## 28 Jul 2017

### Priest (7.2) An Introduction to Non-Classical Logic, ‘Many-valued Logic: The General Structure,’ summary

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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other distracting mistakes, because I have not finished proofreading.]

Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

7. Many-valued Logics

7.2. Many-valued Logic: The General Structure

Brief summary:

In providing the general structure for many-valued 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 many-valued logics by naming all the components, including the parts relevant for truth and validity evaluations. In its most condensed form, the structure of many-valued logics is:

V, D, {fc; 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 fc 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 ≡)

fc; 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 n-tuple corresponding to that connective:

if c is an n-place connective,

v(c(A1, ... , An)) = fc(v(A1), ... , v(An))

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 one-place function such that f¬(0) = 1 and f¬(1) = 0;

f is a two-place 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

(120-121)

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(¬(pq)) = 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 many-valued 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 truth-value. 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 many-valued when V has a finite number of values in it; and when V has n members, we say that it is an n-valued 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 nm possible cases to consider.

(122)

Summary

7.2.1

[A B is defined as (A B) ∧ (B A).]

[For simplicity, we will define material equivalence using conjunction and the material conditional (see section 1.2.1). But I do not know yet how that simplifies things, other than making one fewer connective to define.]

Let us start with the general structure of a many-valued logic. To simplify things, we take, henceforth, A B to be defined as (A B) ∧ (B A).

(120)

7.2.2

[The basic structure for many-valued logic is described by first filling it out with classical values. It is composed of a set of truth values {1, 0}, a set of designated values (the values preserved in valid inferences): {1}, and a set of connectives with their corresponding truth functions. The negation and conjunction truth functions assign values here in the classical bivalent way. The structure in its condensed form is: ⟨V, D, {fc; c C}⟩. V is the set of assignable truth values, D is the set of designated values, C is the set of connectives, c is some particular connective, and fc is the truth function corresponding to the connective that operates on the truth values of the formula in question.]

[Priest will now give the structure of many-valued logics. That structure is a set of symbols and functions that it uses. In its most condensed form, it is: ⟨V, D, {fc; c C}⟩. Let us unpack that. V stands for the set of truth values that can be assigned in our many-valued semantics. Here, those values are found in the set {1,0}. We might at this point be confused, because there are only two values. Either we are setting up a bivalent logic, and later we add more values, or, as in Nolt’s account, we will assign both or no values as members of a set. (From what happens later, it seems we begin here with a bivalent logic as a template that we will modify according to the many-valued logics we construct.) The D is the set of designated values, which are those that are preserved in valid inferences. Here there is just one value, 1, hence so far it seems to be a classical structure. We also have connectives, and as we will see, they are ¬ (negation), ∧ (conjunction), ∨ (disjunction), and ⊃ (material conditional). We think then of the connective’s truth functionally, which seems to mean that we think of the connectives as functions that assign a truth value based on the value they operate upon. So if a formula is assigned the value 1, the truth function operator for negation will assign it 0. Priest also defines conjunction here in the usual bivalent, classical way.]

Let C be the class of connectives of classical propositional logic {∧,∨,¬, ⊃}. The classical propositional calculus can be thought of as defined by the structure ⟨V, D, {fc; c C}⟩. V is the set of truth values {1,0}. D is the set of designated values {1}; these are the values that are preserved in valid inferences. For every connective, c, fc is the truth function it denotes. Thus, f¬ is a one-place function such that f¬(0) = 1 and f¬(1) = 0; f is a two-place function such that f(x, y) = 1 if x = y = 1, and f(x, y) = 0 otherwise; and so | on. These functions can be (and often are) depicted in the following ‘truth tables’.

 f¬ 1 0 0 1

 f∧ 1 0 1 1 0 0 o o

(120-121)

7.2.3

[The interpretation function v maps propositional parameters to V, and thus assigning either 1 or 0. Truth functions apply to those V values recursively in complex formulas. Inferences are semantically valid iff “there is no interpretation that assigns all the premises a value in D, but assigns the conclusion a value not in D.”]

[Now we wonder about assigning truth values to formulas. We do so with an interpretation function, v, which maps the propositional parameter to V, meaning, to either 1 or o. He says the truth functions are applied recursively. For this notion, first recall the idea of “scope indicators” and “main operators” from Agler’s Symbolic Logic section 2.2.3. The recursive procedure seems to be that we substitute the v function for the main connective, while putting that connective’s truth function in for where the v function just was. If this presents us with a new main connective, then we keep repeating until all the connectives are replaced with connective truth functions. This is his example:

v(¬(pq)) = f¬(v(p q)) = f¬(f(v(p), v(q))).

(So if v(p) = 1 and v(q) = 0, v(¬(p q)) = f¬(f(1, 0)) = f¬(0) = 1.)

Let us go part by part. We want to know the value for this formula:

v(¬(pq))

We see that in order to know its value, we need to negate the formulation on the next lowest scale or scope, upon which the negation is operating. So we put the negation function in for v, and we now put v in for the negation connective, because we now need to know the value of the negation of the formula in parenthesis.

f¬(v(p q))

But we notice now that to determine this value, we need to see how the conjunction operates on the next lowest scale or scope, so we do the same thing.

f¬(f(v(p), v(q)))

So the v in v(p) and v(q) will assign a 1 or a 0 to these propositional parameters. In the example assignments, v(p) = 1 and v(q) = 0, that gives us:

f¬(f(1, 0)

We then calculate our evaluations moving from the lowest to the highest scope. Recall our rule for the behavior of the f function.

f is a two-place function such that f(x, y) = 1 if x = y = 1, and f(x, y) = 0 otherwise

Not both are 1, so the function assigns the pair the value 0.

f¬(0)

We next look at our rule for the negation truth function.

f¬ is a one-place function such that f¬(0) = 1 and f¬(1) = 0

Since the given value is 0, the function assigns it the value 1.

f¬(0) = 1

Priest next defines semantic validity, which follows the definition we have for it already (see section 1.1 and section 1.3). We must note that for validity, the value being preserved must be in D, the designated values. This will be important later when third values can be in D. “an inference is semantically valid just if there is no interpretation that assigns all the premises a value in D, but assigns the conclusion a value not in D”.]

An interpretation, v, is a map from the propositional parameters to V. An interpretation is extended to a map from all formulas into V by applying the appropriate truth functions recursively. Thus, for example, v(¬(pq)) = f¬(v(p q)) = f¬(f(v(p), v(q))). (So if v(p) = 1 and v(q) = 0, v(¬(p q)) = f¬(f(1, 0)) = f¬(0) = 1.) Finally, an inference is semantically valid just if there is no interpretation that assigns all the premises a value in D, but assigns the conclusion a value not in D.

(121)

7.2.4

[Many-valued logics keep this structure, but the contents of each set can vary from the classical template we made above. For example, it may have different connectives, different truth values, or different designated values.]

[We can then obtain a many-valued logic by broadening the components of this structure. We may have the same or a different set of connectives in set C. And V may contain any number of truth values, so long as there is at least one value. D can even have different values. Our connectives are normally either one or two place, but we can also have connectives with more places.]

A many-valued logic is a natural generalisation of this structure. Given some propositional language with connectives C (maybe the same as those of the classical propositional calculus, maybe different), a logic is defined by a structure ⟨V, D, {fc; c C}⟩. V is the set of truth values: it may have any number of members (≥ 1). D is a subset of V, and is the set of designated values. For every connective, c, fc is the corresponding truth function. Thus, if c is an n-place connective, fc is an n-place function with inputs and outputs in V.

(121)

7.2.5

[Connectives can take any number of places more than two, and they are evaluated accordingly using the connective functions operating on the series of values. A set of formulas semantically entails some conclusion when there is no interpretation that assigns designated values to the premises but not to the conclusion. A formula is a logical truth (tautology) when every evaluation assigns it a designated value.]

[An interpretation is (still) understood as a mapping from propositional parameters to values in V, by means of the function v. And again we use truth functions recursively, as above. The truth functions for connectives will have some number of places, and the assignment will be based on the values given at each place. Valid inferences are only those where, when all the premises have a designated value, the conclusion does too (or as it is formulated here, when there is no interpretation v making the premises have a designated value but the conclusion not have a designated value). A tautology or logical truth is a formula where every interpretation assigns it a designated value. It is also defined as a formula that is a semantic consequence of the empty set (for discussion on this formulation of tautology, see the comments in section 1.3.4 and section 2.3.11.]

An interpretation for the language is a map, v, from propositional parameters into V. This is extended to a map from all formulas of the language to V by applying the appropriate truth functions recursively. Thus, if c is an n-place connective, v(c(A1, ... , An)) = fc(v(A1), ... , v(An)). Finally, Σ ⊨ A iff there is no interpretation, v, such that for all B ∈ Σ, v(B) ∈ D, but v(A) ∉ D. A is a logical truth iff φ A, i.e., iff for every interpretation v(A) ∈ D.

(121)

7.2.6

[“If V is finite, the logic is said to be finitely many-valued. If V has n members, it is said to be an n-valued logic.”]

[Later we will examine fuzzy logics where our assignable values are all the real numbers between zero and one, and thus we would have an infinite set of possible values to assign. But here we are dealing with a finite number of values.]

If V is finite, the logic is said to be finitely many-valued. If V has n members, it is said to be an n-valued logic.

(121)

7.2.7

[We can check an argument for validity by computing the values for all the premises and conclusions for every possible truth assignment for the propositional parameters. If there is at least one instance where all the premises obtain a designated value but the conclusion does not, then it is invalid. Otherwise, it is valid.]

[Recall from Agler’s Symbolic Logic section 3.6 the truth table method to evaluate arguments for validity. We fill out all the possible truth value assignments for the atomic formulas, and then we compute the values for the premises and conclusion to see if there is any line where all the premises are true and the conclusion false. Here we can do the same, looking for lines (or just simply for evaluation cases, if we are not using tables) where all the premises have a designated value and the conclusion does not.]

For any finitely many-valued logic, the validity of an inference with finitely many premises can be determined, as in the classical propositional calculus, simply by considering all the possible cases. We list all the possible combinations of truth values for the propositional parameters employed. | Then, for each combination, we compute the value of each premise and the conclusion. If, in any of these, the premises are all designated and the conclusion is not, the inference is invalid. Otherwise, it is valid. We will have an example of this procedure in the next section.

(122)

7.2.8

[This exhaustive checking procedure can become impractical in many-valued logic as we increase the number of propositional parameters. “For if there are m propositional parameters employed in an inference, and n truth values, there are nm possible cases to consider.”]

[Recall from Nolt’s Logics section 16.3.21 an argument we checked for validity in a four-valued semantics. It was disjunctive syllogism:

P ∨ Q, ~P ⊢ Q (Nolt 445)

We have two propositional parameters, P and Q. But we have four possible value assignments for any parameter. In order to evaluate every possibility, as you can see, we need 16 rows, or 24. As you can imagine, the number of rows can get impractically long as we add more parameters.]

This method, though theoretically adequate, is often impractical because of exponential explosion. For if there are m propositional parameters employed in an inference, and n truth values, there are nm possible cases to consider. This grows very rapidly. Thus, if the logic is 4-valued and we have an inference involving just four propositional parameters, there are already 256 cases to consider!

(122)

Priest, Graham. 2008 . An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

Also cited:

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

.

## 26 Jul 2017

### Priest (2.5) An Introduction to Non-Classical Logic, ‘Possible Worlds: Representation,’ summary

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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other distracting mistakes, because I have not finished proofreading.]

Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

2. Basic Modal Logic

2.5. Possible Worlds: Representation

Brief summary:

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?

Summary

2.5.1

[We now ask philosophically what modal semantics mean.]

The remainder of chapter 2 will ask the philosophical question of what do modal semantics mean.

2.5.2

[They might mean nothing more than the mathematical machinery underlying them.]

[Recall from section 2.3 how the modal semantics were formulated. And note as well how the propositional language it was built on is formulated (section 1.2); and while we are at it, recall from section 0.1 the set theoretical notation that at times comes into play. Without any of the intuitive elaborations on what the formulations mean, we might be able to simply say that all of this has no other meaning than its bare mathematical machinery.]

One might suggest that they do not mean anything. They are simply a mathematical apparatus – interpretations comprise just bunches of objects (W) furnished with some properties and relations – to be thought of purely instrumentally as delivering an appropriate notion of validity.

(28)

2.5.3

[But the mathematical machinery generates the right kinds of answers for our philosophical concerns, and so its philosophical value must be more than its bare mathematical machinery.]

But Priest notes that the machinery produces what seems to be the right answers. The reason why it does so is perhaps because it accurately models logical aspects of reality. “There must be some relationship between how it works and reality, which explains why it gets things right” (28). Thus it is not satisfying to simply say that the philosophical meaning of modal semantics is nothing more than its mathematical machinery. It must have some greater philosophical value.

2.5.4

[Probably modal semantics represents something fundamental to validity itself.]

[Specifically the mathematics behind modal semantics models validity in the way we want it to. We might say then that the mathematics represents something about validity.]

The most obvious explanation in this context is that the mathematical structures that are employed in interpretations represent something or other which underlies the correctness of the notion of validity.

(28)

2.5.5

[One example of how our mathematicized logical machinery represents real things of philosophical importance is how we represent truth with the symbol ‘1’ (and ‘o’ for falsity). The truth-functional semantics that calculates these symbols models the way validity actually works.]

Priest then gives an example for how the mathematical structures of logic correspond to real things that they model. He notes how we use the number 1 to represent truth; the truth-functional semantics that we use to assign and calculate truth values corresponds to the properties of truth itself.

In the same way, no one supposes that truth is simply the number 1. But that number, and the way that it behaves in truth-functional semantics, are able to represent truth, because the structure of their machinations corresponds to the structure of truth’s own machinations. This explains why truth-functional validity works (when it does).

(28).

2.5.6

[We then ask, what do the possible worlds and the machinery of possible world semantics represent, philosophically speaking?]

Priest ends by asking: “what exactly, in reality, does the mathematical machinery of possible worlds represent? Possible worlds, of course (what else?). But what are they?” (28).

Priest, Graham. 2008 . An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

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## 25 Jul 2017

### Sambursky (CBS) Physics of the Stoics, collected brief summaries

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Collected Brief Summaries for

Samuel Sambursky

Physics of the Stoics

Ch.1. The Dynamic Continuum

1.1 Pneuma and Coherence [selective summary]

The Stoics held that the cosmos is a continuous whole surrounded by a void. The continuity of the whole is a dynamic continuity resulting from a cohering activity of a very rarified substrate called pneuma. Pneuma is composed of a mixture of the active elements, Air and Fire, while Water and Earth are the other two passive elements that pneuma serves to bind. Pneuma has two physical features that give it this cohering role: {1} tonos, the tension which holds things together, and {2} gravitational neutrality, which means it gathers neither too high nor too low but instead equally pervades all domains of the cosmos.

1.2 The Physical State of a Body [selective summary]

Pneuma’s function is two-fold: it both coheres the parts of a thing and it also serves as a field carrying the properties of the thing. There are three hierarchical orders of compositional organizations for things. {1a} Discrete non-denumerable entities, where there is a disordered assembly of bodies that are too disorganized for us to count, as in a crowd of people. {1b} Discrete denumerable entities where the elements are arranged in an ordered way allowing for us to count them, as with an army formation. {2} Contiguous structures, whose elements are combined, as with links in a chain. Both discrete and contiguous structures have the property that any part can survive if all the others are destroyed, as they are constituted by an additive principle. The situation is different for {3} unified structures, which are organized by hexis. Hexis is what organizes the parts of inorganic objects, like physis for plants and psyche for animals. Here the units of the thing are not merely the parts but rather the different properties of the thing, which interpenetrate such that a change in one leads to a change in the others, as a result of the “sympathy” holding between the properties. But if it is the same thing, namely, pneuma, that carries the properties of various things, then how do we explain why different things have different properties? In Sambursky’s interpretation, a hexis of a thing is composed of many pneumata, one for each property. What differentiates the pneumata is that they have their own compositional ratio of mixture of Air and Fire. All such pneumata are combined in the thing but without each losing its own identity, hence their particular properties are expressed. But these pneumata are connected as well such that a change in one creates a change in the others.

1.3 The Problem of Mixture [selective summary]

Pneumata combine with substantial parts, binding them together, to compose whole things and to provide them with their qualities. We can characterize the sort of mixture pneumata make with physical parts as being a special kind of mixture that the Stoics invented. Note first that they regarded there being three types of mixtures. {1} Mingling or mechanical mixtures, which are granular in that its smallest parts sit side by side like a mosaic. {2} Fusions, which are like chemical compounds where the properties of the component parts are lost, while the new whole compound gains its own unique properties. {3} Mixtures proper (krasis for liquids and mixis for non-liquids), where the components interpenetrate entirely and thoroughly such that there is no mosaic-like distribution on the smallest scale. Yet somehow despite this constituent homogeneity, each part retains its own properties and can be separated out again. This is the sort of mixture pneumata make with the other physical parts of a thing so to form its hexis.

1.4 The Four Categories [selective summary]

The Stoics held that all things can be metaphysically classified under four hierarchical categories, all of which fall under the concept of the something. In their consecutive order they are: substratum, quality, state, and relative state. The fourth category is divided into two subcategories. {a} A relative state, which is defined by something outside it, like the father-son relation. And {b} a relative, which is something capable of undergoing change between states by matters of degree that are measure by comparing the two states, like being at a level of two degrees of sweetness compared to bitterness. A hexis is an example of a relative, because it is comprised of pneumata that each express a quality such that a continuous variation in the composition of the pneuma will result in a continuous variation in its quality. The four categories fit within the Stoic theory of the dynamic continuum. The substratum is the pneuma which binds the parts of all things and which endows them with qualities. The qualities are determined by the [physical] states of the pneuma, which are always in relative states, given that they are constantly under variation as their pneumata alter their compositions.

Sambursky, Samuel. 1973. Physics of the Stoics. Westport, Connecticut: Greenwood. [First published 1959, London: Routledge and Kagen Paul.]

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### Priest (2.3) An Introduction to Non-Classical Logic, ‘Modal Semantics’, summary

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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other distracting mistakes, because I have not finished proofreading.]

Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

2. Basic Modal Logic

2.3. Modal Semantics

Brief summary:

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.

νwA) = 1 if νw(A) = 0, and 0 otherwise.

νw(AB) = 1 if νw(A) = νw (B) = 1, and 0 otherwise.

νw(AB) = 1 if νw(A) = 1 or νw (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 wW:

νw(◊A) = 1 if, for some w′W such that wRw′, νw′(A) = 1; and 0 otherwise.

νw(□A) = 1 if, for all w′ ∈ W such that wRw′, νw′(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 □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 = {w1, w2, w3}

w1Rw2, w1Rw3, w3Rw3

vw1 (p) = 0, vw1 (q) = 0;

vw2 (p) = 1, vw2 (q) = 1,

vw3 (p) = 1, vw3 (q) = o,

This is depicted as:

xxxxxxxxxxxxw2xxpxxq

xxxxxxxxxx

¬px¬qxxw1

xxxxxxxxxxx

xxxxxxxxxxxxw3xxpxx¬q

Each world (w1, w2, w3) 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 w3) 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:

xxxxxxxxxxxxxxxxxxxw2xxxxxpxxxxxq

xxxxxxxxxxxxxxxxxxxxxxxxpqxxx¬q

¬pxxxxx¬qxxxxxw1

pq xxxpxxxxxx

xxxxxxxxxxxxxxxxxxw3xxxxxpxxxxx¬q

xxxxxxxxxxxxxxxxxxxxxxxxpxxxxxxxxxx

¬◊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 truth-preserving 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 νw(B) = 1 for all B ∈ Σ, then νw(A) = 1.

A iff φA, i.e., for all interpretations ⟨W, R, v⟩ and all w W, νw(A) = 1.

(23)

Summary

2.3.1

[We will add to our propositional language the modal operators □ for ‘necessarily the case that’ and ◊ for ‘possibly the case that’.]

We will add two operators to our propositional language: □ and ◊. [See section 1.2 for a description of this language.] Priest writes:

Intuitively, □A is read as ‘It is necessarily the case that A’; ◊A as ‘It is possibly the case that A’.

(21)

2.3.2

[In our grammar, any wff with a modal operator added in front is also a wff.]

[Recall the  grammar in section 1.2.2:

The (well-formed) formulas of the language comprise all, and only, the strings of symbols that can be generated recursively from the propositional parameters by the following rule:

If A and B are formulas, so are ¬A, (A ∨ B), (A ∧ B), (A ⊃ B), (A ≡ B).

(Priest 4)

] Priest says we will augment the grammar of section 1.2.2 with this new rule:

If A  is a formula, so are □A and ◊A.

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2.3.3

[The interpretation in our modal semantics will involve the truth-valuation function v, but it also requires that we specify the world  in question (from the set of worlds, W); and for the modal operators, we specify the accessibility relation (R) of the worlds in question. ‘uRv’ means either, “world v is accessible from u” or “in relation to u, situation v is possible” (or “world u accesses world v.”) The interpretation then takes the form: ⟨W, R, v⟩. ]

[Recall from section 1.1.5 that an interpretation can be understood “crudely” as “a way of assigning truth values. And recall from section 1.3.1 that: “An interpretation of the language is a function, ν, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such as v(p) = 1 and v(q) = 0” (5). Priest will define the interpretation for our modal logic. It will have the valuation function. But now we need to specify two additional things. The first is the world we are referring to. The formula true or false in which possible world? The other is world relativity or access. I am not certain yet, but this might only be an issue for formulations taking the modal operators. The idea is that one world is accessible from another, meaning that it is possible in relation to it. Nolt in section 12.1 of his Logics discusses this relation. We write: uRv, and we say either, “world v is accessible from u” or “in relation to u, situation v is possible”. (Here it seems that ‘situation’ means ‘possible world’, but I am not sure.) (I have the impression from parts below that uRv can also be read as “world u accesses world v.”]

An interpretation for this language is a triple ⟨W, R, v⟩. W is a non-empty set. Formally, W is an arbitrary set of objects. Intuitively, its members are possible worlds. R is a binary relation on W (so that, technically, R W×W). Thus, if u and v are in W, R may or may not relate them to each other. If it does, we will write uRv, and say that v is accessible from u. Intuitively, R is a relation of relative possibility, so that uRv means that, relative to u, situation v is possible. υ is a function that assigns a truth value (1 or 0) to each pair comprising a world, w, and a propositional parameter, p. We write this as νw(p) = 1 (or νw(p) = 0). Intuitively, this is read as ‘at world w, p is true (or false)’.

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[Let us try to understand part reading: “R is a binary relation on W (so that, technically, R W×W).”  The subset relation was defined in section 0.1.6 in the following way: “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 XY” (xxviii). Next recall from section 01.1.10 the idea of Cartesian Product: “Given n sets X1, . . . , Xn, their cartesian product, X1×· · ·×Xn, is the set of all n-tuples, the first member of which is in X1, the second of which is in X2, etc.” (xvii). Later in section 01.1.11, an example is given, and Priest writes: “Let N be the set of numbers. Then N × N is the set of all pairs of the form ⟨n,m⟩, where n and m are in N. If R = {⟨2, 3⟩, ⟨3, 2⟩} then R N × N and is a binary relation between N and itself.” So the idea with the Cartesian product seems to be that it would list every possible combination from the one set to the other, but I am not sure. But supposing that to be the case, then R, which would be a set of couples relating one world to another, would then be in the Cartesian product, since the Cartesian product contains all possible combinations of worlds, including each world with itself.]

2.3.4

[The valuation function for negation, conjunction, and disjunction works the same as in classical propositional logic, except now we must specify in which world the valuation holds.]

The valuation function for negation, conjunction, and disjunction operates like we saw for classical propositional logic (section 1.3.2). But now we need to specify in which world the valuation holds.

νwA) = 1 if νw(A) = 0, and 0 otherwise.

νw(AB) = 1 if νw(A) = νw (B) = 1, and 0 otherwise.

νw(AB) = 1 if νw(A) = 1 or νw (B) = 1, and 0 otherwise.

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2.3.5

[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.]

The valuation function for the modal operators, however, will make use of worlds and their relativities.

For any world wW:

νw(◊A) = 1 if, for some w′W such that wRw′, νw′(A) = 1; and 0 otherwise.

νw(□A) = 1 if, for all w′ ∈ W such that wRw′, νw′(A) = 1; and 0 otherwise.

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[Recall ‘uRv’ means either, “world v is accessible from u” or “in relation to u, situation v is possible”. So let us look at the first formulation. “νw(◊A) = 1 if, for some w′W such that wRw′, νw′(A) = 1; and 0 otherwise.” So a formula is possibly true in one world if it is true in another world that is possible in relation to the first. Now: “νw(□A) = 1 if, for all w′ ∈ W such that wRw′, νw′(A) = 1; and 0 otherwise.” A formula is necessarily true if it is true in all worlds that are possible in relation to it.]

In other words, ‘It is possibly the case that A’ is true at a world, w, if A is true at some world, possible relative to w. And ‘It is necessarily the case that A’ is true at a world, w, if A is true at every world, possible relative to w.

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2.3.6

[If a world has no other related worlds, then any ◊A formulation will be false in that world, and any □formulation will be true.]

Priest then makes the following note. [Something is not possibly true in a world if there are no other worlds possible in relation to it where it is true. This means if there are no other worlds that are possible in relation to the first, then no statement with a possibility operator can be true. The next point is a bit tricky. The informal wording for the necessity operator was: “‘It is necessarily the case that A’ is true at a world, w, if A is true at every world, possible relative to w.” Now suppose there are no worlds relative to w, and with regard to w we have □A. We ask, is A true in all worlds relative to A? The answer here is, yes, because there are no worlds relative to A. If we stick literally to the valuation’s formulation, then all formulas of the form □A are true in w when it accesses no other world.]

Note that if w accesses no worlds, everything of the form ◊A is false at w – if w accesses no worlds, it accesses no worlds at which A is true. And if w accesses no worlds, everything of the form □A is true at w – if w accesses no worlds, then (vacuously) at all worlds that w accesses A is true.

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2.3.7

[We can diagram the interpretation by giving each world it place in the diagram, where the formulations that are true in it are listed, and by using arrows to indicate world accessibility.]

Priest will now show us how we can make a diagram to represent an interpretation where there is a finite number of possible worlds involved. The diagram will show the accessibility relations, and it will state the formulas that are true. [So if the valuation function says a certain formula is false, then we write its negation in the diagram. It seems each world gets its own line or at least a spatial place, and we use arrows for the accessibility relation. Suppose you have an arrow going from world 1 to world 2. This means that world 2 is possible in relation to world 1, or that world 2 is accessible from world 1.] Here is our interpretation:

W = {w1, w2, w3}

w1Rw2, w1Rw3, w3Rw3

vw1 (p) = 0, vw1 (q) = 0;

vw2 (p) = 1, vw2 (q) = 1,

vw3 (p) = 1, vw3 (q) = o,

[Note, in the last three lines, where it is formulated for example like vw1 , the numeral is given as an additional subscript, which I cannot reproduce here. It looks like: .] Here is how we would diagram it:

xxxxxxxxxxxxw2xxpxxq

xxxxxxxxxx

¬px¬qxxw1

xxxxxxxxxxx

xxxxxxxxxxxxw3xxpxx¬q

[Note the rounded arrow above world 3. It is set a bit high, but it means that world 3 has access to itself.]

2.3.8

[Using the valuation rules, we can write other formulas that are true for the depicted worlds in the diagram.]

The diagram can show truth values for complex formulas, notably ones with modal operators. Let us keep our model from above.

xxxxxxxxxxxxxxxxxxxw2xxxxxpxxxxxq

xxxxxxxxxxxxxxxx

¬pxxxxx¬qxxxxxw1

xxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxw3xxxxxpxxxxx¬q

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

We notice that p and q are true in world 2. That means pq is true in world 2 as well. So we write that formula in the area for world 2.

xxxxxxxxxxxxxxxxxxxw2xxxxxpxxxxxq

xxxxxxxxxxxxxxxxxxxxxxxxpqxxxxxxx

¬pxxxxx¬qxxxxxw1

xxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxw3xxxxxpxxxxx¬q

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

But, world 2 is possible in relation to world 1: w1Rw2. That means ◊pq holds in world 1. So let us write that formula in the area for world 1.

xxxxxxxxxxxxxxxxxxxw2xxxxxpxxxxxq

xxxxxxxxxxxxxxxxxxxxxxxxpqxxxxxxx

¬pxxxxx¬qxxxxxw1

pq xxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxw3xxxxxpxxxxx¬q

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

[One thing we should note is that while maybe in most cases a world should be able to access itself,] the only world that is possible in relation to itself here is world 3. [Recall the rule for necessity:

νw(□A) = 1 if, for all w′ ∈ W such that wRw′, νw′(A) = 1; and 0 otherwise.

] Since p  is true in all worlds possible in relation to world 3, that means it is necessary in world 3. So let us note that.

xxxxxxxxxxxxxxxxxxxw2xxxxxpxxxxxq

xxxxxxxxxxxxxxxxxxxxxxxxpqxxxxxxx

¬pxxxxx¬qxxxxxw1

pq xxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxw3xxxxxpxxxxx¬q

xxxxxxxxxxxxxxxxxxxxxxxxpxxxxxxxxxx

Now, although world 1 is not possible in relation to world 3 (that is, world 3 does not access world 1), world 3 is nonetheless possible in relation to world 1 (w1Rw3, that is, world 1 accesses world 3). So since it is necessary that p in world 3, that means p is possible that it is necessarily true in world 1. [I am not sure what that would mean for something to be possible that it is necessary. I guess the idea is that in a world that is possible in relation to our own, it is necessary, so it could be necessary in our own too, even though it is not.] We will write that in for world 1, then:

xxxxxxxxxxxxxxxxxxxw2xxxxxpxxxxxq

xxxxxxxxxxxxxxxxxxxxxxxxpqxxxxxxx

¬pxxxxx¬qxxxxxw1

pq xxx◊□pxxxxxx

xxxxxxxxxxxxxxxxxxw3xxxxxpxxxxx¬q

xxxxxxxxxxxxxxxxxxxxxxxxpxxxxxxxxxx

Note that world 2 accesses no other world and not even itself. [Again recall how we evaluate possibility:

νw(◊A) = 1 if, for some w′W such that wRw′, νw′(A) = 1; and 0 otherwise.

Since there are no other worlds such that world 2 has access to them, that means there are no formulas that are true in accessible worlds. That furthermore means that formula with a possibility operator will be false.] This means that in world 2, q is false. [But we only write true formulas, so] that means we write its negation in the area for world 2.

xxxxxxxxxxxxxxxxxxxw2xxxxxpxxxxxq

xxxxxxxxxxxxxxxxxxxxxxxxpqxxx¬q

¬pxxxxx¬qxxxxxw1

pq xxxpxxxxxx

xxxxxxxxxxxxxxxxxxw3xxxxxpxxxxx¬q

xxxxxxxxxxxxxxxxxxxxxxxxpxxxxxxxxxx

2.3.9

[¬◊A at any world is equivalent to □¬A.]

The next point is that ¬◊A in some world is equivalent □¬A. [I will quote the reasoning below, as I will probably missummarize it. Intuitively, the idea might be that if some formula is not possible in some world, that means it is not true in any other accessible world. That would seem to mean that its negation is true in all these other worlds. That then means that in all accessible worlds, the formula’s negation is true. That therefore means that the negation of the formula is necessarily true in the first world, since it is so in all accessible worlds. But let us look at the formulation, and I will try to work through it below.]

Observe that the truth value of ¬◊A at any world, w, is the same as that of □¬A. For:

vw(¬◊A) = 1 iff vw(◊A) = 0

iff for all w′ such that wRw′, vw′(A) = 0

iff for all w′ such that wRw′, vw′A) = 1

iff vw(□¬A) = 1

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[We first have:

vw(¬◊A) = 1 iff vw(◊A) = 0

I am not sure, but it might be derived from this rule:

νwA) = 1 if νw(A) = 0, and 0 otherwise.

Next is:

iff for all w′ such that wRw′, vw′(A) = 0

So maybe the idea here is that if vw(◊A) = 0, that means there is no other related world where A is true, thus it is false in all other related worlds.

iff for all w′ such that wRw′, vw′A) = 1

If A is false in other related worlds, then perhaps by the same negation rule we can say that in these other worlds, ¬A is true.

iff vw(□¬A) = 1

And if ¬A is true in all these other worlds, then it is necessarily true in the first world.]

2.3.10

[¬□A at any world is equivalent to ◊¬A.]

“Similarly, the true value of ¬□A at a world is the same as that of ◊¬A” (23).

2.3.11

[An inference is valid (as a semantic consequence) if it is truth-preserving 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.]

Priest will now define validity, semantic consequence, and logical truth. [I may not summarize this correctly, so see the quotation to follow. The first notion is valid inference. He says that an inference is valid if it preserves truth in all worlds. I would assume that this is a matter then of semantic validity (see section 1.1.5), and I am not entirely sure if this notion of valid inference is in any way different from the notion of semantic consequence formulated below. What is interesting to note is that an inference is valid only if it (semantically) preserves truth in all worlds of all interpretations and not for example just in worlds that are possible in relation to one another. The way semantic consequence is defined formally is something like the following. We have a premise or premises, and a conclusion. If in all worlds of all interpretations, whenever the premises are true the conclusion is true, then the conclusion is a semantic consequence of the premises. And a formula is logically true (tautological, see section 1.3.4) if it is true in all worlds of all interpretations. (I do not understand the part reading, “⊨ A iff φ ⊨ A). In section 0.1.4, φ was defined as the empty set. I discuss this issue in section 1.3.4. There I wondered if the idea was the following. If no valuation can make a formula false, then it can never be that any premises can be true while that formula, understood as a conclusion, is false. So in other words, on the basis of no premises at all (the empty set), we could derive the formula.)]

An inference is valid if it is truth-preserving at all worlds of all interpretations. Thus, if Σ is a set of formulas and A is a formula, then semantic consequence and logical truth are defined as follows:

Σ ⊨ A iff for all interpretations ⟨W, R, v⟩ and all w W: if νw(B) = 1 for all B ∈ Σ, then νw(A) = 1.

A iff φA, i.e., for all interpretations ⟨W, R, v⟩ and all w W, νw(A) = 1.

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Priest, Graham. 2008 . An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

Also cited:

Nolt, John. Logics. 1997. Belmont, CA: Wadsworth.

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