Y − X if and only if a ∈ Y but a ∉ X.
ordered pair
An ordered pair, ⟨a, b⟩, is a set whose members occur in the order shown, so that we know which is the first and which is the second. Similarly for an ordered triple, ⟨a, b, c⟩, quadruple, ⟨a, b, c, d⟩, and, in general, ntuple, ⟨x_{1}, . . . , x_{n}⟩.
cartesian product
Given n sets X_{1}, . . . , X_{n}, their cartesian product, X_{1}×· · ·×X_{n}, is the set of all ntuples, the first member of which is in X_{1}, the second of which is in X_{2}, etc. Thus, ⟨x_{1}, . . . , x_{n}⟩ ∈ X_{1}×· · ·×X_{n} if and only if x_{1 }∈ X_{1 }and . . . and x_{n }∈ X_{n}.
subset relation
A relation, R, between X_{1}×· · ·×X_{n} is any subset of X_{1}×· · ·×X_{n}.  ⟨x_{1}, . . . , x_{n}⟩ ∈ R is usually written as Rx_{1 }. . . x_{n}.
ternary and binary relations
If n is 3, the relation is a ternary relation. If n is 2, the relation is a binary relation, and Rx_{1 }x_{2 }is usually written as x_{1}Rx_{2}.
function
A function from X to Y is a binary relation, f , between X and Y, such that for all x ∈ X there is a unique y ∈ Y such that xfy. More usually, in this case, we write: f(x) = y.
(Priest xxviixxix)
Classical Logic and the Material Conditional
Introduction
The main purpose of logic is to provide an account of validity, which determines what follows from what. The account is given in a metalanguage for a formal, object language. There are two types of validity: {1} Semantic validity (symbolized ⊨) which preserves truth: every interpretation that makes the premises true also makes the conclusion true. {2} Prooftheoretic validity (symbolized ⊢) which is determined by means of a procedure operating on a symbolization of the inference. Most contemporary logicians think that semantic validity is more fundamental than prooftheoretic, but it is good nonetheless to provide a prooftheoretic notion of validity to correspond with a semantic notion. A prooftheory is sound when “every prooftheoretically valid inference is semantically valid (so that ⊢ entails ⊨)” and it is complete when “every semantically valid inference is prooftheoretically valid (so that ⊨ entails ⊢)” (4).
The Syntax of the Object Language
Our object language has a formalized syntax with the following notations.
Propositional parameters (propositional variables):
p_{0}, p_{1}, p_{2}, ....
Connectives:
¬ (negation), ∧ (conjunction), ∨ (disjunction), ⊃ (material conditional), ≡ (material equivalence)
Punctuation:
(, )
Arbitrary indistinct formulas:
A, B, C, ...
Arbitrary distinct formulas:
p, q, r, ...
Arbitrary sets of formulas:
Σ, Π, ...,
Empty set:
φ
Outer parentheses around complex formulas and curly brackets around finite sets are omitted. Wellformed formulas are either propositional parameters or complex formulas built up upon propositional parameters by means of the connectives.
Semantic Validity
An interpretation of an object language is a function, written v, that assigns truth values to formulas, as for example: ν(p) = 1 and ν(q) = 0. For our classical logic semantics, the interpretation function assigns values for the connectives in the following way:
ν(¬A) = 1 if ν(A) = 0, and 0 otherwise.
ν(A ∧ B) = 1 if ν(A) = ν(B) = 1, and 0 otherwise.
ν(A ∨ B) = 1 if ν(A) = 1 or ν(B) = 1, and 0 otherwise.
ν(A ⊃ B) = 1 if ν(A) = 0 or ν(B) = 1, and 0 otherwise.
ν(A ≡ B) = 1 if ν(A) = ν(B), and 0 otherwise.
A conclusion A is a semantic consequence of a set of the premises Σ (that is, Σ ⊨ A) only if there is no interpretation that makes all the members of Σ true and A false, that is, only if every interpretation that makes all the members of Σ true makes A true as well. ‘Σ ⊭ A’ means there is not semantic consequence. A logical truth or tautology is a formula that is true under every evaluation, written for example as: ⊨ A. This also means it is a semantic consequence of the empty set of premises: φ ⊨ A.
Tableaux
We construct structures called tableaux to test for certain properties of arguments and formulas, especially validity and prooftheoretic consequence. The tableau has a structure of branches from a root down to tips. The structure can be displayed in the following way:
↓∙↙ ↘∙ ∙↓ ↙ ↘∙ ∙ ∙
↓∙↙ ↘∙ ∙↓ ↙ ↘∙ ∙ ∙
Double Negation Development (¬¬D) 
............¬¬A .............↓ .............A 
Conjunction Development (∧D) 
...........A ∧ B .............↓ .............A .............↓ .............B 
Negated Conjunction Development (¬∧D) 
¬(A ∧ B) ↙ ↘ ¬A ¬B 
Disjunction Development (∨D) 
...........A ∨ B ..........↙.....↘ ........A.........B 
Negated Disjunction Development (¬∨D) 
.........¬(A ∨ B) .............↓ ............¬A .............↓ ............¬B 
Conditional Development (⊃D) 
...........A ⊃ B ..........↙.....↘ .......¬A.........B 
Negated Conditional Development (¬⊃D) 
..........¬(A ⊃ B) ..............↓ ..............A ..............↓ .............¬B 
Biconditional Development (≡D) 
...........A ≡ B ..........↙.....↘ ........A........¬A ........↓.........↓ ........B........¬B 
Negated Biconditional Development (¬≡D) 
.........¬(A ≡ B) ..........↙.....↘ ........A........¬A ........↓.........↓ .......¬B.........B 
Conditionals
The Material Conditional
Subjunctive and Counterfactual Conditionals
(1.8.1) A strong objection to the semantics of the material conditional and its application to natural language conditionals are sentences with similar antecedents and consequents but on account of subtle grammatical differences have opposite truth values. Priest’s examples are: {1} If Oswald didn’t shoot Kennedy someone else did. (which is true), and {2} If Oswald hadn’t shot Kennedy someone else would have. (which is false). (1.8.2) One common way to deal with the apparent inconsistency in the above examples is to distinguish them in terms of grammatical properties and say that one type is not a material conditional. When a conditional sentence is indicative, it could be material, but when it is subjunctive or counterfactual, often using “would,” it is not material. (1.8.3) The English conditional is probably not ambiguous between subjunctive and indicative moods, on account of explicit syntactical differences that maintain a clear distinction. (1.8.4) Conditionals are subjunctive when they articulate a temporal perspective located before the stated event or fact, and they are indicative if they articulate a temporal perspective where that event or fact is established.
More CounterExamples
(1.9.1) There are three other counterexamples to the material conditional, and they present damning objections to the claim that the English conditional is material. {1} (A ∧ B) ⊃ C ⊢ (A ⊃ C) ∨ (B ⊃ C); for example, “If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on.” {2} (A ⊃ B) ∧ (C ⊃ D) ⊢ (A ⊃ D) ∨ (C ⊃ B); for example, “If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.” And {3} ¬(A ⊃ B) ⊢ A; for example, “It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god” (1415). (1.9.2) We cannot dismiss these counterexamples on grammatical grounds, because they are all in the indicative mood. And we cannot dismiss them on conversational implicature grounds, because none break the rule of assert the strongest. (1.9.3) We cannot object that in fact the above counterexamples really are valid, provided we stipulate that the English conditional is material in those cases. For, by making that stipulation, we are admitting that naturally the English conditional is not material and is only artificially so. But the whole point of these objections is to show that the English conditional is naturally material.
Arguments for ⊃
(1.10.1) Even though the material conditional, ⊃, is not properly suited to describe the functioning of the English conditional, it had come to be regarded as such on account of there only being standard truthtable semantics until the 1960s, and the only plausible candidate in that semantics for “if” formations would be the material conditional. (1.10.2) However, there are notable arguments that the material conditional can be used to understand the English conditional, and they construe that relation in the following way: “‘If A then B’ is true iff ‘A ⊃ B’ is true.” (1.10.3) ‘If A then B’ is true then ¬A ∨ B is true. (1.10.4) Suppose A and ¬A ∨ B are true. By disjunctive syllogism: A, ¬A ∨ B ⊢ B. This fulfills (*), when we take ¬A ∨ B as the C term. [Now, since A, ¬A ∨ B ⊢ B fulfills the definition of the English conditional, and since A, ¬A ∨ B ⊢ B also gives us the (modus ponens) logic of the conditional (given the equivalence of A ⊃ B and ¬A ∨ B), that means the logic of the English conditional is adequately expressed by A ⊃ B.] (1.10.5) Suppose A and ¬A ∨ B are true. By disjunctive syllogism: A, ¬A ∨ B ⊢ B. This fulfills (*), when we take ¬A ∨ B as the C term. (Now, since A, ¬A ∨ B ⊢ B fulfills the definition of the English conditional, and since A, ¬A ∨ B ⊢ B also gives us the (modus ponens) logic of the conditional (given the equivalence of A ⊃ B and ¬A ∨ B), that means the logic of the English conditional is adequately expressed by A ⊃ B.) (1.10.6) What later proves important in the above argumentation is the use of disjunctive syllogism.
Basic Modal Logic
Introduction
We will examine possibleworld semantics and the most basic modal logic, K.
Necessity and Possibility
Modal logic deals with “the modes in which things may be true/false.” Such modes include possibility, necessity and impossibility. Modal semantics can employ the concept of possible worlds, which may be understood provisionally as a world situation that is a variation on our own, with it having slightly (or remarkably) different features. One world is possible relative to another if for example the one could actually become an outcome of the other.
Modal Semantics
In our modal semantics, we add to our propositional language two modal operators, □ for ‘necessarily the case that’ and ◊ for ‘possibly the case that’. An interpretation in our modal semantics takes the form ⟨W, R, v⟩, with W as the set of worlds, R as the accessibility relation, and v as the valuation function. ‘uRv’ can be understood as either, “world v is accessible from u,” “in relation to u, situation v is possible,” or “world u access world v.” Negation, conjunction, and disjunction are evaluated (assigned 0 or 1) just as in classical propositional logic, except here we must specify in which world the valuation holds.
ν_{w}(¬A) = 1 if ν_{w}(A) = 0, and 0 otherwise.
ν_{w}(A ∧ B) = 1 if ν_{w}(A) = ν_{w} (B) = 1, and 0 otherwise.
ν_{w}(A ∨ B) = 1 if ν_{w}(A) = 1 or ν_{w} (B) = 1, and 0 otherwise.
(21)
A formula is possibly true in one world if it is also true in another world that is possible in relation to the first. A formula is necessarily true in a world if it is also true in all worlds that are possible in relation to it.
For any world w ∈ W:
ν_{w}(◊A) = 1 if, for some w′ ∈ W such that wRw′, ν_{w′}(A) = 1; and 0 otherwise.
ν_{w}(□A) = 1 if, for all w′ ∈ W such that wRw′, ν_{w′}(A) = 1; and 0 otherwise.
(22)
Given these definitions, we can conclude that if a world has no other related worlds, then any ◊A formulation will be false in that world (for, it cannot be true in any related world, as there are none), and any □A formulation will be true (for, it is the case that it is true in every accessible world, as there are no accessible worlds). We can diagram the interpretation. Consider this example of an interpretation:
W = {w_{1}, w_{2}, w_{3}}
w_{1}Rw_{2}, w_{1}Rw_{3}, w_{3}Rw_{3}
v_{w1 }(p) = 0, v_{w1 }(q) = 0;
v_{w2 }(p) = 1, v_{w2 }(q) = 1,
v_{w3 }(p) = 1, v_{w3 }(q) = o,
This is depicted as:
xxxxxxxxxxxxw_{2}xxpxxq
xxxxxxxxxx↗
¬px¬qxxw_{1}
xxxxxxxxxx↘x↷
xxxxxxxxxxxxw_{3}xxpxx¬q
Each world (w_{1}, w_{2}, w_{3}) is given its own place on the diagram. Arrows from one world to another indicate the accessibility of the first to the second. The rounded arrow (high above w_{3}) thus means the accessibility of a world to itself. And all the true propositions in a world are listed in that world’s place on the diagram (so if a formula is valuated as 0, its negation is listed). Then, on the basis of our rules, we can infer the following other formulas for each world:
xxxxxxxxxxxxxxxxxxxw_{2}xxxxxpxxxxxq
xxxxxxxxxxxxxxxx↗xxxxxxxxp∧qxxx¬◊q
¬pxxxxx¬qxxxxxw_{1}
◊p∧q xxx◊□pxxxxx↘x↷
xxxxxxxxxxxxxxxxxxw_{3}xxxxxpxxxxx¬q
xxxxxxxxxxxxxxxxxxxxxxxx□pxxxxxxxxxx
¬◊A at any world is equivalent to □¬A. And, ¬□A at any world is equivalent to ◊¬A. An inference is valid (as a semantic consequence) if it is truthpreserving in all worlds of all interpretations, (that is, if in all worlds in all interpretations, whenever the premises are true, so too is the conclusion). A logical truth (or tautology) is a formula that is true in all worlds of all interpretations.
Σ ⊨ A iff for all interpretations ⟨W, R, v⟩ and all w ∈ W: if ν_{w}(B) = 1 for all B ∈ Σ, then ν_{w}(A) = 1.
⊨ A iff φ ⊨ A, i.e., for all interpretations ⟨W, R, v⟩ and all w ∈ W, ν_{w}(A) = 1.
(23)
Modal Tableaux
(2.4.1) Tableaux in modal logic take the same branching node structure as those for propositional logic. However, the nodes themselves have a different structure, and there are two possible ones. {1} A, i, where A is a formula and i is a natural number indicating the world in which the formula holds, or {2} irj, where i is a natural number for a world that accesses world j, also given as a natural number (the r stays as r). (2.4.2) We test for validity by setting the premises to true in world 0 and the negation of the conclusion to true in world 0. (2.4.3) The tableaux rules for modal logic are the same as for nonmodal propositional logic, except we indicate the worlds involved, and the branches inherit the world indicators from above (they are listed below with the new ones). (2.4.4) There are four new tableaux rules for modal operators. (Here we list all rules together):
Double Negation Development (¬¬D) 
¬¬A,i ↓ A,i 
Conjunction Development (∧D) 
A ∧ B,i ↓ A,i B,i 
Negated Conjunction Development (¬∧D) 
¬(A ∧ B),i ↓ ¬A ∨ ¬B,i 
Disjunction Development (∨D) 
A ∨ B,i ↙ ↘ A,i B,i 
Negated Disjunction Development (¬∨D) 
¬(A ∨ B),i ↓ ¬A,i ↓ ¬B,i 
Conditional Development (⊃D) 
A ⊃ B,i ↙ ↘ ¬A,i B,i 
Negated Conditional Development (¬⊃D) 
¬(A ⊃ B),i ↓ A,i ↓ ¬B,i 
Negated Necessity Development (¬□D) 
¬□A,i ↓ ◊¬A,i 
Negated Possibility Development (¬◊D) 
¬◊A,i ↓ □¬A,i 
Relative Necessity Development (□rD) 
□A,i irj ↓ A,j
(both □A,i and irj must occur somewhere on the same branch, but in any order or location) 
Relative Possibility Development (◊rD) 
◊A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) 
(24)
(2.4.5) Branches close when there are contradictions in the same world. (2.4.6) Priest provides examples to show how the tableaux are made. (2.4.7) We make countermodels using completed open branches. We assign worlds in accordance with the i numbers. We assign R relations in accordance with irj formulations. And nodes of the form p, i we assign v_{wi}(p) = 1. And for nodes of the form ¬p, i, we assign v_{wi}(p) = 0. If there are neither of these two, then v_{wi}(p) can be given any value we want. (2.4.8) Priest shows how to make a countermodel with an example. (2.4.9) These tableaux are both sound and complete.
Possible Worlds: Representation
Possible world semantics is mathematical machinery. But it represents certain real features of truth and validity. We wonder, what exactly do possible worlds and their semantics represent, philosophically speaking?
Modal Realism
(2.6.1) Modal realism is the view that possible worlds are real worlds that exist at different times and/or places. (2.6.2) The fact that modal realism is mindboggling should not be a problem, because we allow modern physics to boggle our minds. (2.6.3) An objection to modal realism is that other possible worlds that are real would have to be physically related to our real world and thus be extensions of our world or copartitions of one ultimate real world. The reply is that the fact that the other possible worlds are not spatially, temporally, or causally connected to our means they have no physical connection and thus cannot be extensions or copartitions with our world. (2.6.4) Still, the objector can give the example of black holes where it is conceivable that there is a part of this world that is not spatially, temporally, or causally related to the rest of our world. (2.6.5) It can also be objected that we should not define possibility itself in terms of alternate reality or physically disconnected actuality, because intuitively we do not think that some actuality in our present world demonstrates its possibility in other times or places of our world.
Modal Actualism
(2.7.1) Under modal actualism, possible worlds are understood not as physically real entities, like in modal realism, but rather as abstract entities, like numbers. (2.7.2) One version of modal actualism understands a possible world as a set of propositions or other languagelike entities and as being “individuated by the set of things true at it, which is just the set of propositions it contains” (29). (2.7.3) One problem with the propositional understanding of possible worlds is that there are many sorts of sets of propositions, but not all constitute worlds. For example, “a set that contains two propositions but not their conjunction could not be a possible world” (29). (2.7.4) A big problem with the propositional understanding of possible worlds is that in order for propositions to form a world, we need to know which inferences follow validly from others. Then, after knowing that, we can apply the mathematical machinery to explain which inferences are valid. But as you can see, the mathematical machinery, which is was we are trying to substantiate with this propositional account, is made useless, as it is what is supposed to determine validity, not take validity readymade and redundantly confirm it. (2.7.5) To avoid this problem of validity, there is another sort of modal actualism called combinatorialism. Here a possible world is understood as “the set of things in this world, rearranged in a different way. So in this world, my house is in Australia, and not China; but rearrange things, and it could be in China, and not Australia” (30). (2.7.6) Because arrangements are abstract objects, combinatorialism is a sort of modal actualism. And because combinations can be explained without the notion of validity, combinatorialism avoids the problems of validity that the propositional understanding suffered from. (2.7.7) One big problem with combinatorialism is that it is unable to generate all possible worlds. For, there could be objects in other possible worlds not found in our world or in any other possible world obtained by rearranging the objects in our world.
Meinongianism
(2.8.1) In modal realism, possible worlds and their members are concrete objects, and in modal actualism, they are abstract objects. In both cases, they are existing objects. Now we will consider the idea that they are nonexistent objects (a position called Meinongianism). (2.8.2) We are already familiar with such nonexistent things as Santa Claus and phlogiston. We can think of possible worlds in the same light. (2.8.3) Meinong famously held that there are nonexistent objects, and the arguments against his position are not especially cogent. (2.8.4) An example of an uncogent argument against Meinongism is that nonexistent possible worlds cannot causally interact with us, and thus we can know nothing about them. Yet, this objection would also hold for modal actualism and modal realism too. (2.8.5) That same objection also fails to take into account the fact that we do know facts about certain nonexistent objects on account of these facts being stipulated, for example: “Holmes lived in Baker Street – and not Oxford Street – because Conan Doyle decided it was so” (31). (2.8.6) Priest ends by noting that {1} the aforementioned ideas do not settle the matter, as there are more suggestions to consider, and {2} there are more objections to consider.
Normal Modal Logics
Introduction
In this chapter, we examine some extensions of the modal logic K. We also address the question of which modal logics are most suitable for certain sorts of necessity, and we end by examining tense logics with more than one pair of modal operators.
Semantics for Normal Modal Logics
We distinguish the types of modal logic by subscripting their name to the turnstile, as for example: ⊨_{K}. There are different classes of modal logics. Normal logics are the most important class, and K is the most basic of them. The different modal logics are defined according to certain constraints on the accessibility relation, R, including:
ρ (rho), reflexivity: for all w, wRw.
σ (sigma), symmetry: for all w_{1}, w_{2}, if w_{1}Rw_{2}, then w_{2}Rw_{1}.
τ (tau), transitivity: for all w_{1}, w_{2}, w_{3}, if w_{1}Rw_{2} and w_{2}Rw_{3}, then w_{1}Rw_{3}.
η (eta), extendability: for all w_{1}, there is a w_{2} such that w_{1}Rw_{2}.
An interpretation in which R satisfies conditions ρ (or σ, etc.) is a ρinterpretation (or a σinterpretation, etc.). A logic defined in terms of truth preservation over all worlds of all ρinterpretations is called Kρ (or Kσ, etc). The consequence relation of such a logic is written ⊨_{Kρ} (or ⊨_{Kσ}, etc.). So we would say for example that Σ ⊨_{Kρ} A if and only if for all ρinterpretations ⟨W, R, v⟩, and all w ∈ W, if v_{w}(B) = 1 for all B ∈ Σ, then v_{w}(A) = 1. We can combine the R conditions to get additional sorts of interpretations, like a ρσinterpretation for example. Then, the logic Kστ is the consequence relation defined over all στinterpretations. There are some conventional names for certain various such logics, like S5 for Kρστ, S4 for Kρτ, and B for Kρσ. In nearly all cases, the conditions on R are independent, and they can be mixed and matched at will. Every normal modal logic, L, is an extension of K, in the sense that if Σ ⊨_{K} A then Σ ⊨_{L} A. A restricted K modal logic will have fewer interpretations than K on account of many of the K interpretations not meeting the restriction’s criterion. However, these restrictions also happen to allow the restricted K logics to make more inferences valid. Thus there is an inverse relation between inferences and interpretations with respect to the effects of the restrictions. For this reason Kρσ is an extension of Kρ; Kρστ is an extension of Kρσ, and so on.
Tableaux for Normal Modal Logics
(3.3.1) To make tableaux for other normal modal logics, we will add rules regarding the R accessibility relation. (3.3.2) The tableaux for the different normal modal logics take rules reflecting the properties of the accessibility relations that characterize them.
Tableaux Rules for Kρ, Kσ, and Kτ  
ρ . ↓ iri . . “ρrD”  σ irj ↓ jri . . “σrD”
 τ irj jrk ↓ .irk . “τrD” 
(3.3.3) In the first tableau example for normal modal logics, we learn that □p ⊃ p is valid in Kρ but not in K; thus Kρ is a proper extension of K. (3.3.4) In Priest’s second example, we learn that p ⊃ □◊p is not valid in K but it is valid in Kσ, thus Kσ is a proper extension of K. (3.3.5) In the third of Priest’s examples, we learn that □p ⊃ □□p is not valid in K but it is valid in Kτ, thus Kτ is a proper extension of K. (3.3.6) For compound systems, we must apply the rules for each restriction. When making the tableau, we should apply the ◊rule first. Then secondly we compute and add all the needed new facts about r that then arise. Lastly we should backtrack whenever necessary to apply the □rule in cases of r where it is required. (3.3.7) We make countermodels by assigning worlds in accordance with the i numbers on an open branch, r relations in accordance with the irj formulations, p,i formulations as v_{wi}(p) = 1, ¬p,i formulations as v_{wi}(p) = 0, and if neither of those two cases show for some p, we can assign it any value we want. (3.3.8) These tableaux are both sound and complete.
S5
(3.5.1) The normal modal logic S5 has the universal or υ (upsilon) constraint, meaning that every world relates to every other world: for all w_{1} and w_{2}, w_{1}Rw_{2}. (3.5.2) Given that under an υinterpretation, all worlds access all others, we need not be concerned with the parts of our semantic evaluation rules that mention the R relation. As such, we evaluate necessity and possibility operators in the following way:
v_{w}(□A) = 1 iff for all w′ ∈ W, v_{w′ }(A) = 1
v_{w}(◊A) = 1 iff for some w′ ∈ W , v_{w′}(A) = 1
(3.5.3) We make our tableaux for S5 using the tableau rules for modal logic, but eliminating the r designations; and: “Applying the ◊rule to ◊A,i gives a new line of the form A,j (new j); and in applying the □rule to □A,i, we add A,j for every j” (45).
S5 Relative Necessity Development (□rD) 
□A,i ↓ A,j
(for every j) 
S5 Relative Possibility Development (◊rD) 
◊A,i ↓ A,j
(j must be new: it cannot occur anywhere above on the branch) 
(modified from p.24, section 2.4.4)
(3.5.4) Kρστ and Kυ are equivalent logical systems, because whatever is semantically valid in the one is semantically valid in the other. (3.5.5) S5 stands for both Kυ and Kρστ, on account of their logical equivalence. (3.5.6) S numbering indicates the system’s relative strength.
The Tense Logic K^{t}
(3.6a.1) We will now examine tense logic. (3.6a.2) The semantics of tense logic are the same as modal logic, only with some modifications to reflect certain temporal senses. The notion of succession is modeled with the accessibility relation such that w_{1}Rw_{2} has the intuitive sense: ‘w_{1} is earlier than w_{2}’. “□A means something like ‘at all later times, A’, and ◊A as ‘at some later time, A’,” but “we will now write □ and ◊ as [F] and ⟨F⟩, respectively. (The F is for ‘future’)” (49). (3.6a.3) The tense logic operators for the past are [P] and ⟨P⟩, which correspond semantically to □ and ◊. (3.6a.4) We evaluate the tense operators in the following way:
v_{w}([P]A) = 1 iff for all w′ such that w′Rw, v_{w′ }(A) = 1
v_{w}(⟨P⟩A) = 1 iff for some w′ such that w′Rw, v_{w′ }(A) = 1
v_{w}([F]A) = 1 iff for all w′ such that wRw′, v_{w′ }(A) = 1
v_{w}(⟨F⟩A) = 1 iff for some w′ such that wRw′, v_{w′ }(A) = 1
(50, with the future operator formulations being my guesses.)
(3.6a.5) “If, in an interpretation, R may be any relation, we have the tenselogic analogue of the modal logic, K, usually written as K^{t}” (50). (3.6a.6) The tableaux rules for the tense operators is much like for necessity and possibility only we need to keep in mind the order of r formulations for the different tenses. Priest provides the following tableau rules for the tense operators.
Full Future Development ([F]D) 
[F]A,i irj ↓ A,j
(For all j) 
Partial Future Development (⟨F⟩D) 
⟨F⟩A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) 
Negated Full Future Development (¬[F]D) 
¬[F]A,i ↓ ⟨F⟩¬A,i 
Negated Partial Future Development (¬⟨F⟩D) 
¬⟨F⟩A,i ↓ [F]¬A,i 
Full Past Development ([P]D) 
[P]A,i jri ↓ A,j
(For all j) 
Partial Past Development (⟨P⟩D) 
⟨P⟩A,i ↓ jri A,j
(j must be new: it cannot occur anywhere above on the branch) 
Negated Full Past Development (¬[P]D) 
¬[P]A,i ↓ ⟨P⟩¬A,i 
Negated Partial Past Development (¬⟨P⟩D) 
¬⟨P⟩A,i ↓ ⟨P⟩¬A,i 
(50, with my added names and other data at the bottoms)
(3.6a.7) Priest then gives a tableau example. (3.6a.8) Priest then shows how to construct a countermodel in tense logic, using an example. (We use the same procedure given in section 2.4.7.) (3.6a.9) We can think of time going in reverse, from the future, moving backward through the past, by taking the converse R relation (yRx becomes xŘy) (and/or by converting all F’s to P’s and vice versa ).
Extensions of K^{t}
(3.6b.1) We can apply constraints on the accessibility relation to obtain extensions of our modal tense logic K^{t}. (3.6b.2) These constraints on R condition the way ‘x is before y’ behaves. For example, the transitivity constraint makes beforeness transitive, and we can represent the beginninglessness or endlessness of time using the extendability constraint. (3.6b.3) Priest next notes some natural constraints for tense logic. {1} denseness (δ): if xRy then for some z, xRz and zRy, which places a moment between any two others; {2} forward convergence (ϕ): if xRy and xRz then (yRz or y = z or zRy); that is to say, when two moments come after some given moment, then they cannot belong to two distinct futures but must instead fall along the same timeline; and {3} backward convergence (β): if yRx and zRx then (yRz or y = z or zRy); in other words, two moments coming before some given moment must fall along a single succession. (3.6b.4) Priest next gives the tableau rules for constrained tense logics. (For convenience, I have here added ones in later sections to keep all the rules in one place.
Double Negation Development (¬¬D) 
¬¬A,i ↓ A,i 
Conjunction Development (∧D) 
A ∧ B,i ↓ A,i B,i 
Negated Conjunction Development (¬∧D) 
¬(A ∧ B),i ↓ ¬A ∨ ¬B,i 
Disjunction Development (∨D) 
A ∨ B,i ↙ ↘ A,i B,i 
Negated Disjunction Development (¬∨D) 
¬(A ∨ B),i ↓ ¬A,i ↓ ¬B,i 
Conditional Development (⊃D) 
A ⊃ B,i ↙ ↘ ¬A,i B,i 
Negated Conditional Development (¬⊃D) 
¬(A ⊃ B),i ↓ A,i ↓ ¬B,i 
(p.24, section 2.4.3 and 2.4.4)
Full Future Development ([F]D) 
[F]A,i irj ↓ A,j
(For all j) 
Partial Future Development (⟨F⟩D) 
⟨F⟩A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) 
Negated Full Future Development (¬[F]D) 
¬[F]A,i ↓ ⟨F⟩¬A,i 
Negated Partial Future Development (¬⟨F⟩D) 
¬⟨F⟩A,i ↓ [F]¬A,i 
Full Past Development ([P]D) 
[P]A,i jri ↓ A,j
(For all j) 
Partial Past Development (⟨P⟩D) 
⟨P⟩A,i ↓ jri A,j
(j must be new: it cannot occur anywhere above on the branch) 
Negated Full Past Development (¬[P]D) 
¬[P]A,i ↓ ⟨P⟩¬A,i 
Negated Partial Past Development (¬⟨P⟩D) 
¬⟨P⟩A,i ↓ ⟨P⟩¬A,i 
(p.50, section 3.6a.6, with my added names and other data at the bottoms)
α= World Equality (α=D) 
α(i) j=i ↓ α(j) . α(i) i=j ↓ α(j) .
where α(i) is a line of the tableau containing an ‘i’. α(j) is the same, with ‘j’ replacing ‘i’. Thus: if α(i) is A, i, α(j) is A, j if α(i) is kri, α(j) is krj if α(i) is i = k, α(j) is j = k 
(53; with my naming additions and copied text at the bottom)
ρ, Reflexivity (ρrD) 
ρ . ↓ iri 
σ, Symmetry (σrD) 
σ irj ↓ jri 
τ, Transitivity (τrD) 
τ irj jrk ↓ irk 
η, Extendability (ηrD) 
η . ↓ irj .
It is applied to any integer, i, on a branch, provided that there is not already something of the form irj on the branch, and the j in question must then be new. 
δ, Denseness (δrD) 
δ irj ↓ irk krj
.
where k is new to the branch. 
ϕ, Forward Convergence (δrD) 
ϕ irj irk ↙ ↓ ↘ jrk j=k krj .
where i, j and k are distinct. 
β, Backward Convergence (βrD) 
β jri kri ↙ ↓ ↘ jrk j=k krj .
where i, j and k are distinct. 
(53; with my naming additions and copied text at the bottom)
(3.6b.5) In order to give the tableau rules for the ϕ and β constraints, we first need the rules for world equality. (See the table “α= World Equality (α=D)” above.) (3.6b.6) Priest next gives the rules for the ϕ and β constraints. (See them above.) (3.6b.7) Priest next gives an example tableau. (3.6b.8) We make countermodels in the following way. “For each number, i, that occurs on the branch, there is a world, w_{i}; w_{i}Rw_{j }iff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, v_{wi}(p) = 1, if ¬p, i occurs on the branch, v_{wi}(p) = 0 (and if neither, v_{wi}(p) can be anything one wishes)” (p.27); however, “whenever there is a bunch of lines of the form i = j, j = k, . . . , we choose only one of the numbers, say i, and ignore the others” (54). (3.6b.9) “The tableaux for K^{t} and its various extensions are sound and complete with respect to their semantics” (55). (3.6b.10) The past cannot be altered, so there can only be one timeline of past events. But supposing that the future can be altered, then there are numerous timelines of sequences of future events. “Thus, one might suppose, time satisfies the condition β of backward convergence, but not the condition ϕ of forward convergence” (5556). (3.6b.11) But just because there are two possible futures – ◊⟨F⟩p ∧ ◊⟨F⟩¬p) – does not mean there are two actual futures – ⟨F⟩p∧⟨F⟩¬p.
NonNormal Modal Logics; Strict Conditionals
Introduction
NonNormal Worlds
(4.2.1) We will first examine the technical elements of nonnormality. (4.2.2) Our interpretations of nonnormal modal logics take the structure ⟨W, N, R, v⟩. W is the set of worlds. R is the accessibility relation. v is the valuation function. And N is the set of normal worlds, with all the remaining worlds in W being nonnormal ones. (4.2.3) The semantics are the same for nonnormal worlds, except that at nonnormal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. For, in nonnormal worlds, nothing is necessary and all is possible. (4.2.4) At every world, including nonnormal ones, ¬□A and ◊¬A have the same truth value. ¬◊A and □¬A do too. (4.2.5) Inferences are valid only if they preserve truth in all interpretations at all normal worlds. (4.2.6) Nonnormal modal logics with the structure ⟨W, N, R, v⟩ in which R is a binary relation on W are called N, with such R constraints as ρ, σ, τ etc. creating extensions of N like Nρ, etc. (So here we have N for nonnormal modal logics where we previously had K and its extensions for normal modal logics.) And, “As for normal logics, Nρτ is an extension of Nρ, which is an extension of N, etc.” (4.2.7) Nρ = S2; Nρτ = S3; and Nρστ = S3.5, with the first two S’s being “Lewis systems” and the last one being a “nonLewis system”. (4.2.8) Although nonnormal worlds originally were fashioned solely for technical reasons, in fact they have a philosophical meaning too.
Tableaux for NonNormal Modal Logics
(4.3.1) The tableau rules for nonnormal modal logics N are mostly the same as for normal modal logics K. We need however to add the following exception: “If world i occurs on a branch of a tableau, call it □inhabited if there is some node of the form □B,i on the branch. The rule for ◊A,i is activated only when i = 0 or i is □inhabited” (65). (4.3.2) The ◊rule (that if you have ◊A,i on one node you can obtain that there is an accessible world where A holds) applies only to normal worlds, because possibility in nonnormal worlds does not require it holding in an accessible world; rather, simply all possibilities are true regardless of other worlds. (4.3.3) Priest gives an example showing how the ◊rule is applied when dealing with world 0. (4.3.4) Priest gives another example where we see that as world 1 is not □inhabited, we do not apply the ◊rule to a case in world 1 where there is the possibility operator. (4.3.5) We form counterexamples while keeping in mind which worlds are nonnormal. We assign worlds in accordance with the i numbers. We assign R relations in accordance with irj formulations. And nodes of the form p, i we assign v_{wi}(p) = 1. And for nodes of the form ¬p, i, we assign v_{wi}(p) = 0. If there are neither of these two, then v_{wi}(p) can be given any value we want . (4.3.6) Priest gives an example of a countermodel. When depicting nonnormal worlds, we place the world designator in a box but write the true formulas in that world above the box. (4.3.7) Tableaux for Nρ, Nρτ, etc. use the same additional rules as for Kρ, Kρτ,etc. (4.3.8) “The tableaux for N and its extensions are sound and complete” (67).
The Properties of NonNormal Logics
(4.4.1) K interpretations are special cases of N interpretations, where all the worlds are normal (W = N). This means that K is an extension of N, because “if truth is preserved at all worlds of all Ninterpretations, it is preserved at all worlds of all Kinterpretations” (67). (4.4.2) The extensions of K are also extensions of their respective N extensions, such that for example “Kστ is an extension of Nστ and so on” (67). (4.4.3) Each Klogic is a proper extension of its corresponding Nlogic. (4.4.4) Kρστ is the strongest logic we have seen so far. It is a normal logic, and every other normal logic we have seen is contained in it. Moreover, every nonnormal system we have seen is contained in its corresponding normal system (with none yet being stronger than Kρστ and thus every nonnormal system is weaker than Kρστ). In fact, “N is the weakest system we have met. It is contained in every nonnormal system, and also in K, and so in every normal system” (68). (4.4.5) If we now instead define logical validity as truth preservation over all worlds, including nonnormal ones, then certain formulas like □(A ∨ ¬A) will no longer be valid, because no necessary formulations are true in nonnormal worlds. (4.4.6) The Rule of Necessitation is: for any normal system, ℒ, if ⊨_{ℒ}_{ }A then ⊨_{ℒ}_{ }□A. (4.4.7) The Rule of Necessitation fails in nonnormal systems, because it will not work when applied doubly on the same formula. (4.4.8) On account of the failure of the Rule of Necessitation in nonnormal systems, “Nonnormal worlds are, thus, worlds where ‘logic is not guaranteed to hold’” (69).
S0.5
(4.4a.1) L is a type of nonnormal modal logic. Modal formulas are “sentences of the form □A and ◊A.” And in L, “modal formulas are assigned arbitrary truth values at nonnormal worlds” (69). (4.4a.2) In L, the evaluation function v “assigns each modal formula a truth value at every nonnormal world” (69). (4.4a.3) The tableau rules for L are the same as for N, except “there are no rules applying to modal formulas or their negations at worlds other than 0. That is, the rules of 2.4.4 apply at world 0 and world 0 only” (69). (4.4a.4) Priest then gives an example of how to create the tableau for a valid formula and another for an invalid one. (4.4a.5) We fashion countermodels from open branches in tableaux in L in the following way. Worlds in W are assigned according to i numbers: “For each number, i, that occurs on the branch, there is a world, w_{i}” (p.27, section 2.4.7). World 0 is the only normal world, and all others are nonnormal: “N = {w_{0}}” (70). Accessibility relations follow the irj formulations: “w_{i}Rw_{j }iff irj occurs on the branch” (p.27, section 2.4.7). Atomic propositional formulas are true for their indicated world, and negated atomic formulas are false for their world: “for every propositional parameter, p, if p,i occurs on the branch, v_{wi}(p) = 1, if ¬p,i occurs on the branch, v_{wi}(p) = 0 (and if neither, v_{wi}(p) can be anything one wishes” (p.27, section 2.4.7). All necessity and possibility operated formulas, no matter how complex, in nonnormal worlds are assigned truth values in the same way: “if i > 0, and □A,i, is on the branch, v_{wi} (□A) = 1; if ¬□A,i, is on the branch, v_{wi} (□A) = 0; similarly for ◊A” (70). (4.4a.6) Priest then provides an example of a countermodel. (4.4a.7) We obtain extensions of L by applying constraints on the accessibility relation, as for example ρ (reflexivity), σ (symmetry), and τ (transitivity). (4.4a.8) “The tableaux for L and its extensions are sound and complete with respect to their semantics” (70). (4.4a.9) On account of the historical development of modal logics, S0.5^{0 }and S0.5 are L and Lρ respectively, only without the possibility operator used in them. (4.4a.10) We cannot define the possibility operator for L in the same way as for K and N, i.e., ‘◊A’ as ‘¬□¬A’, because these formulas do not necessarily have the same truth value in all worlds for L. (4.4a.11) “If we wish to make ◊ behave in L as it does when it is defined, we have to add an extra constraint: for every world, w, v_{w}(◊A) = v_{w}(¬□¬A) (that is, v_{w}(◊A) = 1 − v_{w}(□¬A)” (71). (4.4a.12) But this equivalence breaks down in nonnormal worlds in L. (4.4a.13) N is a proper extension of L, and L is the weakest modal logic we have seen so far. (4.4a.14) Since L has nonnormal worlds, the Rule of Necessitation fails in L. Thus, “that ‘logic need not hold’ at nonnormal worlds in L is patent: if A is a logical truth, □A can behave any old way at such a world” (71). (4.4a.15) “□A is valid in L (and Lρ) iff A is a truthfunctional tautology, or, more accurately, is valid in virtue of its truthfunctional structure” (71).
Strict Conditionals
(4.5.1) We now will examine the conditional in the context of modal logic. (4.5.2) There are contingently true material conditionals that we would not want to say are true on account of their contingency, like, “The sun is shining ⊃ Canberra is the federal capital of Australia.” For, “Things could have been quite otherwise, in which case the material conditional would have been false” (72). To remedy this, we could define the conditional as: “‘if A then B’ as □(A ⊃ B), where □ expresses an appropriate notion of necessity” (72). (4.5.3) This definition of the conditional using modal logic is called the strict conditional, symbolized as A⥽B and defined as □(A⊃B). (4.5.4) The strict conditional does not validate the problematic counterexamples (that we have seen) for the material conditional.
The Paradoxes of Strict Implication
(4.6.1) We wonder if the definition of the strict conditional – A⥽B is defined as □(A⊃B) – is adequate. But first we need to address the matter of its variance under different systems of modal logic. (4.6.2) To model conditionality in general and the strict conditional in particular, we need modus ponens to hold, as it is a basic inferential principle that should hold when the conditional has its normal semantics. But in systems without the ρconstraint (reflexivity), modus ponens will fail. Thus our system at least needs the ρconstraint . (4.6.3) We need not narrow our systems down any further than systems with the ρconstraint, because no matter what, they will all lead to the paradoxes of strict implication: ‘□B ⊨ A ⥽ B’, ‘¬◊A ⊨ A ⥽ B’; and also ‘⊨ A ⥽ (B ∨ ¬B) ’, ‘⊨ (A ∧ ¬A) ⥽ B’.
The Explosion of Contradictions
(4.8.1) One of the paradoxes of the strict conditional is: ⊨ (A ∧ ¬A) ⥽ B. By modus ponens we derive: (A∧¬A)⊨B. In other words, contradictions entail everything (any arbitrary formula whatsoever). But this is counterintuitive, and there are counterexamples that we will consider. (4.8.2) The first counterexample: Bohr knowingly combined inconsistent assumptions in his model of the atom, but on that account the model functioned well. However, explosion does not hold here, because we cannot on the basis of the contradiction infer everything else, like electronic orbits being rectangles. (4.8.3) The second counterexample: we can have inconsistent laws without their contradiction entailing everything. (4.8.4) The third counterexample: there are perceptual illusions that give us inconsistent impressions without giving us all impressions. For example, the waterfall illusion gives us the impression of something moving and not moving, but it does not thereby also give us every other impression whatsoever. The fourth counterexample: there can be fictional situations where contradictions hold but that thereby not all things hold as well.
Lewis’ Argument for Explosion
(4.9.1) Strict conditionals do not require relevance, as we see for example with: ⊨ (A ∧ ¬A) ⥽ B. So we might object to them on this basis. (4.9.2) C.I. Lewis argues that (A ∧ ¬A) ⥽ B is intuitively valid, because from A ∧ ¬A it is intuitively valid to infer A and ¬A; from ¬A it is intuitively valid to infer ¬A ∨ B, and from A and ¬A ∨ B it is intuitively valid, by disjunctive syllogism, to derive B. [Now, if each step has a connection on the basis of its intuitive validity, that means the final conclusion B should have a connection, by extension, to A ∧ ¬A on the basis of the intuitively valid steps leading from the premise to the final conclusion. So despite objections to the contrary, there is a connection between the antecedent and consequent in (A ∧ ¬A) ⥽ B, according to Lewis. (4.9.3) C.I. Lewis also formulates an argument for the connection between antecedent and conclusion for A ⥽ (B ∨ ¬B), but this argument is a bit less convincing than the one for (A ∧ ¬A) ⥽ B.
Conditional Logics
Introduction
(5.1.1) We look now at conditional logics, which are modal logics with “a multiplicity of accessibility relations of a certain kind” (82). (5.1.2) We will also consider some more problematic inferences involving the conditional.
Some More Problematic Inferences
(5.2.1) There are three inferences involving the conditional that are valid in classical logic and for the strict conditional, but as we will see in the next section, they are problematic. They are: {1} Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B; {2} Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C; and Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A. (5.2.2) Here are the problematic counterexample illustrations. {1} Antecedent strengthening: A ⊃ B ⊨ (A ∧ C) ⊃ B; “If it does not rain tomorrow we will go to the cricket. Hence, if it does not rain tomorrow and I am killed in a car accident tonight then we will go to the cricket.” {2} Transitivity: A ⊃ B , B ⊃ C ⊨ A ⊃ C; “If the other candidates pull out, John will get the job. If John gets the job, the other candidates will be disappointed. Hence, if the other candidates pull out, they will be disappointed.” {3} Contraposition: A ⊃ B ⊨ ¬B ⊃ ¬A; “If we take the car then it won’t break down en route. Hence, if the car does break down en route, we didn’t take it.” (5.2.3) One might reply to the above objections by saying that they are enthymemes and thus would be valid were we to supply the right relevant information among the premises. (5.2.4) When we supply additional relevant material to the premises of these counterexample illustrations, they show their validity. (5.2.5) But since in such illustration counterexamples we cannot explicitly list all circumstances in the premises that are needed for the argument to be purely nonenthymemic, then this objection does not work absolutely sufficiently yet. (5.2.6) But in fact we can capture all of these infinitely many needed additional deenthymemizing clauses by simply saying for all of them, “other things being equal,” which is called a ceteris paribus clause. (5.2.7) Ceteris paribus clauses {1} are conditioned by the other antecedent term they are conjoined with, because that term might require particular clauses be implied while others be excluded and {2} are contextdependent. (5.2.8) ‘A > B’ means a conditional with a ceteris paribus clause. And, “A > B is true (at a world) if B is true at every (accessible) world at which A ∧ C_{A} is true” (84).
Intuitionistic Logic
Introduction
In this chapter we will examine intuitionistic logic. It arose from intuitionism in mathematics, and it has a natural possible world semantics. We will also examine its philosophical foundations and its account of the conditional.
Intuitionism: The Rationale
(6.2.1) To understand the original rationale for intuitionism, we should note that we can understand strange sentences that we never heard before, like, “Granny had led a sedate life until she decided to start pushing crack on a small tropical island just south of the Equator.” (6.2.2) We can understand such complex unfamiliar sentences on account of compositionality, which says that “the meaning of a sentence is determined by the meanings of its parts, and of the grammatical construction which composes these” (103). (6.2.3) An orthodox view of meaning is that the meaning of a statement is given by its truth conditions (“the conditions under which it is true”). On account of compositionality, statements built up using connectives are determined on the basis of the connectives’ truthfunctional operation on the truth conditions of the constituent statements. (6.2.4) The common notion of truth is that it is a correspondence between what a linguistic formulation says and an extralinguistic reality in which that said circumstance in fact holds. But we think there are mathematical truths and meaningful formulations, yet the idea of an extralinguistic reality is problematic in mathematics, as we will see. (6.2.5) Mathematical realists hold that there is an extralinguistic reality corresponding to the truths of mathematical formulations like “2 + 3 = 5;” they think for example that there are “objectively existing mathematical objects, like 3 and 5.” Intuitionists however see this as a sort of mystical view and think rather that we should not apply the correspondence theory of truth to mathematical formulations. (6.2.6) Intuitionism expresses a statement’s meaning on the basis of its proof conditions, which are the conditions under which the sentence is proved. (6.2.7) The proof condition of a simple sentence is whatever we would take to be a sufficient proof [as for example a sufficient mathematical proof for a mathematical formula.] The proof conditions for complex sentences built up using connectives will be similar to the normal conditions only now using the notion of proof (note that ⇁ and ⊐ symbolize negation and the conditional):
A proof of A ∧ B is a pair comprising a proof of A and a proof of B.
A proof of A ∨ B is a proof of A or a proof of B.
A proof of ⇁A is a proof that there is no proof of A.
A proof of A ⊐ B is A construction that, given any proof of A, can be applied to give a proof of B.
(104)
(6.2.8) These proof conditions cannot validate excluded middle, because there are formulas that cannot be proved nor can it be proven that there is no proof for them.
Possible Worlds Semantics for Intuitionism
(6.3.1) We will examine a certain sort of possible world semantics to capture the ideas of intuitionistic logic. (6.3.2) The only connectives in our intuitionist logic are ∧, ∨, ⇁ and ⊐ (with the last two being negation and the conditional, respectively). (6.3.3) Our intuitionistic possible worlds semantics takes the structure ⟨W, R, v⟩. It is mostly the same as logic K_{ρτ}, meaning that it is a normal modal logic in which the R accessibility relation is reflexive (all worlds have access to themselves) and transitive (whenever a first world has access to a second and that second to a third, then the first has access to that third as well.) There is one additional constraint, called the heredity condition, which means that when a proposition is true in one world, it it is true in all other worlds that are accessible from it. (6.3.4) By means of certain rules we evaluate molecular formulas. Negation and the conditional involve accessible worlds. (6.3.5) The heredity condition holds not just for propositional parameters but for all formulas. (6.3.6) To see how the above interpretation captures intuitionist ideas, we first conceive of the way that information accumulates over time as being like one world (like our world at one moment) as being a set of proven things and another world accessible from the first having the same proven things and maybe more (like our world progressing later into a world perhaps with more information). (6.3.7) The possible world semantics for intuitionism captures the ideas in the proof conditions. (6.3.8) We define validity in intuitionistic logic as truth preservation over all worlds of all interpretations, and we write intuitionistic logical consequence as ⊨_{I}. (6.3.9) If there is only one world, the intuitionistic interpretation is equivalent to a classical one. And intuitionistic logic is a sublogic of classical logic, because everything that is intuitionistically valid is classical valid, but not everything classical valid is intuitionistically valid. (6.3.10) By adding constraints the R accessibility relation in intuitionistic logics, we can generate stronger ones.
Tableaux for Intuitionistic Logic
(6.4.1) Our tableaux for intuitionistic logic will build from those for modal logic, but with some modifications. In modal logic, the nodes take one of two forms: {1} A,i, where A is a formula and i is a natural number indicating the world in which the formula holds, or {2} irj, where i is a natural number for a world that accesses world j, also given as a natural number (the r stays as r). For our intuitionistic tableaux, “The first modification is that a node on the tableau is now of the form A,+i or A,−i. The ﬁrst means, intuitively, that A is true at world i; the second means that A is false at i” (107) Previously we did not need this information in the tableaux about truth and falsity, because A’s being false in a world was equivalent to its negation being true, and so we would represent that with ¬A,i. (It seems then that being “false” or at least lacking a proof, here in intuitionistic systems, means either that {1} within some world, there is a disproof (a proof that there is no proof) for a formula, in other words, that for instance ⇁A in world 0 (maybe written as v_{wo} (⇁A) = 1 and) symbolized as ⇁A,+0 in the tableaux, which means that it is the case that there is a disproof for A in world 0, or that {2} there is currently neither a proof for a formula nor a disproof for that formula (I am not sure how that is written normally, maybe for instance as v_{wo} (A) = 0, but it is) written as A,−0 in the tableaux, meaning that it is not the case that there is a proof for A in world 0. (If there were a disproof, and thus if v_{wo} (⇁A) = 1, then I think still you would thereby have v_{wo} (A) = 0).) (6.4.2) We form the initial list of our tableaux by setting all premises to true in world 0, thus as: B,+0. And the conclusion is set to false in world 0, thus as: A,−0. (6.4.3) We close a branch on our tableau when we obtain a contradiction, that is, “just when we have nodes of the form A,+i and A,−i. (6.4.4) Priest then gives the tableaux rules (see below. The list includes the accessibility rules from the next section also).
Conjunction Development, True (∧D,+) 
A ∧ B,+i ↓ A,+i B,+i 
Conjunction Development, False (∧D,−) 
A ∧ B,−i ↙ ↘ A,−i B,−i 
Disjunction Development, True (∨D,+) 
A ∨ B,+i ↙ ↘ A,+i B,+i 
Disjunction Development, False (∨D,−) 
A ∨ B,−i ↓ ¬A,−i ↓ ¬B,−i 
Conditional Development, True (⊐D,+) 
A ⊐ B,+i irj ↙ ↘ A,−j B,+j
applied for every j on the branch 
Conditional Development, False (⊐D,−) 
A ⊐ B,−i ↓ irj A,+j ↓ B,−j
the j is new 
Negation Development, True (⇁D,+) 
⇁A,+i irj ↓ A,−j
applied for every j on the branch 
Negation Development, False (⇁D,−) 
⇁A,−i ↓ irj A,+j
the j is new 
Heredity, True (hD,+) 
p,+i irj ↓ p,+j . p is any propositional parameter, applied to every j (distinct from i) 
(modified from p.108, section 6.4.4)
ρ, Reflexivity (ρrD) 
ρ . ↓ iri 
τ, Transitivity (τrD) 
τ irj jrk ↓ irk 
Priest has us “Note that, in particular, we can never ‘tick off’ any node of the form A ⊐ B,+i or ⇁A,+i, since we may have to come back and reapply the rule if anything of the form irj turns up” (108109). (6.4.5) We also have the ρ reflexivity and τ transitivity accessibility rules (shown in the listing above). (6.4.6) Priest next gives an example tableau to show that ⊢_{I} p ⊐ ⇁⇁p. (6.4.7) Priest next gives another example tableau that shows that p ⊐ q ⊬_{I} ⇁p ∨ q. (6.4.8) “Countermodels are read off from an open branch of a tableau in a natural way. The worlds and accessibility relation are as the branch of the tableau specifies. If a node of the form p,+i occurs on the branch, p is set to true at w_{i} ; otherwise, p is false at w_{i }. (In particular, if a node of the form p,−i occurs on the branch, p is set to false at w_{i} )” (110). (6.4.9) Priest then gives a more visual portrayal of the countermodel from above section 6.4.8. Here “We indicate the fact that p is true (at a world) by +p, and the fact that it is false by −p” (110). (6.4.10) “The tableaux are sound and complete with respect to the semantics” (111). (6.4.11) Priest then gives an example of an infinite open tableau. (6.4.12) Priest then shows how it is easier to directly make a countermodel in cases of infinite tableaux.
Manyvalued Logics
Introduction
Manyvalued logics have more than two truth values. We will examine the semantics of propositional manyvalued logics in this chapter along with other philosophical and logical issues related to manyvaluedness.
Manyvalued Logic: The General Structure
In providing the general structure for manyvalued logics, we first simplify our system by defining material equivalence in the following way:
A ≡ B is defined as (A ⊃ B) ∧ (B ⊃ A)
We will articulate the structure of manyvalued logics by naming all the components, including the parts relevant for truth and validity evaluations. In its most condensed form, the structure of manyvalued logics is:
⟨V, D, {f_{c}; c ∈ C}⟩
V is the set of assignable truth values. D is the set of designated values, which are those that are preserved in valid inferences (like 1 for classical bivalent logic). C is the set of connectives. c is some particular connective. And f_{c} is the truth function corresponding to some connective, and it operates on the truth values of the formula in question. In a classical bivalent logic,
V = {1, 0}
D = {1}
C = {¬, ∧, ∨, ⊃, ≡} (but recall we have redefined ≡)
f_{c}; c ∈ C = {f_{¬}, f_{∧}, f_{∨}, f_{⊃}}
We also have an interpretation function v that assigns values to the propositional parameters, and the connective truth functions operate recursively on the assigned propositional parameter values to compute the values of the complex formulas. The connective truth functions are defined in terms of the series of values for the places in the ntuple corresponding to that connective:
if c is an nplace connective,
v(c(A_{1}, ... , A_{n})) = f_{c}(v(A_{1}), ... , v(A_{n}))
For example, we could consider a classical bivalent system where V = {1, 0}, and we could define the connective functions for negation and conjunction in the following way.
f_{¬} is a oneplace function such that f_{¬}(0) = 1 and f_{¬}(1) = 0;
f_{∧} is a twoplace function such that f_{∧}(x, y) = 1 if x = y = 1, and f_{∧}(x, y) = 0 otherwise [...]
f_{¬}  
1  0 
0  1 
f_{∧}  1  0 
1  1  0 
0  o  o 
(120121)
The connective evaluations are done recursively. We substitute the connective truth functions in for the connectives themselves by working from greatest to least scope. For example:
v(¬(p∧q)) = f_{¬}(v(p ∧ q)) = f_{¬}(f_{∧}(v(p), v(q)))
Consider the following value assignments for the above formula:
v(p) = 1 and v(q) = 0
Using our connective truth function definitions from above, we would recursively evaluate by going from least to greatest scope, so:
v(¬(p ∧ q)) = f_{¬}(f_{∧}(1, 0)) = f_{¬}(0) = 1
Semantic entailment, validity, and tautology (logical truth) are defined using D, the set of designated values. A set of formulas semantically entails some conclusion when there is no interpretation that assigns designated values to the premises while not assigning a designated value to the conclusion.
Σ ⊨ A iff there is no interpretation, v, such that for all B ∈ Σ, v(B) ∈ D, but v(A) ∉ D
Thus a valid inference is one where there is no interpretation in which all the premises have designated values but the conclusion does not. A formula is a logical truth (tautology) when every evaluation assigns it a designated value.
A is a logical truth iff φ ⊨ A, i.e., iff for every interpretation v(A) ∈ D
In order to craft a manyvalued system of our choosing, we can modify the components of this structure. We of course will want to expand V to include three or more possible assignments for truthvalue. We might also want to restructure validity by adding designated values. Additionally, we could change the types of connectives or alter the evaluations for their truth functions. We say that a logic is finitely manyvalued when V has a finite number of values in it; and when V has n members, we say that it is an nvalued logic. We can evaluate an argument for validity by computing the values for the premises and conclusions for every possible set of assignments for the propositional parameters. When there is an interpretation where all the premises have a designated value but the conclusion does not, then it is invalid, and valid otherwise. The number of possible sets of assignments can become unmanageable for such validity evaluations, because they increase exponentially with each additional propositional parameter.
if there are m propositional parameters employed in an inference, and n truth values, there are n^{m }possible cases to consider.
(122)
The 3valued Logics of Kleene and Łukasiewicz
The structure of manyvalued logics can be formulated as:
⟨V, D, {f_{c}; c ∈ C}⟩
V is the set of assignable truth values. D is the set of designated values, which are those that are preserved in valid inferences (like 1 for classical bivalent logic). C is the set of connectives. c is some particular connective. And f_{c} is the truth function corresponding to some connective, and it operates on the truth values of the formula in question. In classical logic: the assignable truth values of V are true and false, or 1 and 0; the designated values are just 1, and the connectives are: f_{¬}, f_{∧}, f_{∨}, f_{⊃}. [A ≡ B we are defining as (A ⊃ B) ∧ (B ⊃ A)] And finally, the connective functions operate on truth values in accordance with certain rules (displayed often as the truth tables for connectives that we are familiar with. [This was covered in the previous section.] K_{3} (strong Kleene threevalued logic) and Ł_{3 }are two sorts of threevalued logics. Both keep D as {1}, and both extend V to {1, i, 0}. 1 is true , 0 is false, and i is neither true nor false. We have the same connectives (excluding for simplicity the biconditional), and the connective functions associated with them are defined in the following way. K_{3} in particular has these assignments for the connective functions:
f_{¬}  
1  0 
i  i 
0  1 
f_{∧}  1  i  0 
1  1  i  0 
i  i  i  0 
0  0  0  0 
f_{∨}  1  i  0 
1  1  1  1 
i  1  i  i 
0  1  i  0 
f_{⊃}  1  i  0 
1  1  i  0 
i  1  i  i 
0  1  1  1 
One problem with K_{3} is that every formula can obtain the undesignated value i by assigning all of its propositional parameters the value of i. (Simply look at the tables above where the input values are i. You will see in all cases the output is i too.) This means there are no logical truths in K_{3}. One remedy for making the law of identity a logical truth is by changing the value assignment for the conditional such that when both antecedent and consequent are i, the whole conditional is 1.
f_{⊃}  1  i  0 
1  1  i  0 
i  1  1  i 
0  1  1  1 
This new system, where everything else is identical to K_{3}except for the above alternate valuation for the conditional, is called Ł_{3} (Łukasiewicz’ threevalued logic).
LP and RM_{3}
Last time we examined K_{3}, which was defined as:
V = {1, i, 0}
D = {1}
f_{c}; c ∈ C = {f_{¬}, f_{∧}, f_{∨}, f_{⊃}}
(note, A ≡ B we are defining as (A ⊃ B) ∧ (B ⊃ A) )
f_{¬} 1 0 i i 0 1
f_{∧} 1 i 0 1 1 i 0 i i i 0 0 0 0 0
f_{∨} 1 i 0 1 1 1 1 i 1 i i 0 1 i 0
f_{⊃} 1 i 0 1 1 i 0 i 1 i i 0 1 1 1
Here i means neither true nor false. LP has the same structure as K_{3}, except in LP we have D = {1, i}. And in LP, the 1 is understood to mean true and true only, 0 to mean false and false only, and i to mean both true and false. The connective functions then follow our intuition regarding this alternate sense for the V values. Suppose for A∧B that A is 1 and B is i. Since B is at least true, then A∧B is at least true. And since B is also at least false, A∧B is also at least false. So A∧B is both true and false, or i, which is what the truth tables calculate it to be. There are two notable advantages of LP over K_{3}. {1} In LP, unlike in K_{3}, the law of excluded middle holds:
⊭_{K3 }p ∨ ¬p
⊨_{LP}_{ }p ∨ ¬p
And the principle of explosion, or the inference rule ex falso quodlibet, are not valid in LP, unlike in K_{3}:
p ∧ ¬p ⊨_{K3 }q
p ∧ ¬p ⊭_{LP }q
But there is one disadvantage of LP compared with K_{3}. In LP, modus ponens is not valid:
p, p ⊃ q ⊨_{K3 }q
p, p ⊃ q ⊭_{LP }q
This can be solved by changing the evaluation for the conditional connective in the following way, thereby creating RM_{3}:
f_{⊃}  1  i  0 
1  1  0  0 
i  1  i  o 
0  1  1  1 
Manyvalued Logics and Conditionals
(7.5.1) We will now examine the conditional operator in manyvalued logics. (7.5.2) We will assess whether or not some problematic inferences using conditionals are valid in K_{3}, Ł_{3}, LP_{3}, and RM_{3}, by making a table. (In the table below, a ‘✓’ means the inference or formula is valid in the given system, and an ‘×’ means it is not valid.)

 K_{3}  Ł_{3}  LP  RM_{3} 
1  q ⊨ p ⊃ q  ✔  ✔  ✔  × 
2  ¬p ⊨ p ⊃ q  ✔  ✔  ✔  × 
3  (p ∧ q) ⊃ r ⊨ (p ⊃ r) ∨ (q ⊃ r)  ✔  ✔  ✔  ✔ 
4  (p ⊃ q) ∧ (r ⊃ s) ⊨ (p ⊃ s) ∨ (r ⊃ q)  ✔  ✔  ✔  ✔ 
5  ¬(p ⊃ q) ⊨ p  ✔  ✔  ✔  ✔ 
6  p ⊃ r ⊨ (p ∧ q) ⊃ r  ✔  ✔  ✔  ✔ 
7  p ⊃ q, q ⊃ r ⊨ p ⊃ r  ✔  ✔  ×  ✔ 
8  p ⊃ q ⊨ ¬q ⊃ ¬p  ✔  ✔  ✔  ✔ 
9  ⊨ p ⊃ (q ∨ ¬q)  ×  ×  ✔  × 
10  ⊨ (p ∧ ¬p) ⊃ q  ×  ×  ✔  × 
(7.5.3) Generally speaking, the manyvalued logics still validate many of the problematic inferences using the conditional. (7.5.4) We have the intuitions that in finitely manyvalued logics, the following two things should hold:
(i) if A (or B) is designated, so is A ∨ B
(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).
Only in K_{3} does (ii) not hold. (7.5.5) Given these two rules, suppose we have a manyvalued logic with one more formula than there are truthvalues; that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated. (7.5.6) But there are counterexamples to this claim (and in these counterexamples, the intuitive sense of the sentences does not allow for any true biconditional combinations of two different sentences, even though technically they should evaluate as true). For instance, “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true” (127). (So suppose we have a threevalued logic, and John has 1 hair on his head. That means “John has 2 hairs on his head if and only if John has 3 hairs on his head” is true (or at least true, or ‘designated’ whatever way), on account of both sides being false, even though the intuitive sense of the formulation would make the biconditional false (or at least senseless); for, John’s having x number of hairs on his head should not be conditional on his having x ± 1 hairs on his head. Thus, finitely manyvalued logics will always be potentially vulnerable to the following problem: because the disjunction of all biconditionals should be true, then at least one must be true, meaning that in the case of propositions like “John has x number of hairs”, there must be at least one true one that reads “John has x number of hairs on his head only if John has x + 1 number of hairs on his head.” But that is senseless even though it would be evaluated as true.)
Truthvalue Gluts: Inconsistent Laws
(7.6.1) We will now examine philosophical motivations for advocating for multivalued logics with truthvalue gaps or gluts. (7.6.2) In this chapter subsection, Priest will elaborate on the issue of inconsistent laws. (7.6.3) For example, consider if long ago there were the laws {1} that no aborigines have the right to vote, but {2} all propertyholders have that right. At the time it was unthinkable for aborigines to own property, but later in history they do. Thus in the legal system, later on in history, aborigines both have and do not have the right to vote. (7.6.4) In cases of insistent laws, normally they are rectified to make them consistent. Nonetheless, they will remain inconsistent for some time until that change will be made. (7.6.5) Priest next considers a possible objection, namely, that such seemingly contradictory laws are actually consistent, because there is always some other law that clarifies which of the two contradicting laws takes precedent; “for example lex posterior (that a later law takes precedence over an earlier law), or that constitutional law takes precedence over statute law, which takes precedence over case law. One might insist that all contradictions are only apparent” (128). (7.6.6) Priest’s reply to this objection is that while it may be that in actual fact there are many cases where additional laws dissolve the apparent legal contradiction, in principle it is still possible, as for example were both laws made at the same rank.
Truthvalue Gluts: Paradoxes of Selfreference
(7.7.1) We will now consider paradoxes of selfreference as motivation for advocating for truthvalue gluts. (7.7.2) One paradox of selfreference is the liar’s paradox. For example, ‘this sentence is false’. “Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false” (129). (7.7.3) Another paradox of selfreference is Russell’s Paradox: “Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false. “ (7.7.4) There are many such arguments that come to a conclusion of the form A∧¬A, and supposing they are sound, that makes the conclusions true and thus means there really are truthvalue gluts. (7.7.5) We will now examine briefly a couple claims that these paradoxical arguments are not sound. (7.7.6) Objection 1: All selfreferential sentences are meaningless. Reply 1: But, there are many such meaningful sentences, like, ‘this sentence has five words’. (7.7.7) Objection 2: The liar sentence is neither true nor false. Thus our logical assumptions remove excluded middle, and we cannot develop the argument as, “either it is true or false; if false, then thus; if true then false; thus ...”. For, now we have a third situation, that it is neither. (7.7.8) Reply 2: “Extended Paradoxes” still present a contradiction. For example: “This sentence is either false or neither true nor false”. If true, it is either false or neither value. Either way, it is not true, which contradicts our assumption that it is true. If it is either false or neither valued (meaning that it is not true), then its value is what it claims to be, and thus it is true, which contradicts what we assumed. Reply 3: Some paradoxes of selfreference, like Berry’s paradox, do not invoke the law of excluded middle.
Truthvalue Gaps: Denotation Failure
(7.8.1) One motivation for arguing for truthvalue gaps are intuitionistic situations where neither A nor ¬A can be verified. We discussed intuitionism previously, so we turn instead to two other arguments for gaps. (7.8.2) The first sort of argument for truthvalue gaps are “sentences that contain noun phrases that do not appear to refer to anything, like names such as ‘Sherlock Holmes’, and descriptions such as ‘the largest integer’ (there is no largest)” (130). (7.8.3) Frege claimed that “all sentences containing such terms are neither true nor false;” but this is too strong of a claim, because we would want for example for the following sentence to be true: “Sherlock Homes does not really exist” even though it has a nondenoting term. (7.8.4) But there are sorts of sentences with nondenoting terms, called “truths of fiction,” that would seem to really be true, false, or neither on account of the fictional world they are statements about. For example, “Holmes lived in Baker Street” would be true, because that is where the author Conan Doyle says Homes lives; “Holmes’ friend, Watson, was a lawyer,” would be false, because Doyle says that Watson was a doctor, and “Holmes had three maiden aunts” would be neither true nor false, because Doyle never says anything about Holmes’ aunts or uncles. (7.8.5) But some say that fictional truth sentences are really shorthand for sentences beginning with “In the play/novel/film (etc.), it is the case that...”. So, “in Doyle’s stories, it is the case that Holmes lived in Baker Street;” “in Doyle’s stories, it is not the case that Watson was a lawyer;” and “in Doyle’s stories, it is not the case that Holmes had three maiden aunts, and it is not the case that he did not” (thereby making all such sentences true.) (7.8.6) “Another sort of example of a sentence that can plausibly be seen as neither true nor false is a subject/predicate sentence containing a nondenoting description, like ‘the greatest integer is even’” (131). (7.8.7) But it is not necessary to say that nondenoting descriptions are neither true nor false, because they fulfill this function when being just false. (7.8.8) And in fact, in many cases nondenoting descriptions would work better being simply false. For example, let “Father Christmas” be “the old man with a white beard who comes down the chimney at Christmas bringing presents,” and thus the following is simply false: “The Greeks worshipped Father Christmas.” (7.8.9) Nonetheless, even Russell’s view that nondenoting descriptions are false does not help for cases when we would say they should be true; “For example, it appears to be true that the Greeks worshipped the gods who lived on Mount Olympus” (131132). (7.8.10) So although we have reason to pursue nondenotation as a motivation for truthvalue gaps, we see that it is problematic.
Truthvalue Gaps: Future Contingents
(7.9.1) Another motivation for holding that there are truthvalue gaps are future contingents, which are statements about the future that can be uttered now but for which there presently are no facts that make them true or false, as for example: “The first pope in the twentysecond century will be Chinese” (132). (7.9.2) So some might claim that future contingents are really either true or false, and we do not know which yet. However, we will now examine Aristotle’s argument that this cannot be so. (7.9.3) Aristotle’s argument for future contingents is that they cannot be either true or false, because that would mean the futures they describe are certain and necessary, while in fact they are not. Take sentence S: “The first pope in the twentysecond century will be Chinese.” Suppose it is true. That means the first pope in the twentysecond century will in fact be Chinese. But that cannot be necessarily true, because we do not know yet. Suppose instead S is false. Then that pope will not be Chinese, necessarily, which also cannot be correct for the same reason. Either way, the future outcome would need to be necessary, but it is not, because for right now it is still contingent. Thus future contingents cannot be either true or false. (7.9.4) Objection: Aristotle’s argument, which can be illustrated as: “If S were true now, then it would necessarily be the case that the first pope in the twentysecond century will be Chinese” is ambiguous between □(A ⊃ B) (“‘if it is true now that the first pope in the twentysecond century will be Chinese, then it necessarily follows that the first pope in the twentysecond century will necessarily be Chinese”) or A ⊃ □B (“if it is true now that the first pope in the twentysecond century will be Chinese, then that the first pope in the twentysecond century will be Chinese is true of necessity.”) (7.9.5) If we take the first interpretation, □(A ⊃ B), then it would be true (because in a world, if something about the future is true now, then it cannot be otherwise that it will be false in the future of that world), but we cannot from A, □(A ⊃ B) infer that □B (because A or the conditional might not hold in other worlds and thus B may not be true in all other worlds). If we take the second interpretation, (A ⊃ □B) then we can from A, (A ⊃ □B) validly infer □B (by modus ponens), but we would not feel that it is justified to say (A ⊃ □B) in the first place (because we do not want to imply that B will happen no matter what anyway, fatalistically, regardless of A.) (My parenthetical explanations are faulty here and will be revised after the elaborations in section 11a.7.)
Supervaluations, Modality and Manyvalued Logic
(7.10.1) We turn now to two matters that are related to Aristotle’s argument for truthvalue gaps on the basis of future contingents. (7.10.2) We will probably not want all statements about the future to be valueless, as many statements can be determinable now as true or false. Thus we need excluded middle to hold in many cases for statements about the future. And since it does not hold in “K_{3} or Ł_{3}, these logics do not appear to be the appropriate ones for future statements” (133). (7.10.3) We can use a technique called supervaluation to produce a logic that is better suited to accommodate both valued and valueless statements about the future. “Let v be any K_{3} interpretation. Define v ≤ v′ to mean that v′ is a classical interpretation that is the same as v, except that wherever v(p) is i, v′(p) is either 0 or 1. (So v′ ‘fills in all the gaps’ in v.) Call v′ a resolution of v. Define the supervaluation of v, v^{+}, to be the map such that for every formula, A:
v^{+}(A) = 1 iff for all v′ such that v ≤ v′, v′(A) = 1
v^{+}(A) = 0 iff for all v′ such that v ≤ v′, v′(A) = 0
v^{+}(A) = i otherwise
The thought here is that A is true on the supervaluation of v; just in case however its gaps were to get resolved (and, in the case of future contingents, will get resolved), it would come out true. We can now define a notion of validity as something like ‘truth preservation come what may’, Σ ⊨^{S} A (supervalidity), as follows:
Σ ⊨^{S} A iff for every v, if v^{+}(B) is designated for all B ∈ Σ ⊨^{S}, v^{+}(A) is designated
(where the designated values here are as for K_{3}),” namely, just 1 (pp.133134). (7.10.4) “A fundamental fact is that Σ ⊨^{S} A iff A is a classical consequence of Σ. (In particular, therefore, ⊨^{S} A ∨ ¬A even though A may be neither true nor false!)” (134). (7.10.5) Classical validity and supervaluational validity hold when conclusions are understood to be a singular formula, but it does not hold for multipleconclusion validity. For instance,
A ∨ B ⊨ A, B
is classically valid but not supervaluationally valid. (7.10.5a) Priest next shows how we can avoid the misalignment of classical and supervaluational validity for multiple conclusions by redefining supervaluational validity in the following way: “Define an inference to be valid iff, for every K_{3} interpretation, v, every resolution of v that makes every premise true makes some (or, in the single conclusion case, the) conclusion true. Since the class of resolutions of all K_{3} interpretations is exactly the set of classical evaluations, this gives exactly classical logic (single or multiple conclusion, as appropriate)” (134135). (7.10.5b) To give an LP logic corresponding to the K_{3} logic from supervaluation, we use a technique called subvaluation: “we will use ⊨_{S} instead of ⊨^{S} (and call this subvalidity). This time, A ⊨_{S} Σ iff the multiple conclusion inference from A to Σ is classically valid (and a fortiori for single conclusion inferences)” (135). (7.10.5c) Priest next notes that the above subvaluational technique of LP does not work for multipremise inferences. For example, A, B ⊨ A ∧ B is classically valid but not subvaluationally valid. (7.10.5d) The different super/subvaluational techniques render different notions of validity, and so we need to ask, “In the case of future contingents, for example, are we interested in preserving actual truth value, truth value we can ‘predict now’, or ‘eventual’ truth value?” (136) Priest notes that our answer can depend on why we think that gaps or gluts arise in such situations and on the sort of application we have in mind. (7.10.6) For Łukasiewicz, a statement about a future contingent says something that possibly may happen, but it can be otherwise. Thus that statement of the future contingent with the possibility operator is true but with the necessity operator is false.
f◊ 1 1 i 1 0 0
Defining □A in the standard way, as ¬◊¬A, gives it the truth table:
f□ 1 1 i 0 0 0 (136)
(7.10.7) The above definitions for the modal operators give us a modal logic that captures some of Aristotle’s thinking on future contingency, like p ⊨_{Ł3} □p, but it betrays others, like ◊A, ◊B ⊨_{Ł3} ◊(A ∧ B). (7.10.8) “[N]one of the modal logics that we have looked at (nor conditional logics, nor intuitionist logic) is a finitely manyvalued logic” (137). (7.10.9) By using uniform substitution, we can render every logic into an infinitely manyvalued logic. “A uniform substitution of a set of formulas is the result of replacing each propositional parameter uniformly with some formula or other (maybe itself). Thus, for example, a uniform substitution of the set {p, p ⊃ (p ∨ q)} is {r ∧ s, (r ∧ s) ⊃ ((r ∧ s) ∨ q)}. A logic is closed under uniform substitution when any inference that is valid is also valid for every uniform substitution of the premises and conclusion. All standard logics are closed under uniform substitution” (137). (7.10.10) “[E]very logical consequence relation, ⊢, closed under uniform substitution, is weakly complete with respect to a manyvalued semantics. That is, ⊢A iff A is logically valid in the semantics” (137).
First Degree Entailment
Introduction
In first degree entailment (FDE), interpretations are not formulated as functions that assign truth values, standard or not, to propositional parameters. Rather, in FDE, interpretations are formulated as relations between formulas and standard truth values. In this chapter we examine FDE, along with an alternate possible world semantics for it, and we discuss the issues of explosion and disjunctive syllogism.
The Semantics of FDE
In our semantics for First Degree Entailment (FDE), our only connectives are ∧, ∨ and ¬ (with A ⊃ B being defined as ¬A ∨ B.) FDE uses relations rather than functions to evaluate truth. So a truthvaluing interpretation in FDE is a relation ρ between propositional parameters and the values 1 and 0. We write pρ1 for p relates to 1, and pρ0 for p relates to 0. This allows a formula to have one of the following four valueassignment situations: just true (1, e.g.: pρ1), just false (0, e.g.: pρ0), both true and false (1 and 0, e.g.: pρ1, pρo), and neither true nor false (no such valuing formulations). In FDE, being false (that is, relating to 0) does not automatically mean being untrue (that is, not relating to 1), because it can still be related to 1 along with 0. For formulas built up with connectives, we use the same criteria as in classical logic to evaluate them, only here we can have formulas taking both values. In FDE, semantic consequence is defined as:
Σ ⊨ A iff for every interpretation, ρ, if Bρ1 for all B ∈ Σ then Aρ1
(144)
and logical truth or tautology as:
⊨ A iff φ ⊨ A, i.e., for all ρ, Aρ1
(144)
Tableaux for FDE
The tableaux for First Degree Entailment will allow us to evaluate arguments for validity. The rules are modeled after those of classical logic, but are made more complex with the inclusion of a new addition to the formulations, namely, after each formula, we add a comma and then + for true ones and − for false ones, like:
p,+
and
¬p,−
We set up the tableaux to test for valid inference by first stacking the premises and lastly the negated conclusion. We then use the tableau rules to develop the branches until we can determine them as being either open or closed.
Double Negation Development, True (¬¬D,+) 
¬¬A,+ ↓ A,+ 
Double Negation Development, False (¬¬D,−) 
¬¬A,− ↓ A,− 
Conjunction Development, True (∧D,+) 
A ∧ B,+ ↓ A,+ B,+ 
Conjunction Development, False (∧D,−) 
A ∧ B,− ↙ ↘ A,− B,− 
Negated Conjunction Development, True (¬∧D,+) 
¬(A ∧ B),+ ↓ ¬A ∨ ¬B,+ 
Negated Conjunction Development, False (¬∧D,−) 
¬(A ∧ B),− ↓ ¬A ∨ ¬B,− 
Disjunction Development, True (∨D,+) 
A ∨ B,+ ↙ ↘ A,+ B,+ 
Disjunction Development, False (∨D,−) 
A ∨ B, ↓ A, B, 
Negated Disjunction Development, True (¬∨D, +) 
¬(A ∨ B),+ ↓ ¬A ∧ ¬B,+ 
Negated Disjunction Development, False(¬∨D, ) 
¬(A ∨ B), ↓ ¬A ∧ ¬B, 
Branches close when they contain nodes of the form A,+ and A,−. Open branches indicate countermodels: to any p,+, we assign pρ1; and to any ¬p,+ we assign pρ0. And we make no other assignments than that. This technique will make the premises true and the conclusion not true, when it is an invalid inference. The tableaux are sound and complete.
FDE and Manyvalued Logics
First Degree Entailment’s evaluating relation allows for four value situations, namely, a formula being valued as 1, as 0, as having a relation to 1 and to 0, and as having no such value relations. We can thus alternatively think of First Degree Entailment as having four singular values each as their own: 1 (for just true), 0 (for just false), b (for both), and n (for neither). The value assignments are then structured so that they match the outcomes for the First Degree Entailment rules. The truth tables for the connectives in fourvalued FDE semantics are thus:
f_{¬} 1 0 b b n n o 1
f_{∧} 1 b n o 1 1 b n 0 b b b 0 0 n n 0 n 0 o 0 0 0 0
f_{∨} 1 b n o 1 1 1 1 1 b 1 b 1 b n 1 1 n n o 1 b n 0
We can make a shorthand of these valuations with the following diamond lattice (Hasse diagram):
1 
↗ ↖ 
b n 
↖ ↗ 
0 
[Here is a version that I modified for my own purposes of visual demonstration, so it is not Priest’s, and it is probably flawed:
1  
↙↗  ↑  ↖↘ 
⟳ b    n ⟲ 
↘↖  ↓  ↗↙ 
0 
Negation toggles 0 and 1, and it maps n to itself and b to itself:
Negation ⇅ and ⥁
1  
↙↗  ↑  ↖↘ 
⟳ b    n ⟲ 
↘↖  ↓  ↗↙ 
0 
For conjunction, we take the greatest lower bound for both of the conjunct values, that is to say, when moving upward, we find the highest place we can start from in order to arrive at both of the two conjunct values (we can also start and arrive at the same place, if we begin at our destination).
Conjunction ↑ >↓
(greatest lower bound, reading upwards)
1  
↙↗  ↑  ↖↘ 
⟳ b    n ⟲ 
↘↖  ↓  ↗↙ 
0 
For disjunction, we look for the least upper bound: when moving downward, we seek the lowest place we can start from to arrive at both values.
Disjunction ↓ < ↑
(least upper bound, reading downwards)
1  
↙↗  ↑  ↖↘ 
⟳ b    n ⟲ 
↘↖  ↓  ↗↙ 
0 
] The designated values in FDE are 1 and b, so an inference in FDE is valid only if there is no interpretation that assigns all the premises 1 or b and the conclusion 0 or n. We can place constraints on FDE in order to obtain other logics, like the threevalued logics K_{3 }and LP and also Classical Logic. One such constraint is exclusion, which prevents there from being a formula that is both 1 and 0:
Exclusion: for no p, pρ1 and pρ0
FDE under the constraint of exclusion is K_{3}. Now, to make K_{3} sound and complete, we use modified FDE tableaux rules, namely, we can (additionally) also close a branch when it contains A,+ and ¬A,+. Another constraint is exhaustion, which prevents there from being any formulas with no values:
Exhaustion: for all p, either pρ1 or pρ0
FDE under the exhaustion constraint is LP. To make LP sound and complete, we modify the FDE tableaux rules such that {1} a branch also closes if it has formulas of the form A,− and ¬A, −, and {2} we obtain countermodels from open branches by using the following rule: “if p,− is not on the branch (and so, in particular, if p,+ is), set pρ1; and if ¬p,− is not on the branch (and so, in particular, if ¬p,+ is), set pρ0” (149). And then, FDE under both the exhaustion and exclusion constraints is Classical Logic. Note that FDE is a proper sublogic of K_{3 }and LP, because every interpretation of K_{3 }or LP is an interpretation in FDE .
The Routley Star
FDE can be given an equivalent, twovalued, possible world semantics in which the negation is an intensional operator, meaning that it is defined by means of related possible worlds. In this case, we use Routley’s star worlds. We have a star function, *, which maps a world to is “star” or “reverse” world (and back again to the first one, if applied yet another time. That bringing back to the first is what defines the function). So for any world w, the * function gives us its companion star world w^{∗}. We evaluate conjunctions and disjunctions based on values in the given world. But what is notable in Routley Star semantics is that a negated formula in a world w is valued true in that world not on the basis of it unnegated form being false in that world w, but rather on the basis of its unnegated form being false in the star world w^{∗} (so a negated formula is 1 in w if its unnegated form is 0 in w*.) Validity is defined as truth preservation for all worlds and interpretations. For constructing tableaux in Routley Star logic, we designate not just the truthvalue for a formula but also the world in which that formula has that value. This is especially important for negation, where the derived formulas are found in the companion star world. Here is how the tableau rules work for Routley Star logic [quoting Priest, except for the rules tables, where I add my own names and abbreviations, following David Agler]:
Nodes are now of the form A,+x or A,−x, where x is either i or i#, i being a natural number. (In fact, i will always be 0, but we set things up in a slightly more general way for reasons to do with later chapters.) Intuitively, i# represents the star world of i. Closure occurs if we have a pair of the form A,+x and A,−x. The initial list comprises a node B,+0 for every premise, B, and A,−0, where A is the conclusion. The tableau rules are as follows, where x is either i or i#, and whichever of these it is, x̄ is the other.
Conjunction Development, True (∧D,+x) 
A ∧ B,+x ↓ A,+x B,+x 
Conjunction Development, False (∧D,−x) 
A ∧ B,−x ↙ ↘ A,−x B,−x 
Disjunction Development, True (∨D,+x) 
A ∨ B,+x ↙ ↘ A,+x B,+x 
Disjunction Development, False (∨D,−x) 
A ∨ B,x ↓ A,x B,x 
Negation Development, True (¬D,+x) 
¬A,+x ↓ A,x̄ 
Negation Development, False (¬D,−x) 
¬A,x ↓ A,+x̄ 
(152, Note, names and abbreviations are my own and are not in the text.)
We test for validity first [as noted above] by setting every premise to true in the nonstar world and the conclusion to false in the nonstar world. We then apply all the rules possible, and if all the branches are closed [recall from above that closure occurs if we have a pair of the form A,+x and A,−x] then it is valid, and invalid otherwise [so it is invalid if any branches are open]. We then can make countermodels using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world (that is to say, “ if p,+x occurs on the branch, v_{wx}(p) = 1, and if p,−x occurs on the branch, v_{wx}(p) = 0.”) The equivalence between Routley Star semantics and FDE becomes apparent when we make the following translation: v_{w}(p) = 1 iff pρ1; v_{w}_{∗}(p) = 0 iff pρ0.
Paraconsistency and the Disjunctive Syllogism
(8.6.1) On account of truthvalue gluts, p ∧ ¬p ⊢ q is not valid in FDE, and thus FDE does not suffer from explosion (which happens when contradictions entail any arbitrary formula and thus a contradiction entails everything). (8.6.2) Both FDE and LP are paraconsistent logics, because in them it is invalid to infer any arbitrary formula from a contradiction. (8.6.3) Disjunctive syllogism (p, ¬p ∨ q ⊨ q) fails in FDE (set p to b and q to 0; b is a designated value but is not preserved), and it fails in LP (set p to i and q to 0.; i is a designated value, but it also is not preserved ). (8.6.4) Arguments for the material and strict conditional that use disjunctive syllogism are thus faulty on account of its invalidity (in FDE and LP). (8.6.5) Because disjunctive syllogism fails for the material conditional in FDE, so too does modus ponens fail for it as well, given their equivalence. This suggests that the material conditional does not adequately represent the real conditional. (8.6.6) Those who argue that disjunctive syllogism is intuitively valid can do so only by showing that truthvalue gluts are invalid. They think that by saying one of two disjuncts is false (in a true disjunction) necessities the other disjunct be true. But we can also have the intuition that certain formulas should be both true and false. And suppose one of the disjuncts is ¬p, and suppose that p is both true and false. That does not necessitate that ¬p be just false; for, it would also be both true and false. In other words: “The truth of p does not rule out the truth of ¬p: both may hold” (154). Since ¬p is at least true, it does not necessitate that the other disjunct be true, and so we cannot infer that the other disjunct is true. For, only one needs to be at least true. So if we start with the intuition that there can be truthvalue gluts, then disjunctive syllogism is intuitively invalid. (8.6.7) A more convincing defense of the disjunctive syllogism is that we rely on it for reasoning well. Often times we know either of two things can be true; when one proves false, we know it must be the other one. (8.6.8) Even though disjunctive syllogism is invalid, it still functions quite well for normal everyday reasoning. It only fails when there is a truthvalue glut. Otherwise, our daily life presents us normally with consistencies, so it will still deliver correct inferences usually. We just need to be careful to distinguish those cases with gluts and remember not to use it then. (8.6.9) There is precedent for this sort of discrimination of situations for appropriate inference uses in mathematics, so we should not feel too uncomfortable with it in cases of logical reasoning. For example, when dealing with finite sets, if one set is a proper subset of another, we can infer that it is smaller. But for infinite sets, we cannot draw that inference. For example, the set of even numbers is a proper subset of the set of natural numbers, but both sets have the same size. (8.6.10) Since we are wiling to accept inference discrimination in mathematics, we can surely accept it in logic, and so we can set aside the objection that we must reject truthvalue gluts (or that we need the material conditional) simply because we need disjunctive syllogism to reason properly.
Logics with Gaps, Gluts and Worlds
Introduction
(9.1.1) In this chapter we will examine ways that we can combine the techniques of both modal logic and manyvalued logic, especially with logics that involve strict conditional world semantics and First Degree Entailment. (9.1.2) We will also further elaborate on the notion of nonnormal worlds. (9.1.3) At the end of this chapter we will examine logics of constructible negation and connexive logics.
Adding →
(9.2.1) In order to introduce a wellfunctioning conditional into FDE, we could build a possible world semantics upon it. “To effect this, let us add a new binary connective, →, to the language of FDE to represent the conditional. By analogy with Kυ, a relational  interpretation for such a language is a pair ⟨W, ρ⟩, where W is a set of worlds, and for every w ∈ W, ρ_{w} is a relation between propositional parameters and the values 1 and 0” (163164). (9.2.2) We will use the symbol → for the conditional operator in our possible worlds FDE semantics. We still use the ρ relation to assign truthvalues. But we also will specify the worlds in which that value holds. (9.2.3) The evaluation rules for ∧, ∨ and ¬ and just like those for FDE, only now with worlds specified.
A ∧ Bρ_{w}1 iff Aρ_{w}1 and Bρ_{w}1
A ∧ Bρ_{w}0 iff Aρ_{w}0 or Bρ_{w}0
(164)
A ∨ Bρ_{w}1 iff Aρ_{w}1 or Bρ_{w}1
A ∨ Bρ_{w}0 iff Aρ_{w}0 and Bρ_{w}0
¬Aρ_{w}1 iff Aρ_{w}0
¬Aρ_{w}0 iff Aρ_{w}1
(not in the text)
(9.2.4) In our possible worlds FDE, a conditional is true if in all worlds, whenever the antecedent is true, so is the consequent. And it is false if there is at least one world where the antecedent is true and the consequent false.
A → Bρ_{w}1 iff for all w′ ∈ W such that Aρ_{w}_{′}1, Bρ_{w}_{′}1
A → Bρ_{w}0 iff for some w′ ∈ W, Aρ_{w}_{′}1 and Bρ_{w}_{′}0
(9.2.5) In our possible worlds FDE, “semantic consequence is defined in terms of truth preservation at all worlds of all interpretations:
Σ ⊨ A iff for every interpretation, ⟨W, ρ⟩, and all w ∈ W: if Bρ_{w}1 for all B ∈ Σ, Aρ_{w}1
(164)
(9.2.6) “A natural name for this logic would be Kυ_{4}. We will call it, more simply, K_{4}” (164).
Tableaux for K_{4}
(9.3.1) We will formulate the tableau procedures for K_{4} by modifying those for FDE. (9.3.2) Nodes in our possible worlds FDE tableaux take the “form A,+i or A,−i, where i is a natural number.” To test for validity, we compose our initial list by formulating our premise nodes as “B,+0” and our conclusion as “A,−0”. “A branch closes if it contains a pair of the form A,+i and A,−i” (164). (9.3.3) The tableau rules for the extensional connectives (∧, ∨ and ¬) in K_{4} are the same as for FDE except “i is carried through each rule.”[Below I include the conditional rules from the next section. Note that the rules for double negation and disjunction are not in the text and are probably mistaken.]
Double Negation Development, True (¬¬D,+) 
¬¬A,+i ↓ A,+i 
Double Negation Development, False (¬¬D,−) 
¬¬A,−i ↓ A,−i 
Conjunction Development, True (∧D,+) 
A ∧ B,+i ↓ A,+i B,+i 
Conjunction Development, False (∧D,−) 
A ∧ B,−i ↙ ↘ A,−i B,−i 
Negated Conjunction Development, True (¬∧D,+) 
¬(A ∧ B),+i ↓ ¬A ∨ ¬B,+i 
Negated Conjunction Development, False (¬∧D,−) 
¬(A ∧ B),−i ↓ ¬A ∨ ¬B,−i 
Disjunction Development, True (∨D,+) 
A ∨ B,+i ↙ ↘ A,+i B,+i 
Disjunction Development, False (∨D,−) 
A ∨ B,i ↓ A,i B,i 
Negated Disjunction Development, True (¬∨D, +) 
¬(A ∨ B),+i ↓ ¬A ∧ ¬B,+i 
Negated Disjunction Development, False(¬∨D, ) 
¬(A ∨ B),i ↓ ¬A ∧ ¬B,i 
Conditional Development, True (→D,+) 
A → B,+i ↙ ↘ A,j B,+j .
j is every number that occurs on the branch 
Conditional Development, False (→D,−) 
A → B,i ↓ A,+j B,j .
j is a new number 
Negated Conditional Development, True (¬→D, +) 
¬(A → B),+i ↓ A,+j ¬B,+j .
j is a new number 
Negated Conditional Development, False(¬→D, ) 
¬(A → B),i ↙ ↘ A,j ¬B,j .
j is every number that occurs on the branch 
(165, titles for the rules are my own additions)
(9.3.4) To the above rules we add those for the conditional [see the rules just above, where they were moved to.] (9.3.5) Priest then gives a tableau example showing a valid inference. (9.3.6) Priest next gives an example of an inference that is not valid. (9.3.7) We make countermodels from open branches in the following way: “There is a world w_{i }for each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρ_{wi}1; if ¬p,+i occurs on the branch, set pρ_{wi}0. ρ relates no parameter to anything else” (166). (9.3.8) (It can be proven that the possible worlds FDE tableaux are sound and complete with respect to the semantics.)
Nonnormal Worlds Again
(9.4.1) On account of the potential for truthvalue gaps and gluts, the conditional in our possible worlds First Degree Entailment system K_{4} does not suffer from the following paradoxes of the strict conditional: ⊨ p → (q ∨ ¬q), ⊨ (p ∧¬p) → q. (9.4.2) In K_{4}, if ⊨ A then ⊨ B → A. That means ⊨ p → (q → q) is valid, because ⊨ q → q is valid. (9.4.3) Even though ⊨ p → (q → q) , which contains the law of identity, is valid, we can think of a paradoxical instance of this that shows how the law of identity can fail: “if every instance of the law of identity failed, then, if cows were black, cows would be black. If every instance of the law failed, then it would precisely not be the case that if cows were black, they would be black” (167). (9.4.4) As we noted, the conditional should be able to express things that go against the laws of logic, like the law of identity. We should be able to formulate sentences in which the antecedent supposes some law of logic not to hold, and then the consequent would express what sorts of things would follow from that. Nonnormal worlds are ones where the normal laws of logic may fail; so we should implement nonnormal worlds: “we need to countenance worlds where the laws of logic are different, and so where laws of logic, like the law of identity, may fail. This is exactly what nonnormal worlds are” (167). (9.4.5) We thus need to consider nonnormal worlds where the laws of logic fail and, given how the conditionals express those laws, where the conditional takes on values different than it would in normal worlds (K_{4}). (9.4.6) At a nonnormal world, A → B might be able to take on any sort of value, because the laws of logic may change in that world. (9.4.7) We “take an interpretation to be a structure ⟨W, N, ρ⟩, where W is a set of worlds, N ⊆ W is the set of normal worlds (so that W − N is the set of nonnormal worlds), and ρ does two things. For every w, ρ_{w} is a relation between propositional parameters and the truth values 1 and 0, in the usual way. But also, for every nonnormal world, w, ρ_{w} is a relation between formulas of the form A → B and truth values” (167). (9.4.8) The truth conditions for connectives in our nonnormal worlds K_{4} are the same as for K_{4}, except in nonnormal worlds, the conditional is assigned its value not recursively but in advance by the ρ relation. Here are the truth conditions for normal worlds:
A ∧ Bρ_{w}1 iff Aρ_{w}1 and Bρ_{w}1
A ∧ Bρ_{w}0 iff Aρ_{w}0 or Bρ_{w}0
(p.164, section 9.2.3)
A ∨ Bρ_{w}1 iff Aρ_{w}1 or Bρ_{w}1
A ∨ Bρ_{w}0 iff Aρ_{w}0 and Bρ_{w}0
¬Aρ_{w}1 iff Aρ_{w}0
¬Aρ_{w}0 iff Aρ_{w}1
(not in the text)
A → Bρ_{w}1 iff for all w′ ∈ W such that Aρ_{w}_{′}1, Bρ_{w}_{′}1
A → Bρ_{w}0 iff for some w′ ∈ W, Aρ_{w}_{′}1 and Bρ_{w}_{′}0
(p.164, section 9.2.4)
(9.4.9) Our nonnormal worlds FDE system will be called N_{4}, and it will define validity in the same way as for K_{4}, namely, as truth preservation at all normal worlds of all interpretations.
Tableaux for N_{4}
_{}
(9.5.1) The tableau rules for N_{4} are the same as for K_{4}, except the rules for → will apply only at world 0.
Double Negation Development, True (¬¬D,+) 
¬¬A,+i ↓ A,+i 
Double Negation Development, False (¬¬D,−) 
¬¬A,−i ↓ A,−i 
Conjunction Development, True (∧D,+) 
A ∧ B,+i ↓ A,+i B,+i 
Conjunction Development, False (∧D,−) 
A ∧ B,−i ↙ ↘ A,−i B,−i 
Negated Conjunction Development, True (¬∧D,+) 
¬(A ∧ B),+i ↓ ¬A ∨ ¬B,+i 
Negated Conjunction Development, False (¬∧D,−) 
¬(A ∧ B),−i ↓ ¬A ∨ ¬B,−i 
Disjunction Development, True (∨D,+) 
A ∨ B,+i ↙ ↘ A,+i B,+i 
Disjunction Development, False (∨D,−) 
A ∨ B,i ↓ A,i B,i 
Negated Disjunction Development, True (¬∨D, +) 
¬(A ∨ B),+i ↓ ¬A ∧ ¬B,+i 
Negated Disjunction Development, False(¬∨D, ) 
¬(A ∨ B),i ↓ ¬A ∧ ¬B,i 
Conditional Development, True (→D,+) 
A → B,+i ↙ ↘ A,j B,+j .
j is every number that occurs on the branch (and this rule applies only to world 0) 
Conditional Development, False (→D,−) 
A → B,i ↓ A,+j B,j .
j is a new number. (Here i will always be 0 and j will be 1) 
Negated Conditional Development, True (¬→D, +) 
¬(A → B),+i ↓ A,+j ¬B,+j .
j is a new number. (Here i will always be 0 and j will be 1) 
Negated Conditional Development, False(¬→D, ) 
¬(A → B),i ↙ ↘ A,j ¬B,j .
j is every number that occurs on the branch (and this rule applies only to world 0. So i will always be 0) 
(165, titles for the rules are my own additions. Note that the rules for double negation and disjunction are not in the text and are probably mistaken. Also, I am guessing about the conditionals, too.)
(9.5.2) Priest then gives an example of a formula that is valid in N_{4} but not in K_{4}. (9.5.3) We construct countermodels from open branches in the following way. There is a world w_{i }for each i on the branch. For all propositional parameters, p, in every world (normal or not) and for conditionals, A → B, at nonnormal worlds only, if p,+i or A → B,+i occurs on the branch, set pρ_{wi }1 or A → Bρ_{wi }1; if ¬p,+i or ¬(A → B),+i occurs on the branch, set pρ_{wi}0 or A → Bρ_{wi }o. There are no other facts about ρ. (9.5.4) “N_{4} is a sublogic of K_{4}, but not the other way around,” because all valid formulas of N_{4} are valid in K_{4}, but not all valid formulas of K_{4} are valid in N_{4}. (9.5.5) “The tableaux for N_{4} are sound and complete with respect to the semantics” (169).
Star Again
(9.6.1) We can apply the N_{4} constructions to Routley Star ∗ semantics. (9.6.2) To the Routley semantics that we have seen before, we now add the rule for the conditional →, which gives us K_{∗}. Here is the formalization:
Formally, a Routley interpretation is a structure ⟨W, ∗, v⟩, where W is a set of worlds, ∗ is a function from worlds to worlds such that w^{∗∗} = w, and v assigns each propositional parameter either the value 1 or the value 0 at each world. v is extended to an assignment of truth values for all formulas by the conditions:
v_{w}(A ∧ B) = 1 if v_{w}(A) = v_{w} (B) = 1, otherwise it is 0. .
v_{w}(A ∨ B) = 1 if v_{w}(A) = 1 or v_{w} (B) = 1, otherwise it is 0.
v_{w}(¬A) = 1 if v_{w*}(A) = 0, otherwise it is 0.
 Note that v_{w*}(¬A) = 1 iff v_{w**}(A) = 0 iff v_{w}(A) = 0. In other words, given a pair of worlds, w and w* each of A and ¬A is true exactly once. Validity is defined in terms of truth preservation over all worlds of all interpretations.
(p.151152, section 8.5.3)
Let ⟨W, ∗, v⟩ be any Routley interpretation (8.5.3). This becomes an interpretation for the augmented language when we add the following truth condition for →:
v_{w}(A → B) = 1 iff for all w′ ∈ W such that v_{w}_{′} (A) = 1, v_{w}_{′ }(B) = 1
Call the logic that this generates, K_{∗}.
(169)
(9.6.3) Priest then supplies the tableau rules for K_{∗}.
Conjunction Development, True (∧D,+x) 
A ∧ B,+x ↓ A,+x B,+x 
Conjunction Development, False (∧D,−x) 
A ∧ B,−x ↙ ↘ A,−x B,−x 
Disjunction Development, True (∨D,+x) 
A ∨ B,+x ↙ ↘ A,+x B,+x 
Disjunction Development, False (∨D,−x) 
A ∨ B,x ↓ A,x B,x 
Negation Development, True (¬D,+x) 
¬A,+x ↓ A,x̄ 
Negation Development, False (¬D,−x) 
¬A,x ↓ A,+x̄ 
Conditional Development, True (→D,+x) 
A → B,+x ↙ ↘ A,y B,+y . where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch 
Conditional Development, False (→D,−x) 
A → B,x ↓ A,+j B,j . where x is either i or i#; y is anything of the form j or j#, where one or other (or both) of these is on the branch. j must be new. 
(last two are based on p.169, and those above from p.152, section 8.5.4, with names and bottom text added, possibly mistakenly; please consult the original text)
(9.6.4) Priest then gives an example tableau of an invalid formula: that p ∧ ¬q ⊬ ¬(p → q). (9.6.5) We make countermodels using completed open branches. On the basis of the world indicators in the branches, we assign to the formulas the values indicated by the true (+) and false (−) signs for the respective world. When there is negation, however, we need to use values in the starcompanion world. “W is the set of worlds which contains w_{x} for every x and x̄ that occurs on the branch. For all i, w^{*}_{i} = w_{i# }and w^{*}_{i#}= w_{i}. v is such that if p,+x occurs on the branch, v_{x}(p) = 1, and if p,−x occurs on the branch, v_{x}(p) = 0” (170). (9.6.6) In K_{∗}, we still have the problematic valid formula: ⊨ p → (q → q). We can remedy this by adding nonnormal worlds to get N_{∗}. “An interpretation is a structure ⟨W, N, ∗, v⟩, where N ⊆ W; for all w ∈ W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A → B at every nonnormal world. The truth conditions are exactly the same as for K_{∗}, except that the truth conditions for → apply only at normal worlds; at nonnormal worlds, they are already given by v. Validity is defined in terms of truth preservation at normal worlds. Call this logic N_{∗}” (170). (9.6.7) We make our tableau for N_{∗ }the same way as for for K_{∗}, only now the rules for the conditional → only apply for world 0. We generate countermodels the same way too. (9.6.8) The tableaux for K_{∗} and N_{∗} are sound and complete. (9.6.9) K_{4} and N_{4} are not equivalent to K_{∗} and N_{∗}. For example, K_{∗} and N_{∗} validate contraposition: p → q ⊨ ¬q → ¬p, but K_{4} and N_{4} do not. (9.6.10) Additionally, K_{4} and N_{4} verify p ∧ ¬q ⊨ ¬(p → q), but K_{∗} and N_{∗} do not.
Impossible Worlds and Relevant Logic
(9.7.1) We will now discuss philosophical matters regarding K_{4} , N_{4} , K_{∗} , and N_{∗}. (9.7.2) We will now call nonnormal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different. (9.7.3) Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds. (9.7.4) We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular nonclassical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail.” (9.7.5) Objections to logically impossible worlds do not work. For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling. (9.7.6) In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normallyimpossible physical events can take place, but for contingent reasons happen not to. (9.7.7) Logically impossible worlds can also in fact be ones where laws of logic indeed are broken. (9.7.8) Relevant propositional logics are ones where whenever “A → B is logically valid, A and B have a propositional parameter in common” (172). (9.7.9) But N_{4} is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in nonnormal worlds) in combination with the arbitrarity of their value assignments in nonnormal worlds. (9.7.10) In a similar way, N_{∗} is also a relevant logic. (9.7.11) Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters. (9.7.12) There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173). (9.7.13) Relevance in our systems here however is not conditions added on top of classical validity. (9.7.14) If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @.
Logics of Constructible Negation
(9.7a.1) We will now examine logics of constructible negation, which add an account of negation to the negationless part of intuitionistic logic (or positive intuitionistic logic). The important feature of these logics is that “unlike intuitionist logic, they treat truth and falsity evenhandedly” (175). (9.7a.2) We will “Consider interpretations of the form ⟨W, R, ρ⟩, where W is the usual set of worlds, R is a reflexive and transitive binary relation on W, and for every w ∈ W, and propositional parameter, p, ρ_{w} relates p to 1, 0, both or neither, subject to the heredity constraints: if pρ_{w}1 and wRw′, then pρ_{w}_{′}1 ; if pρ_{w}0 and wRw′, then pρ_{w}_{′}0 ” (175). (9.7a.3) Priest next provides the truth/falsity conditions for the connectives in our logic of constructible negation.
A ∧ Bρ_{w}1 iff Aρ_{w}1 and Bρ_{w}1
A ∧ Bρ_{w}0 iff Aρ_{w}0 or Bρ_{w}0
_{}
A ∨ Bρ_{w}1 iff Aρ_{w}1 or Bρ_{w}1
A ∨ Bρ_{w}0 iff Aρ_{w}0 and Bρ_{w}0
_{}
¬Aρ_{w}1 iff Aρ_{w}0
¬Aρ_{w}0 iff Aρ_{w}1
A ⊐ Bρ_{w}1 iff for all w′ such that wRw′, either it is not the case that Aρ_{w}_{′}1 or Bρ_{w}_{′}1
A ⊐ Bρ_{w}0 iff Aρ_{w}1 and Bρ_{w}0
(175)
Validity is truth preservation in all worlds of all interpretations. (9.7a.4) Priest next gives the tableau rules for I_{4}.
Double Negation Development, True (¬¬D,+) 
¬¬A,+i ↓ A,+i 
Double Negation Development, False (¬¬D,−) 
¬¬A,−i ↓ A,−i 
Conjunction Development, True (∧D,+) 
A ∧ B,+i ↓ A,+i B,+i 
Conjunction Development, False (∧D,−) 
A ∧ B,−i ↙ ↘ A,−i B,−i 
Negated Conjunction Development, True (¬∧D,+) 
¬(A ∧ B),+i ↓ ¬A ∨ ¬B,+i 
Negated Conjunction Development, False (¬∧D,−) 
¬(A ∧ B),−i ↓ ¬A ∨ ¬B,−i 
Disjunction Development, True (∨D,+) 
A ∨ B,+i ↙ ↘ A,+i B,+i 
Disjunction Development, False (∨D,−) 
A ∨ B,i ↓ A,i B,i 
Negated Disjunction Development, True (¬∨D, +) 
¬(A ∨ B),+i ↓ ¬A ∧ ¬B,+i 
Negated Disjunction Development, False(¬∨D, ) 
¬(A ∨ B),i ↓ ¬A ∧ ¬B,i 
(p.165, section 9.3.3; titles for the rules are my own additions)
Conditional Development, True (⊐D, +) 
A ⊐ B,+i irj ↙ ↘ A,j B,+j . j is any number on the branch 
Conditional Development, False (⊐D,) 
A ⊐ B,i ↓ irj A,+jB,j . j is new to the branch 
(176, titles for the rules are my own additions)
Negated Conditional Development, True (¬⊐D,+) 
¬(A ⊐ B),+i ↓ A,+i¬B,+i 
Negated Conditional Development, False (¬⊐D,) 
¬(A ⊐ B),i ↙ ↘ A,i ¬B,i 
ρ, Reflexivity (ρrD) 
ρ . ↓ iri 
τ, Transitivity (τrD) 
τ irj jrk ↓ .irk 
(see p.38, section 3.3.2; with my naming additions)
Heredity, Unnegated, True (hD) 
p,+i .irj ↓ p,+j . p is any propositional parameter 
Heredity, Negated, True (¬hD) 
¬p,+i .irj ↓ ¬p,+j . p is any propositional parameter 
(176, with my naming additions)
(9.7a.5) Priest then does some example tableaux to show that ⊢_{I4 }¬¬A ⊐ A, and ⊬_{I4 }(p ∧ ¬p) ⊐ q. (9.7a.6) Countermodels are formed in the following way. “There is a world w_{i }for each i on the branch; for propositional parameters, p, if p,+i occurs on the branch, set pρ_{wi}1; if ¬p,+i occurs on the branch, set pρ_{wi}0. ρ relates no parameter to anything else” (p.166). “w_{i}Rw_{j } iff irj occurs on the branch” (p.27). (9.7a.7) We obtain I_{3} by adding the Exclusion Constraint to I_{4}: “for no p and W, pρ_{w}1 and pρ_{w}0.” (This makes it similar to K_{3}.) Our tableaux for I_{3} have the additional branch closure rule that a branch closes when both a propositional parameter and its negation are true in some same world. (9.7a.8) Formulas lacking negation that are valid in I are also valid in I_{4} and I_{3}. (9.7a.9) But negation behaves differently in I than it does in I_{4} and I_{3}. (9.7a.10) By changing I_{4}’s conditional rule for falsity and the tableau rule for negated conditional, we get a logic called W.
A ⊐ Bρ_{w}0 iff A ⊐ ¬Bρ_{w}1 (i.e., for all w′ such that wRw′, either it is not the case that Aρ_{w}_{′}1 or Bρ_{w}_{′} 0).
Negated Conditional
Development, True (¬⊐D,+)
¬(A ⊐ B),±i
↓
A ⊐ ¬B,±i
(178, naming is my own)
(9.7a.11) W is a connexive logic, meaning that there are two particular inferences it has: Aristotle ¬(A ⊐ ¬A) and Boethius (A ⊐ B) ⊐ ¬(A ⊐ ¬B). (9.7a.12) Unlike all other logics we deal with, connexive logics are not sublogics of classical logic; for, not all the inferences that are valid in connexive logics are also valid in classical logic, here especially: Aristotle and Boethius. (9.7a.13) Aristotle and Boethius have intuitive appeal, despite being heterodox principles of conditionality. (9.7a.14) Another feature that W has that the others of this book lack is that its class of logical truths is inconsistent, namely, (p ∧ ¬p) ⊐ ¬(p ∧ ¬p) contradicts Aristotle ¬(A ⊐ ¬A). (9.7a.15) The tableaux for I_{4}, I_{3}, and W are sound and complete.
Relevant Logics
Introduction
(10.1.1) In this chapter Priest will introduce relevant logics, which “are obtained by employing a ternary relation to formulate the truth conditions of →” and which can be made stronger by adding constraints to that ternary relation. (10.1.2) Also we will combine relevant semantics with the semantics of conditional logics “to give an account of ceteris paribus enthymemes” (188).
The Logic B
(10.2.1) We can strengthen relevant logics like N_{4} and N_{∗ }to accommodate certain intuitively correct principles regarding the conditional by incorporating nonnormal worlds and a ternary accessibility relation on worlds, Rxyz. (10.2.2) The intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. (10.2.3) We will focus on the ternary relation ∗ semantics, as they have been the ones studied historically speaking. (10.2.4) The ternary ∗ interpretation is a structure, ⟨W, N, R, ∗, v⟩, where “W is a set of worlds, N ⊆ W is the set of normal worlds (so that W − N is the set of nonnormal worlds)”, “for all w ∈ W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A → B at every nonnormal world,” and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)” (167; 170; 189). (10.2.5) Priest next gives the truth conditions for connectives.
v_{w}(A ∧ B) = 1 if v_{w}(A) = v_{w} (B) = 1, otherwise it is 0.
v_{w}(A ∨ B) = 1 if v_{w}(A) = 1 or v_{w} (B) = 1, otherwise it is 0.
v_{w}(¬A) = 1 if v_{w*}(A) = 0, otherwise it is 0.
(p.151, section 8.5.3; p.169. section 9.6.6, see 9.6.2)
at normal worlds, the truth conditions for → are:
v_{w}(A → B) = 1 iff for all x ∈ W such that v_{x}(A) = 1, v_{x}(B) = 1
The exception is that if w is a nonnormal world:
v_{w}(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if v_{x}(A) = 1, then v_{y}(B) = 1
(189)
(10.2.6) Validity is truth preservation over all normal worlds. (10.2.7) This logic is named B, and it is a sublogic of K_{∗}, while N_{∗} is a sublogic of B. (10.2.8) The normality condition is Rwxy iff x = y. By implementing it in the conditional rule, we can simply it so that it works for all worlds: v_{w}(A → B) = 1 iff for all x ∈ W such that v_{x}(A) = 1, then v_{x}(B) = 1. (10.2.9) Finally Priest notes that “the normality condition falls apart into two halves. From left to right: if Rwxy then x = y and from right to left, since x = x: Rwxx” (190).
The Ternary Relation
Fuzzy Logics
Introduction
(11.1.1) In this chapter we examine fuzzy logic, which assigns to sentences truth values of any real number between 0 and 1. (11.1.2) We will also discuss vagueness, which is one of the main philosophical motivations for fuzzy logic, and we will discuss fuzzy logic’s relation to relevant logics. (11.1.3) We also examine fuzzy conditionals, including how modus ponens fails in fuzzy logic.
Sorites Paradoxes
(11.2.1) Priest first illustrates the sorites paradox. A person begins at age five and is thus a child. One second after that the person is still a child. Thus also one second after that new second the person is still a child. No additional second will cause the child to definitively cease being a child and start being an adult. However, after 30 years, we know that the person is now an adult. (11.2.2) The sorites paradox results from vague predicates like “is a child,” where , “very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate” (221). (11.2.3) Many other vague predicates, like “is tall,” “is drunk,” “is red,” “is a heap,” and even “is dead,” can all be used to construct sorites paradoxes. (11.2.4) We can structure the sorites paradox as a chain of modus ponens inferences where we say that something begins at a certain state at a certain time, and next that if something is so at that time it is so in the next second, and we repeat that indefinitely, never arriving upon the state we know it will change into.
. . . and Responses to Them
(11.3.1) We will now consider responses to the sorites paradox, where for
M_{0}, M_{1}, . . . , M_{k}
M_{0} is definitely true but M_{k} is definitely false. We will try to understand what is going on, logically speaking, between M_{0} and M_{k}. (11.3.2) If we simply think that every sentence is either simply true or simply false, then we can break the paradoxical chain at some point where we say for example that the person is a child at this moment and in the next one they are an adult (“there must be a unique i such that M_{i} is true, and M_{i}_{+1} is false. In this case, the conditional M_{i} → M_{i}_{+1} is false, and the sorites argument is broken” (222).) But, while that solves the paradox, it goes against our intuition that the change is continuous and thus there can be no such discrete leap happening from one instant to the next (and thus no discrete jump from truth to falsity). (11.3.3) Some still think that there are these discrete leaps of truth value in continuous changes, and the reason it strikes us as counterintuitive is simply because we lack the means to know where exactly the change takes place. (11.3.4) Those arguing the above claim – that there is a discrete truthvalue break but we cannot know it – use the following reasoning. We can only know true things, and we can only make judgements from evidence. The discrete truthvalue shift in the actual change will make M_{i} true but M_{i}_{+1} false. However, the evidential basis will be the same for both. This means that when we make the judgment M_{i}_{+1} (on the basis of the misleading evidence that is the same from the prior moment), we are making a false judgment, and so we can never know when the shift happens. (11.3.5) The main problem with this argument is that the real problem with the paradox is not that we cannot know where the change happens but that there could even be such a sharp cutoff point in a continuous change. (11.3.6) Another proposal is that cases of vagueness require that we reject a bivalent dichotomy between simple truth and falsity, and so for sorites changes, there would be a middle part where the sentences are {1} neither true nor false, or {2} both true or false. (11.3.7) One threevalued solution is using K_{3 }(and perhaps in addition supervaluation). “In this case, there is some i, such that M_{i} is true and M_{i}_{+1} is neither true nor false. Again, M_{i} → M_{i}_{+1} is not true, and so the sorites argument fails” (223). (11.3.8) But threevalued solutions suffer from the same counterintuitiveness: it is hard to accept that there is a discrete boundary between truth and the middle value. (11.3.9) So since the changes are continuous, we might want to use a fuzzy logic where the truthvalues come in continuous degrees too. (11.3.10) But even fuzzy logic has this same problem, because somewhere there must be a change from completely true to less than completely true.
The Continuumvalued Logic Ł
(11.4.1) One way to construct a fuzzy logic is as a manyvalued logic with a continuum of values from 0 (completely false) to 1 (completely true), including all real number values between, such that 0.5 is half true, and so on. (11.4.1a) To formulate the semantics for the connectives, we for now will use the oldest and most philosophically interesting means to do so. (11.4.2) Priest next gives the semantics for the connectives in our continuum manyvalued logic.
f_{¬} (x) = 1 − x
f_{∧}(x, y) = Min(x, y)
f_{∨}(x, y) = Max(x, y)
f_{→}(x, y) = x ⊖ y
where Min means ‘the minimum (lesser) of’; Max means ‘the maximum (greater) of’; and x ⊖ y is a function defined as follows:
if x ≤ y, then x ⊖ y = 1
if x > y, then x ⊖ y = 1 − (x − y) (= 1 − x + y)
 Note that we could say ‘x ≥ y’ instead of ‘x > y’ in the second clause, since if x = y, 1 − (x − y) = 1. Note, also, that we could define x ⊖ y equivalently as Min(1, 1 − x + y).
(225226)
(11.4.3) The formulations of the semantic evaluations for the connectives in fuzzy logic hold to the basic intuitions we have about how they should operate. (11.4.4) Priest next notes that: if x ≤ y, then y ⊖ z ≤ x ⊖ z ; and if x ≤ y, then z ⊖ x ≤ z ⊖ y . (11.4.5) Our continuumvalued fuzzy logic “is a generalisation of both classical propositional logic, and Łukasiewicz’ 3valued logic;” for, if we use only 1 and 0, we get the outcomes for classical semantics, and if we use just 0, 0.5, and 1 (with 0.5 understood as i), we get the outcomes for Ł3. (11.4.6) The designated value is context dependent, and so “any context will determine a number, ε, somewhere between 0 and 1, such that the things that are acceptable are exactly those things with truth value x, where x ≥ ε” (226). (11.4.7) Validity is defined as: “Σ ⊨_{ε} A iff for all interpretations, v, if v(B) ≥ ε for all B ∈ Σ, then v(A) ≥ ε” (226). (In other words, an inference is valid under the following condition: whenever the premises are at least as high as the ((contextdetermined designated fuzzy)) value ε, then so too is the conclusion at least as high as ε. (11.4.8) Our fuzzy logic is called Ł, and its contextindependent definition of validity is: Σ ⊨ A iff for all ε, where 0 ≤ ε ≤ 1, Σ ⊨_{ε} A . (11.4.9) A set of truthvalues X can be listed in descending numerical order. Suppose it is in an infinite set following a pattern like {0.41, 0.401, 0.4001, 0.40001, . . .}. Even though there would be no least member, there is still however a number that would be the greatest possible figure that is still less than or equal to all the members, in this case being 0.4. And it is called the greatest lower bound of set X, abbreviated as Glb(X). (11.4.10) A simpler characterization of validity would be: Σ ⊨ A iff for all v, Glb(v[Σ]) ≤ v(A). (11.4.11) Given the semantic evaluation for conjunction and the conditional, we can formulate validity in the following way: {B_{1}, . . . , B_{n}} ⊨ A iff for all v, v((B_{1} ∧ . . . ∧ B_{n}) → A) = 1. “Thus (for a finite number of premises), validity amounts to the logical truth of the appropriate conditional when the set of designated values is just {1}, that is, the logical truth of the conditional in ⊨_{1}. The logic with just 1 as a designated value is usually written as Ł_{ℵ}, and it is called Łukasiewicz’ continuumvalued logic” (227).
Appendix: Manyvalued Modal Logics
Introduction
(11a.1.1) In this chapter we will examine manyvalued modal logics. (11a.1.2) First we examine the general structure of a manyvalued modal logic, illustrated with Łukasiewicz continuumvalued modal logic. (11a.1.3) We will also examine manyvalued modal First Degree Entailment logics, including modal K3 and modal LP. (11a.1.4) We will end the chapter with a discussion of future contingents.
General Structure
(11a.2.1) A “propositional manyvalued logic is characterised by a structure ⟨V, D, {f_{c} : c ∈ C}⟩, where V is the set of semantic values, D ⊆ V is the set of designated values, and for each connective, c, f_{c} is the truth function it denotes. An interpretation, v, assigns values in V to propositional parameters; the values of all formulas can  then be computed using the f_{c}s; and a valid inference is one that preserves designated values in every interpretation” (242243). (11a.2.2) We will assume that the set of truthvalues of V are ordered from lesser to greater (or to equal than: ≤) and that “every subset of the values has a greatest lower bound (Glb) and least upper bound (Lub) in the ordering” (242). (11a.2.3) Our manyvalued modal logic adds to manyvalued logic “the monadic operators, □ and ◊ in the usual way” (242). (11a.2.4) “An interpretation for a manyvalued modal logic is a structure ⟨W, R, S_{L}, v⟩, where W is a nonempty set of worlds, R is a binary accessibility relation on W, S_{L} is a structure for a manyvalued logic, L, and for each propositional parameter, p, and world, w, v assigns the parameter a value, v_{w}(p), in V” (242). (11a.2.5) The value of a connective on formulas in a certain world is determined by the value generated by the corresponding connective function operating recursively on the truth values of those formulas in that particular world: “if c is an nplace connective v_{w}(c(A_{1}, . . . , A_{n})) = f_{c}(v_{w}(A_{1}), . . . , v_{w}(A_{n}))” (242). (11a.2.6) The truthconditions for the modal operators are:
v_{w}(□A) = Glb{v_{w}_{′}(A) : wRw′}
v_{w}(◊A) = Lub{ v_{w}_{′}(A) : wRw′}
(11a.2.7) Validity is defined in the following way: “Σ ⊨ A iff for every interpretation, ⟨W, R, S_{L}, v⟩, and for every w ∈ W, whenever v_{w}(B) ∈ for every B ∈ Σ, v_{w}(A) ∈ ” (242). (11a.2.8) Our manyvalued modal logic is called K_{L}, and we can apply the accessibility relation constraints on it to derive stronger logics, like K_{L}ρ, K_{L}σ, K_{L}ρτ, and so forth.
Modal FDE
(11a.4.1) In our manyvalued modal FDE logic, we have just the connectives ∧, ∨ and ¬, because A ⊃ B is defined as ¬A ∨ B. (11a.4.2) FDE can be formulated as a fourvalued logic, and the connectives can be evaluated with the following diamond lattice (Hasse diagram):
1 
↗ ↖ 
b n 
↖ ↗ 
0 
f_{∧} is the greatest lower bound; f_{∨} is the least upper bound; and f_{¬} maps 1 to 0, vice versa, and each of b and n to itself. (11a.4.3) Our modal FDE logic is called K_{FDE}. “If we ignore the value n in the nonmodal case (that is, we insist that formulas take one of the values in {1, b, 0}) we get the logic LP. In the modal case, we get K_{LP}. If we ignore the value b in the nonmodal case, we get the logic K_{3}. In the modal case, we get K_{K3}” (245). (11a.4.4) The equivalence between fourvalued FDE truthvalues and the four valuesituations of the relational semantics are: v(A) = 1 iff Aρ1 and it is not the case that Aρ0 ; v(A) = b iff Aρ1 and Aρ0 ; v(A) = n iff it is not the case that Aρ1 and it is not the case that Aρ0 ; and v(A) = 0 iff it is not the case that Aρ1 and Aρ0 . The truthfalsity conditions for the connectives in the relational semantics are: A ∧ Bρ1 iff Aρ1 and Bρ1 ; A ∧ Bρ0 iff Aρ0 or Bρ0 ; A ∨ Bρ1 iff Aρ1 or Bρ1 ; A ∨ Bρ0 iff Aρ0 and Bρ0 ; ¬Aρ1 iff Aρ0 ; and ¬Aρ0 iff Aρ1 . “Validity is defined in terms of the preservation of relating to 1” (245). (11a.4.5) K_{FDE} has the same relational semantics but with world designations and with the following rules for the necessity and possibility operators:
v_{w}(A) = 1 iff Aρ_{w}1 and it is not the case that Aρ_{w}0
v_{w}(A) = b iff Aρ_{w}1 and Aρ_{w}0
v_{w}(A) = n iff it is not the case that Aρ_{w}1 and it is not the case that Aρ_{w}0
v_{w}(A) = 0 iff it is not the case that Aρ_{w}1 and Aρ_{w}0
A ∧ Bρ_{w}1 iff Aρ_{w}1 and Bρ_{w}1
A ∧ Bρ_{w}0 iff Aρ0 or Bρ_{w}0
A ∨ Bρ_{w}1 iff Aρ_{w}1 or Bρ_{w}1
A ∨ Bρ_{w}0 iff Aρ_{w}0 and Bρ_{w}0
¬Aρ_{w}1 iff Aρ_{w}0
¬Aρ_{w}0 iff Aρ_{w}1
□Aρ_{w}1 iff for all w′ such that wRw′, Aρ_{w}_{′}1
□Aρ_{w}0 iff for some w′ such that wRw′, Aρ_{w}_{′}0
◊Aρ_{w}1 iff for some w′ such that wRw′, Aρ_{w}_{′}1
◊Aρ_{w}0 iff for all w′ such that wRw′, Aρ_{w}_{′}0
(based on with quotation from p.245246)
(11a.4.6) Next Priest provides the argumentation for why the truth/falsity conditions are formulated the way they are. (11a.4.7) By adding a possible worlds exhaustion constraint to K_{FDE} we can get K_{LP}. (11a.4.8) By adding a possible worlds exclusion constraint to K_{FDE} we can get K_{K3}. (11a.4.9) By adding both the exhaustion and exclusion constraints to K_{FDE}, we get the classical modal logic K. (11a.4.10) We can apply the world accessibility relation constraints (like ρ, σ, τ, etc) to our specific manyvalued modal logics to get such extensions as: K_{FDE}ρ, K_{LP}ρτ, K_{K3}σ and so forth. (11a.4.11) There are no logical truths (tautologies) in K_{FDE} and K_{K3}, because to be a logical truth means that the formula is true under all interpretations. But in K_{FDE} and K_{K3}, under any interpretation, every formula can be valued neither true nor false, and thus no formulas can be logical truths. (11a.4.12) The interpretations for the logics of the family we are considering are monotonic. (11a.4.13) A corollary of this is “that ⊨_{K} A iff ⊨_{KLP}A (and similarly for Kρ and K_{LP}ρ, etc.)” (247).
Future Contingents Revisited
(11a.7.1) We may use manyvalued modal logics for contending with the problem of future contingents. Aristotle’s analysis of them lends to the manyvalued solutions. We begin with the intuition that there are contingent future events for which there are presently no facts that could make them true or false, (as for example, “The first pope in the twentysecond century will be Chinese” (132)). We then suppose that right now a statement about a future contingent is true (or false). That would mean that it cannot be otherwise, and thus fatalism would hold if future contingents are presently either true or false. But that goes against our original intuition that nothing in the present makes such statements true or false, so Aristotle concludes that statements about future contingents cannot have either the value true or false. (11a.7.2) Priest next quotes the important Aristotle passage for this discussion of future contingents, from De Int. 18^{b}10–16.
. . . if a thing is white now, it was true before to say that it would be white, so that of anything that has taken place, it was always true to say ‘it is’ or ‘it will be’. But if it was always true to say that a thing is or will be, it is not possible that it should not be or not come to be, and when a thing cannot not come to be, it is impossible that it should not come to be, and when it is impossible that it should not come to be, it must come to be. All then, that is about to be must of necessity take place. It results from this that nothing is uncertain or fortuitous, for if it were fortuitous it would not be necessary.
(251252, quoting De Int. 18^{b}10–16. Translation from Vol. 1 of Ross (1928).)
(11a.7.3) This argument may be read in the following way. “Let q be any statement about a future contingent event. Let T_{q} be the statement that it is (or was) true that q. Then □(T_{q} ⊃ q). Hence T_{q} ⊃ □q. And since □q is not true, neither is T_{q}. A similar argument can be run for ¬q. So neither T_{q} nor T_{¬q} holds. Read in this way, the reasoning contains a modal fallacy (passing from □(A ⊃ B) to (A ⊃ □B))” (252). (11a.7.4) The above reading is incorrect, because Aristotle holds that the past and present are unchangeable and thus necessary. So the inference from □(T_{q} ⊃ q) should be □T_{q} ⊃ □q, which is valid. (11a.7.5) The above argument can be formulated without the conditional or the T_{q} formula. We have the statement about the future, q. “If q were true, this would be a present fact, and so fixed; that is, it would be necessarily true, that is: q ⊨ □q. Similarly, if it were false, it would be necessarily false: ¬q ⊨ □¬q. Since neither □q nor □¬q holds, neither q nor ¬q holds” (252). (11a.7.6) Aristotle does not allow exceptions to the Law of NonContradiction. So the sort of manyvalued modal logic we use in application to his Future Contingents argument should validate it. Thus we should not use K_{FDE} or K_{LP} but rather K_{K3}, in which there is the option for formulas to be neither true nor false, but not the option for contradictions. (11a.7.7) In our manyvalued modal logic, we indicate futurity with the R accessibility relation:“Think of the accessibility statement wRw′ as meaning that w′ may be obtained from w by some number (possibly zero) of further things happening” (252). Given the nature of time, R is reflexive and transitive but not symmetrical. To capture Aristotle’s assumption that “once something is true/false, it stays so,” we will use a modified heredity constraint called the Persistence Constraint: “for every propositional parameter, p, and world, w:
If pρ_{w}1 and wRw′, pρ_{w}_{′}1
If pρ_{ w}0 and wRw′, pρ_{ w}_{′}0
(11a.7.8) The persistence constraint does not hold for modalized formulas. (11a.7.9) Our manyvalued K_{3}ρτ logic, augmented by the Persistence Constraint, is called A (for Aristotle). “In this logic p ⊨ □p and ¬p ⊨ □¬p. Aristotle’s argument therefore works. But, of course, in A, p ∨ ¬p may fail to be true.” (11a.7.10) For our Aristotle logic A, neither □p not □¬p holds. However, Aristotle thinks that eventually p or ¬p will have to hold, thus he thinks □(p ∨ ¬p). Yet, this does not hold in logic A. (11a.7.11) To allow □(p ∨ ¬p) to hold in logic A, we can take a temporal perspective of the end of time when everything has been decided. “Call a world complete if every propositional parameter is either true or false. A natural way of giving the truth conditions for □ is as follows:
□Aρ_{w}1 iff for all complete w′ such that wRw′, Aρ_{w′}1
□Aρ_{w}0 iff for some complete w′ such that wRw′, Aρ_{w′ }0
The truth/falsity conditions for ◊ are the same with ‘some’ and ‘all’ interchanged. □A may naturally be seen as expressing the idea that A is inevitable. [...] for any complete world, w, Persistence holds for all formulas. It follows that at such a world, A is true iff □A is, and that all formulas are either true or false” (254). (11a.7.12) These above revised truth/falsity conditions for necessity allow us to capture the important assumptions and valid inferences in Aristotle’s argumentation regarding future contingency, namely: “p ⊨ □p, ¬p ⊨ □¬p (so Aristotle’s argument still works), ⊨ □(p ∨ ¬p), but not ⊨ □p ∨ □¬p” (254).
Quantification and Identity
Classical Firstorder Logic
Introduction
In this chapter, we examine the semantics and tableaux of classical firstorder logic, along with problems and certain technical issues involved in it.
Syntax
Our firstorder language has the following vocabulary:
• variables: v_{0}, v_{1}, v_{2}, ...
• constants: k_{0}, k_{1}, k_{2}, ...
• for every natural number n > 0, nplace predicate symbols: P^{0}_{n}, P^{1}_{n}, P^{2}_{n}, ...
• connectives: ∧, ∨, ¬, ⊃, ≡
• quantifiers: ∀, ∃
• brackets: (, )
Specifically we may use:
• x, y, z for arbitrary variables
• a, b, c for arbitrary constants
• P_{n}, Q_{n}, S_{n} for arbitrary nplace predicates
• A, B, C for arbitrary formulas
• Σ, Π for arbitrary sets of formulas
Its grammar includes the following:
• Any constant or variable is a term.
The formulas are specified recursively as follows.
• If t_{1}, ... , t_{n} are any terms and P is any nplace predicate, Pt_{1 ..} t_{n} is an (atomic) formula.
• If A and B are formulas, so are the following:
(A ∧ B), (A ∨ B), ¬A, (A ⊃ B), (A ≡ B).
• If A is any formula, and x is any variable, then ∀xA, ∃xA are formulas. I will omit outermost brackets in formulas.
And regarding quantified formulas:
• An occurrence of a variable, x, in a formula, is said to be bound if it occurs in a context of the form ∃x ... x ... or ∀x ... x ....
• If it is not bound, it is free.
• A formula with no free variables is said to be closed.
• A_{x}(c) is the formula obtained by substituting c for each free occurrence of x in A.
Semantics
Regarding the semantics of classical firstorder logic, we say that:
An interpretation of the language is a pair, ℑ = ⟨D, v⟩. D is a nonempty set (the domain of quantification); v is a function such that:
• if c is a constant, v(c) is a member of D
• if P is an nplace predicate, v(P) is a subset of D^{n }
(D^{n }is the set of all ntuples of members of D, {⟨d_{1}, ..., d_{n}⟩: d_{1}, ..., d_{n }∈ D}. By convention, ⟨d⟩ is just d, and so D^{1 }is D.)
(Priest 264)
All formulas have a truth value. To evaluate them, since they use variables which are substitutable by constants,
we extend the language to ensure that every member of the domain has a name. For all d ∈ D, we add a constant to the language, k_{d}, such that v(k_{d}) = d. The extended language is the language of ℑ, and written L(ℑ). The truth conditions for (closed) atomic sentences are:
v(Pa_{1 }... a_{n}) = 1 iff ⟨v(a_{1}), _{}..., v(a_{n})⟩ ∈ v(P) (otherwise it is 0)
The truth conditions for the connectives are as in the propositional case (1.3.2).
(Priest 265)
v(¬A) = 1 if v(A) = 0, and 0 otherwise.
v(A ∧ B) = 1 if v(A) = v(B) = 1, and 0 otherwise.
v(A ∨ B) = 1 if v(A) = 1 or v(B) = 1, and 0 otherwise.
v(A ⊃ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
v(A ≡ B) = 1 if v(A) = v(B), and 0 otherwise.(Priest 5)
For the quantifiers:
v(∀xA) = 1 iff for all d ∈ D, v(A_{x}(k_{d})) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some d ∈ D, v(A_{x}(k_{d})) = 1 (otherwise it is 0)
(Priest 265)
We define validity in the following way:
Validity is a relationship between premises and conclusions that are closed formulas, and is defined in terms of the preservation of truth in all interpretations, thus: Σ ⊨ A iff every interpretation that makes all the members of Σ true makes A true.
(Priest 265)
We find the following equivalences in classical firstorder logic:
v(¬∃xA) = v(∀x¬A)
v(¬∀xA) = v(∃x¬A)
v(¬∃x(Px ∧ A)) = v(∀x(Px ⊃ ¬A)
v(¬∀x(Px ⊃ A) = v(∃x(Px∧¬A))
(Priest 265)
And lastly we note [something regarding denotation, namely] that
If C is some set of constants such that every object in the domain has a name in C, then:
v(∀xA) = 1 iff for all c ∈ C, v(A_{x}(c)) = 1 (otherwise it is 0)
v(∃xA) = 1 iff for some c ∈ C, v(A_{x}(c)) = 1 (otherwise it is 0)
(Priest 265)
Tableaux
(12.4.1) The tableau rules for classical firstorder logic add four quantifier rules to those from propositional logic.
Double Negation Development (¬¬D) 
¬¬A ↓ A 
Conjunction Development (∧D) 
A ∧ B ↓ A ↓ B 
Negated Conjunction Development (¬∧D) 
¬(A ∧ B) ↙ ↘ ¬A ¬B 
Disjunction Development (∨D) 
A ∨ B ↙ ↘ A B 
Negated Disjunction Development (¬∨D) 
¬(A ∨ B) ↓ ¬A ↓ ¬B 
Conditional Development (⊃D) 
A ⊃ B ↙ ↘ ¬A B 
Negated Conditional Development (¬⊃D) 
¬(A ⊃ B) ↓ A ↓ ¬B 
Biconditional Development (≡D) 
A ≡ B ↙ ↘ A ¬A B ¬B 
Negated Biconditional Development (≡D) 
A ≡ B ↙ ↘ A ¬A ¬B B 
Negated Existential Development (¬∃D) 
¬∃xA ↓ ∀x¬A 
Negated Universal Development (¬∀D) 
¬∀xA ↓ ∃x¬A 
Universal Instantiation Development (UI,D) 
∀xA ↓ A_{x}(a)
where a is any constant on the branch. (If there are not any, we select one at will.) 
Particular Instantiation Development (PI,D) 
∃xA ↓ A_{x}(c)
where c is any constant that does not occur so far on the branch. 
(68; 266, with names and additional text at the bottom made by me)
Note: we should never checkoff lines of the form ∀xA, because it is always possible that we will later introduce a new constant and then need to repeat universal instantiation on it. Yet, it can be useful to write beside the line all the constants it has been instantiated with. (12.4.2) We cannot apply a rule to an internal part of a line but only to the whole line itself. So we should not have:
¬(A ∧ ∀xB)
↓
¬(A ∧ Bx(a))
(12.4.3) Priest will “write Σ ⊢ A to mean that there is a closed tableau whose initial list comprises the members of Σ together with the negation of A” (266267). (12.4.4) It does not matter what order we apply the tableau rules. (12.4.5) Priest then gives a first example tableau for a valid inference. (12.4.6) Priest next gives a second example tableau for a valid inference. (12.4.7) Priest then gives a third example tableau, this time for an invalid inference. (12.4.8) We construct countermodels from open branches in the following way: “take a domain which contains a distinct object, ∂b, for every constant, b, on the branch. v(b) is ∂_{b}. v(P) is the set of ntuples ⟨∂_{b1}, . . . , ∂_{bn}⟩ such that Pb_{1} . . . b_{n} occurs on the branch. Of course, if ¬Pb_{1} . . . b_{n} is on the branch, ⟨∂_{b1}, . . . , ∂_{bn}⟩ ∉ v(P), since the branch is open. (If a predicate or constant does not occur on the branch, the value given to it by v is a don’t care condition: it can be anything one likes)” (268). (12.4.9) We can check which formulas are true by seeing if the model makes them so. (12.4.10) Tableaux can be infinite. One way this happens is if there is an endless generation of new constants on account of repeating applications of instantiation. (12.4.11) When tableaux are infinite, we can use guesswork to formulate countermodels. (12.4.12) Priest next provides an algorithm for ensuring that infinite tableaux are complete: “(1) For each branch in turn (there is only a finite number at any stage of the construction), we run down the formulas on the branch, applying any rule that generates something not already on the branch. (In the case of a rule such as UI, which has multiple applications, we make all the applications possible at this stage.) (2) We then go back and repeat the process” (270). (12.4.13) “The tableaux are sound and complete with respect to the semantics” (270). (12.4.14) In order to show the behavior of quantifiers, Priest lists the ways that the quantifiers interact with the propositional operators.
In classical logic, the interactions are as follows. ‘A ⊣⊢ B’ means ‘A ⊢ B and B ⊢ A’. C is any closed formula. A * at the end of a line indicates that the converse does not hold, in the sense that there are instances that are not valid. (So, for example, in the first line for Negation, if A is Px, we have ¬∀xPx ⊬ ∀x¬Px.) Where the converse does not hold, there is often a restricted version involving a closed formula that does. Where this exists, it is given on the next line. [...]
1. No Operators
(a) ∀xC ⊣⊢ C
(b) ∃xC ⊣⊢ C
2. Negation <