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by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]
[Logic & Semantics, Entry Directory]
[The following collects the brief summaries for sections we examined in Nolt’s Logics. Note, starting with section 16.2, the paragraphs are enumerated, and the larger sections are divided by the numbered paragraphs that are summarized in that post.]
[An entry directory for this text without the brief summaries can be found at:
http://piratesandrevolutionaries.blogspot.com/2016/06/nolt-logics-entry-directory.html]
Collected Brief Summaries for
John Nolt
Logics
Part 3: Classical Predicate Logic
Chapter 6: Classical Predicate Logic: Syntax
Identity
We can use the ‘=’ operator – meaning “is identical to” or “is the same thing as” – to articulate a number of things normally inexpressible in the language of predicate logic. It is an operator, because its meaning remains constant in all cases. But it dually is a two-place predicate, relation, or function, because it relates one term to another in a semantic manner where the extension of one term is said to be no different than the extension of the other. The non-identity of terms is written normally as ‘~a=b’, and so this does not mean ‘not-a is identical to b’ but rather that ‘a is not identical to b’. This is because for atomic formulas, when they take a negation operator, its scope extends over the entire formula. We also can write the non-identity of terms as ‘a≠b’ (but ‘~a=b’ is less problematic and thus preferred). Using the equality operator, we can render the following meanings.
Else
‘God is more perfect than anything else’
∀x(~x = g → Pgx)
Other than
‘Al will go with anyone other than Beth’
∀x(~x = b → Gax)
Except
(for one term)
‘Al will go with anyone except Beth’ [...]
∀x(~x = b → Gax) &~Gab
or
∀x(~x = b ↔ Gax)
Except
(for two terms, where the predication of the terms is not determined)
‘Everyone except Al and Beth is happy’
∀x((~x = a & ~x = b) → Hx)
Except
(for two terms, where the predication is denied to the exceptional terms)
∀x((~x = a & ~x = b) → Hx) & (~Ha & ~Hb)
or
∀x((~x = a & ~x = b) ↔ Hx)
Superlatives
‘Al is faster than all other runners’
Ra & ∀x((Rx & ~x = a) → Fax)
‘There is no largest number.’
~∃x(Nx & ∀y((Ny & ~y = x) → Lxy))
Only
(for one term)
‘Only Al is happy’
Ha & ∀x(Hx → x = a)
Only
(for two terms)
‘Only Al and Beth are happy’
∀x(Hx ↔ (x = a ∨ x = b))
At least
(for one term)
‘There is at least one mind.’
∃xMx
At least
(for two terms)
‘There are at least two minds.’
∃x∃y((Mx&My) & ~x = y)
At least
(for three terms)
‘There are at least three minds.’
∃x∃y∃z(((Mx&My) & Mz) & (~x = y & ~y = z) & ~x = z)
At most
(for one term)
‘ There is at most one mind.’
∀x∀y((Mx & My) → x = y)
or
~∃x∃y((Mx&My) & ~x = y)
or
∃x∀y(My → y = x)
At most
(for two terms)
‘There are at most two minds.’
∃x∃y∀z(Mz → (z = x ∨ z = y)
At most
(for three terms)
‘There are at most three minds.’
∃x∃y∃z∀w(Mw → ((w = x ∨ w = y) ∨ w = z))
At most
(for all things)
‘There is at most just one thing.’
∃x∀y y = x
Numerical Quantifiers
(for one term)
‘There is exactly one mind.’
(∃xMx & ∃x∀y(My → y = x)
or
∃x∀y(My ↔ y = x)
Numerical Quantifiers
(for two terms)
‘There are exactly two minds.’
∃x∃y(~x = y & ∀z(Mz ↔ (z = x ∨ z = y)))
Definite Description
(affirmative)
‘The present king of France is bald’
or
‘There is at least one thing which is presently king of France, it alone is presently king of France, and it is bald.’
∃x((Kx & ∀y(Ky → y = x)) & Bx)
Definite Description
(negating existence of)
‘It is not the case that there is at least one present king of France, who alone is presently king of France, and who is bald.’
~∃x((Kx & ∀y(Ky → y = x)) & Bx)
Definite Description
(negating the predication of)
‘There is at least one present king of France, who alone is presently king of France, and who is not bald.’
∃x((Kx & ∀y(Ky → y = x)) & ~Bx)
A definite description, like “the present king of France,” seems like a name, but in fact it has the more complex logical structure of “the F is G”, or
∃x((Fx & ∀y(Fy → y = x)) & Gx)
When we formulate propositions using definite descriptions for non-existing entities, we can fall into error by not translating the ambiguous English formulations into the unambiguous formalizations. This allows us to avoid paradoxes that really are not inherent to such instances. For example, we have “The present king of France is bald.” We have the first intuition that it is false, because there is no such thing which takes that predicate, so the predication fails. We then have the second intuition that since it is false, its negation must be true. And so we think, “The present king of France is not bald” is true. But we as well have the third intuition, which is basically the same as the first, that the sentence “The present king of France is not bald” is false, because it refers to a non-existing entity and the predication therefore fails. So the same sentence, “The present king of France is not bald” is both true and false, under two equally valid intuitions. But Bertrand Russell’s analysis of definite descriptions shows that in fact we are not affirming and denying the same sentence. Rather we are saying that “There presently is no bald king of France” is true while “There is presently a bald king of France” is false. [See the last two formalization in the above listing.]
Chapter 8:
Classical Predicate Logic: Inference
Identity
There are two identity inference rules for making proofs in predicate logic.
Identity Introduction (=I) Where α is any name, assert α = α.
Identity Elimination (= E) From a premise of the form α = β and a formula Φ containing either α or β, infer any formula which results from replacing one or more occurrences of either of these names by the other in Φ.
(241-242)
Part 4:
Extensions of Classical Logic
Chapter 11:
Leibnizian Modal Logic
Modal Operators
There are certain operators called alethic modifiers. They include deontic (or ethical) modalities (e.g. ‘it ought to be the case that’, ‘it is forbidden that’); propositional attitudes (‘believes that’, ‘knows that’, ‘hopes that’, ‘wonders whether’); and tenses (‘was’, ‘is’ , and ‘will be’). Propositional attitude modifiers are binary, but all the rest are monadic. ‘It is necessary that Φ’ is written □Φ and ‘it is possible that Φ’ is written ◊Φ. Alethic modifiers are often duals meaning that one can be converted into another by adding negations around the symbols, as for example:
□Φ↔~◊~Φ
◊Φ↔~□~Φ
[basic set-up and evaluation of Leibnizian possible worlds]
Evaluating formulas in modal logic involves possible world modeling. A possible world is, intuitively speaking, an alternate world-situation where things are slightly or drastically different from our own. Formally, a possible world is a combination of a domain of objects along with valuative functions that assign names to these objects, assign ordered n-tuples to n-place predicates, and assign truth values to well-formed formulas. When evaluating modal operators, we consider a set of worlds, each with its own valuation schemes and domains. When a formula is true in all such worlds under consideration, it is necessary. And when it is true in some worlds, it is possible. The following defines possible worlds and their valuations.
DEFINITION A Leibnizian valuation or Leibnizian model v for a formula or set of formulas of modal predicate logic consists of the following:
1. A nonempty set Wv of objects, called the worlds of v.
2. For each world w in Wv a nonempty set Dw of objects, called the domain of w.
3. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:
i. If σ is a name, then v(σ) is a member of the domain of at least one world.
ii. If σ is a zero-place predicate (sentence letter), v(σ, w) is one (but not both) of the values T or F. |
iii. If σ is a one-place predicate, v(σ, w) is a set of members of Dw .
iv. If σ is an n-place predicate (n>1), v(σ, w) is a set of ordered n-tuples of members of Dw.
(Nolt 314-315)
Valuation Rules for Leibnizian Modal Predicate Logic
Given any Leibnizian valuation v, for any world w in Wv:
1. If Φ is a one-place predicate and α is a name whose extension v(α) is in Dw, then
v(Φα, w) = T iff v(α) ∈ v(Φ, w);
v(Φα, w) = F iff v(α) ∉ v(Φ, w).
2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in Dw, then
v(Φα1, ... , αn, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = F iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
3. If α and β are names, then
v(α = β, w) = T iff v(α) = v (β);
v(α = β, w) = F iff v(α) ≠ v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = T iff v(Φ, w) ≠ T;
v(~Φ, w) = F iff v(Φ, w) = T.
5 .
v(Φ & Ψ, w) = T iff both v(Φ, w) = T and v(Ψ, w) = T;
v(Φ & Ψ, w) = F iff either v(Φ, w) ≠ T or v(Ψ, w) ≠ T, or both.
6 .
v(Φ ∨ Ψ, w) = T iff either v(Φ, w) = T or v(Ψ, w) = T, or both;v(Φ ∨ Ψ, w) = F iff both v(Φ, w) ≠ T and v(Ψ, w) ≠ T.
7.
v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;
v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.
8 .
v(Φ ↔ Ψ, w) = T iff either v(Φ, w) = T and v(Ψ, w) = T, or v(Φ, w) ≠ T and v(Ψ, w) ≠ T;
v(Φ ↔ Ψ, w) = F iff either v(Φ, w) = T and v(Ψ, w) ≠ T, or v(Φ, w) ≠ T and v(Ψ, w) = T.
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
9 .
v(∀βΦ, w) = T iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T;
v(∀βΦ, w) = F iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) ≠ T;
10 .
v(∃βΦ, w) = T iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) = T;
v(∃βΦ, w) = F iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ T; |
11 .
v(□Φ, w) = T iff for all worlds u in Wv, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u in Wv, v(Φ, u) ≠ T;
12 .
v(◊Φ, w) = T iff for some world u in Wv, v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u in Wv, v(Φ, u) ≠ T.
(Nolt 315-316)
[Possible worlds, actual worlds, redefining logical concepts for possible worlds, and some important theorems in possible world semantics]
In our possible world semantics, no one possible world is uniquely actual. Rather, from the perspective of a world, it is itself actual and the others possible. However, likewise, from the perspective of a second possible world, the first along with the others are possible, and second is itself actual. This is called the indexicality of actuality. For our possible world semantics, we need to revise our definitions for many concepts to incorporate the element of possible worlds.
DEFINITION A formula is valid iff it is true in all worlds on all of its valuations.
DEFINITION A formula is consistent iff it is true in at least one world on at least one valuation.
DEFINITION A formula is inconsistent iff it is not true in any world on any of its valuations.
DEFINITION A formula is contingent iff there is a valuation on which it is true in some world and a valuation on which it is not true in some world.
DEFINITION A set of formulas is consistent iff there is at least one valuation containing a world in which all the formulas in the set are true.
DEFINITION A set of formulas is inconsistent iff there is no valuation containing a world in which all the formulas in the set are true.
DEFINITION Two formulas are equivalent iff they have the same truth value at every world on every valuation of both.
DEFINITION A counterexample to a sequent is a valuation containing a world at which its premises are true and its conclusion is false. |
DEFINITION A sequent is valid iff there is no world in any valuation on which its premises are true and its conclusion is not true.
DEFINITION A sequent is invalid iff there is at least one valuation containing a world at which its premises are true and its conclusion is not true.
(Nolt 319-320)
Some important metatheorems for possible world semantics are:
METATHEOREM: Any sequent of the form □Φ⊦Φ is valid.
(Nolt 320)
[From something being necessarily true we can conclude that it is true in the actual world.]
METATHEOREM: Any formula of the form ◊Φ↔~□~Φ is valid.
(Nolt 321)
[If something is possible then it is not the case that it is necessarily not so. This structure works for necessity to: □Φ↔~◊~Φ.]
METATHEOREM: Every sequent of the form α = β ⊦ □α = β
(Nolt 323)
[Because of rigid designation and transworld identity, if two names designate the same object in the actual world, then they designate the same object in every possible world.]
METATHEOREM: The sequent ‘∃xFx ⊦ ∃x◊Fx’ is valid.
(Nolt 323)
[If there is something with a certain property, then we can conclude that there is something for which it possibly has that property. In other words, whatever actually has a property also possibly has it.]
METATHEOREM: The sequent ‘◊∃xFx ⊦ ∃x◊Fx’ is invalid.
(Nolt 324)
[Suppose it is possible that there is something that takes some predicate. That means in some world, not necessarily our actual one, something does take that predicate. From this we cannot conclude that in our actual world there is something that possibly takes that predicate. For, were it for example the case that nothing in our actual world in fact takes that predicate, it is not possible that something can take it.]
A Natural Model?
A natural model is one that is sufficient to what it is modeling, meaning that its “domain consists of the very objects we mean to talk about,” and its “predicates and names denote exactly the objects of which they are true on their intended meanings” (Nolt 325). In modal logic, a natural model will have an actual world where the domain of objects are actual objects and the predicates are assigned to objects that actually have those predicates. The other possible worlds are variations on the actual world. But name designations remain the same in all worlds in the model. Possibility in possible world modeling can be understood as informal possibility, meaning that neither syntactic nor semantic contradictions are allowed, or formal possibility, where only syntactic contradictions are prohibited. This means that the extensions of predicates can be inconsistent with one another, and for now we use this formal notion of possibility in our possible world modeling. Related to this distinction is the one between the realist and the nominalist view of essences. Realists believe that essences are real things in the world. So in terms of possible worlds, something’s real essence in our actual world will determine what sorts of variations it can undergo while still being identical to itself in other worlds, at least insofar as it is essentially the same even if certain properties vary. The nominalist view thinks that essences are artificially created through linguistic practices, so from this perspective there is no certain way to know what sorts of variations something can undergo while still being essentially the same. To avoid this issue, we do not deal with metaphysical possibility, which would be concerned with the ways and extents something’s essence allows it to vary, but rather we are concerned with logical possibility, which is concerned just with logical contradictions that might hold in a world. And as we noted, we are concerned with formal possibility rather than with informal possibility. That is to say, we only care about syntactical contradictions of the form A and not-A. And we are not concerned with semantic contradictions of the form, for example, of Ra & Ca, where R means ‘is red’ and C means ‘is colorless.’ In other worlds, in some world, ‘a’ can be in both the extensions of R and of C, even though their semantic meanings are incompatible.
Inference in Leibnizian Logic
| DUAL | Duality | From either ◊Φ and ~□~Φ, infer the other; from either □Φ and ~◊~Φ, infer the other. |
| K | K rule | From □(Φ → Ψ), infer (□Φ → □Ψ). |
| T | T rule | From □Φ, infer Φ. |
| S4 | S4 rule | From □Φ, infer □□Φ |
| B | Brouwer rule | From Φ, infer □◊Φ. |
| N | Necessitation | If Φ has previously been proven as a theorem, then any formula of the form □Φ may be introduced at any line of a proof. |
| □= | Necessity of identity | From α = β, infer □α = β. |
Chapter 12:
Kripkean Modal Logic
Kripkean Semantics
Kripkean semantics allows us to model certain logical ideas and principles in modal logic that we are unable to model using Leibnizian semantics. The main problem is that Leibnizian semantics will make certain arguments valid (or invalid) when they should not be for a certain type of modality. For example, physical possibility does not behave the same way as logical possibility. Take for instance the fact that accelerating an object faster than the speed of light is logically possible but not physically possible. So we need to change the way we make models for physical possibility. One way of thinking about this is by comparing what is physically possible in each world. In our world (world 1), objects in space can have either circular or elliptical orbits. Now suppose that in world 2, they only have circular orbits. So it is physically impossible according to the laws of physics in world 2 for objects to have elliptical orbits. Now suppose further we take the perspective of world 2, where elliptical orbits are physically impossible. Were we to consider world 1 from world 2’s perspective, we would say that world 1 is a physically impossible world, because its laws of physics do not obey our own. However, were we to look at world 2 from our perspective, we would say that it is a physically possible world, because it does not break any of our physical laws (it just is more physically restricted than ours). So in order to model physical possibility, we can specify this relation of world relativity. It is called relative possibility, accessibility, or alternativeness. In our example, we would say that world 2 is possible relative to world 1, or that world 2 is an alternative to world 1, or that world 2 is accessible to world 1. However, we cannot invert these formulations. We write world y is accessible to world x as xℛy. We can diagram this with circles and arrows, with an arrow going from a first circle to a second meaning that the second is accessible from the first. Our orbit worlds would be diagramed as:
(Nolt 337)
As we can see, each world is accessible to itself, because each world follows its own physical laws. However, other sorts of modality, like deontic modality, do not guarantee this reflexive self accessibility. So when we use Kripkean semantics, we must stipulate the world relativities by making a set that lists ordered couples of the form <x, y> where y is accessible to (possible relative to) x. So for our example above:
ℛ = {<1, 2>, <1, 1>, <2, 2>}
A Kripkean model is defined in the following way:
DEFINITION A Kripkean valuation or Kripkean model v for a formula or set of formulas of modal predicate logic consists of the following:
1. A nonempty set Wv of objects, called the worlds of v.
2. A relation ℛ, consisting of a set of pairs of worlds from Wv.
3. For each world w in Wv a nonempty set Dw of objects, called the domain of w.
4. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:
i. If σ is a name, then v(σ) is a member of the domain of at least one world.
ii. If σ is a zero-place predicate (sentence letter), v(σ, w) is one (but not both) of the values T or F.
iii. If σ is a one-place predicate, v(σ, w) is a set of members of Dw .
iv. If σ is an n-place predicate (n>1), v(σ, w) is a set of ordered n-tuples of members of Dw.
(337)
And these are the valuation rules for Kripkean semantics:
Valuation Rules for Kripkean Modal Predicate Logic
Given any Leibnizian [or Kripkean?] valuation v, for any world w in Wv:
1. If Φ is a one-place predicate and α is a name whose extension v(α) is in Dw, then
v(Φα, w) = T iff v(α) ∈ v(Φ, w);
v(Φα, w) = F iff v(α) ∉ v(Φ, w).
2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in Dw, then
v(Φα1, ... , αn, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = F iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
3. If α and β are names, then
v(α = β, w) = T iff v(α) = v (β);
v(α = β, w) = F iff v(α) ≠ v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = T iff v(Φ, w) ≠ T;
v(~Φ, w) = F iff v(Φ, w) = T.
5 .
v(Φ & Ψ, w) = T iff both v(Φ, w) = T and v(Ψ, w) = T;
v(Φ & Ψ, w) = F iff either v(Φ, w) ≠ T or v(Ψ, w) ≠ T, or both.
6 .
v(Φ ∨ Ψ, w) = T iff either v(Φ, w) = T or v(Ψ, w) = T, or both;v(Φ ∨ Ψ, w) = F iff both v(Φ, w) ≠ T and v(Ψ, w) ≠ T.
7.
v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;
v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.
8 .
v(Φ ↔ Ψ, w) = T iff either v(Φ, w) = T and v(Ψ, w) = T, or v(Φ, w) ≠ T and v(Ψ, w) ≠ T;
v(Φ ↔ Ψ, w) = F iff either v(Φ, w) = T and v(Ψ, w) ≠ T, or v(Φ, w) ≠ T and v(Ψ, w) = T.
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
9 .
v(∀βΦ, w) = T iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T;
v(∀βΦ, w) = F iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) ≠ T;
10 .
v(∃βΦ, w) = T iff for some potential name α of all objects d in Dw, v(α,d) (Φα/β , w) = T;
v(∃βΦ, w) = F iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ T;
11′ .
v(□Φ, w) = T iff for all worlds u such that wℛu, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u, wℛu and v(Φ, u) ≠ T;
12′ .
v(◊Φ, w) = T iff for some world u, wℛu and v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u such that wℛu, v(Φ, u) ≠ T.
We define validity in Kripkean semantics in the following way:
A sequent is valid relative to a given set of models (valuations) iff there is no model in that set containing a world in which the sequent's premises are true and its conclusion is not true. To say that a sequent is valid relative to Kripkean semantics in general is to say that it has no counterexample in any Kripkean model, regardless of how ℛ is structured.
(340)
[Here the models in the set are restricted in accordance with the sort of world relativity being modeled].
Chapter 13:
Deontic and Tense Logics
13.2
A Modal Tense Logic
[Basic set-up and evaluation of modal tense logic]
Intuitively, time involves a passage extending linearly from past to future. As it moves forward, it could follow one of many possible ramifying branches from the current now point. So the future, under this view, is multiple and undecided. The past, however, cannot become otherwise after it happens. Were we to draw this structural feature of time, we would have a tree-trunk line for history, as it has only one, and a series of branches for the future, expanding out from the present, which stands at the end of the singular past-trunk and before the branching futures.
(Nolt 366)
At each point in time, there are different facts and existing objects, so we can specify what holds at certain time points. But we can also see the branching futures as possible worlds in relation to our own, as these alternate future paths may or may not be realized in our own actual world. So we can speak of situations holding or not holding for certain possible worlds at certain times. Furthermore, we can use a modal tense logic to place these temporally contingent facts into temporal relations with one another. This way, we can say that some situation will hold in a future moment or in all future moments, and so on. We use these following symbols for modal tense operators:
H – it has always been the case that
P – it was (at some time) the case that
G – it will always be the case that
F – it will (at some time) be the case that
(Nolt 367)
Each of these can be combined with themselves and with the others many times over, and they can also be combined with the alethic modal operators, necessity and possibility. The definition of this modal tense logic model is the following:
DEFINITION A model or valuation v for a formula or set of formulas of modal predicate logic consists of the following:
1. A nonempty set ℑ of objects called the times of v.
2. A transitive relation ℰ, consisting of a set of pairs of times from ℑ.
3. A nonempty set Wv of objects, called the worlds of v.
4. Corresponding to each world w, a set ℑw of times called the times in w such that for any pair of times t1 and t2 in this set, either t1ℰt2 or t2ℰt1 or t1 = t2.
5. For each world w and time t in w, a nonempty set D(t,w) of objects called the domain of w at t.
6. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:
i. If σ is a name, then v(σ) is a member of D(t,w) for at least one time t and world w. |
ii. If σ is a zero-place predicate (sentence letter), and t is in w, then v(σ, t, w) is one (but not both) of the values T or F.
iii. If σ is a one-place predicate and t is in w, v(σ, t, w) is a set of members of D(t,w).
iv. If σ is an n-place predicate (n>1), and t is in w, v(σ, t, w) is a set of ordered n-tuples of members of D(t,w).
(Nolt 370-371, boldface in the original)
Note here the ‘earlier than’ relation ℰ that orders the time points in the worlds. Possible worlds that share all moments up to a particular time point are said to be accessible to each other at that time point, meaning that the future path that one takes (the facts that become true) can be the same that the other takes, all while maintaining the same past. This accessibility relation is written ℛ. (When models are not accessible, that means they have different histories, which are not alterable, and thus what is possible for one world is no longer possible for the other.)
DEFINITION Given a model v for a formula or set of formulas, then for any worlds w1 and w2 and time t of v, w1ℛw2t iff
1. t is a time in both w1 and w2, |
2. w1 and w2 contain the same times up to t; that is, for all times t′, if t′ℰt, then t′ is in w1 iff t′ is in w2, and
3. w1 and w2 have the same atomic truths at every moment up to t; that is, for all times t′ such that t′ℰt, D(t′, w1) = D(t′, w2), and for all predicates Φ, v(Φ, t′, w1) = v(Φ, t′, w2).
(Nolt 371-372)
The valuation rules for this modal tense logic are the following:
Valuation Rules for Modal Tense Logic
Given any valuation v of modal tense logic whose set of worlds is Wv, for any world w in Wv and time t in w:
1. If Φ is a one-place predicate and α is a name whose extension v(α) is in D(t,w), then v(Φα, t, w) = T iff v(α) ∈ v(Φ, t, w).
2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in D(t,w), then
v(Φα1, ... , αn, t, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, t, w)
3. If α and β are names, then v(α = β, t, w) = T iff v(α) = v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, t, w) = T iff v(Φ,t, w) ≠ T
5 .
v(Φ & Ψ, t, w) = T iff both v(Φ, t, w) = T and v(Ψ, t, w) = T
6 .
v(Φ ∨ Ψ, t, w) = T iff either v(Φ, t, w) = T or v(Ψ, t, w) = T, or both7.
v(Φ → Ψ, t, w) = T iff either v(Φ, t, w) ≠ T or v(Ψ, t, w) = T, or both
8 .
v(Φ ↔ Ψ, t, w) = T iff v(Φ, t, w) = v(Ψ, t, w)
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and D(t,w) is the domain that v assigns to world w at time t.
9 .
v(∀βΦ, t, w) = T iff for all potential names α of all objects d in D(t,w), v(α,d) (Φα/β , t, w) = T
10 .
v(∃βΦ, t, w) = T iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) = T
11 .
v(□Φ, t, w) = T iff for all worlds u such that wℛut, v(Φ, t, u) = T
12.
v(◊Φ, t, w) = T iff for some world u, wℛut and v(Φ, t, u) = T
13.
v(HΦ, t, w) = T iff for all times t′ in w such that t′ℰt and v(Φ, t′, w) = T
14.
v(PΦ, t, w) = T iff for some time t′ in w, t′ℰt and v(Φ, t′, w) = T
15.
v(GΦ, t, w) = T iff for all times t′ in w such that tℰt′ and v(Φ, t′, w) = T
16.
v(FΦ, t, w) = T iff for some time t′ in w, tℰt′ and v(Φ, t′, w) = T
(Nolt 372-373, boldface in the original)
And important logic terms are defined in the following way for this modal tense logic:
DEFINITION A formula is valid iff it is true at all times in all worlds on all of its valuations.
DEFINITION A formula is consistent iff it is true at at least one time in at least one world on at least one valuation.
DEFINITION A formula is inconsistent iff it is not true at any time in any world on any of its valuations.
DEFINITION A formula is contingent iff there is a valuation on which it is true at some time in some world and a valuation on which it is not true at some time in some world.
DEFINITION A set of formulas is consistent iff there is at least one valuation containing a world in which there is a time at which all the formulas in the set are true.
DEFINITION A set of formulas is inconsistent iff there is no valuation containing a world in which there is a time at which all the formulas in the set are true. |
DEFINITION Two formulas are equivalent iff they have the same truth value at every time in every world on every valuation of both.
DEFINITION A counterexample to a sequent is a valuation containing a world in which there is a time at which its premises are true and its conclusion is not true.
DEFINITION A sequent is valid iff there is no world in any valuation containing a time at which its premises are true and its conclusion is not true.
DEFINITION A sequent is invalid iff there is at least one valuation containing a world in which there is a time at which its premises are true and its conclusion is not true.
(Nolt 373-374, boldface in the original)
Using this model of modal tense logic, we can show that one particular argument for determinism is invalid, and thus there is a philosophical usefulness for this model. However, what is philosophically at issue cannot be settled by the model, but rather, the model can only be based on philosophical assumptions and cannot always prove or disprove them. Nonetheless, the model can at least help us clarify our philosophical conceptions about time. Also, it can establish certain views as consistent or inconsistent, and it shows the deterministic view to at least be consistent.
Chapter 14:
Higher-Order Logics
Higher-Order Logics: Syntax
In first-order logic, we can have quantifiers that quantify over variables that stand for individuals. In second-order logic, we can have quantifiers that quantify over predicates. In this way, we can express the following inference, for example: “Al is a frog. Beth is a frog. Therefore, Al and Beth have something in common.” We can write it as: ‘Fa, Fb ⊢ ∃X(Xa & Xb)’. Here we have the predicate variable ‘X’, which allows us to refer to some unspecified predicate as a variable. Third-order logic allows us to quantify over predicates for properties of individuals. Take this inference for example: “Socrates is snub-nosed. Being snub-nosed is an undesirable property. Therefore, Socrates has a property that has a property.” If we use larger type-face for the third-order predicate variable, we can write this as:
(Nolt 383)
[Here ‘s’ = Socrates, ‘N’ = ‘is snub-nosed, ‘U’ = ‘is undesirable’, ‘X’ is a predicate variable, and ‘Y’ is a predicate variable for another predicate. The conclusion says that there is some property such that Socrates has it and also that this property itself has some property.] Higher-order logics are possible as well. Variables on each order are of a different type. An infinite hierarchy of higher-order logics is called a theory of types. Higher-order logic is especially useful for using logic to formulate the properties of numbers and other concepts in arithmetic. The view that all mathematical ideas can be defined in terms of purely logical ideas, and also that all mathematical truths are logical truths, is called logicism. We can use second-order logic to express a number of important logical ideas. One of them is identity. Leibniz’s law says that objects are identical if and only if they share exactly the same properties. It is written:
Leibniz’s Law
a = b ↔ ∀X(Xa ↔ Xb)
It is analyzable into two subsidiary principles.
The Identity of Indiscernibles
∀X(Xa ↔ Xb) → a = b
This says that if two things are indiscernible, as they share exactly the same properties, then they are identical. The other is
The Indiscernibility of Identicals
a = b → ∀X(Xa ↔ Xb)
This says that if two things are identical, then they share exactly the same properties. We can express an analogy as:
∃Z(Zab & Zcd)
This says that “there is some respect in which a stands to b as c stands to d” (Nolt 386). The idea that ‘All asymmetric relations are irreflexive’ can be written as:
∀Z(∀x∀y(Zxy → ~Zyx) → ∀x~Zxx)
Part 5:
Nonclassical Logics
Chapter 15:
Mildly Nonclassical Logics
Free Logics
Leibnizian semantics for modal logic only worked when we dealt strictly with entities that can be said to exist. Were we to deal with a non-existing entity, then we have problems when using the existential quantifier, since it implies the thing’s existence. A solution for this is using a free logic. It is “free” in the sense that it is free from the restriction from only making models with existing entities. One sort of semantics for a free logic is Meinongian semantics. It has both an “inner” domain of existing entities but also another “outer” domain for all existing and as well all non-existing entities. The extensions of all names are found in the outer domain, but only names of existing things have extensions in the inner domain. And the extension of non-quantified predicates is found in the outer domain of existing and non-existing predicates. This allows us to make true statements about the properties and relations of non-existing entities. For example, we might say that it is true that a unicorn has a horn, even though none exist. For, the extension for the predicate “is horned” would be found in the outer domain where unicorns are listed. However, when we use quantifiers, the extensions of the predicates are only found in the inner domain of existing entities. This way, were we to use an existential quantifier for a non-existing entity, the formula would be false. For, the predication would not be fulfilled, as no entity would belong to its proper extension, and this is because, in our example, unicorns are not found in the inner domain of existing entities, which is where we look for quantified expressions. The definition and rules for Meinongian semantics are the following.
DEFINITION A Meinongian valuation or Meinongian model v for a formula or set of formulas of modal predicate logic consists of the following:
1. A nonempty set Wv of objects, called the worlds of v.
2. A nonempty set ℑ of objects, which is called the outer domain of v,
3. For each world w in Wv a nonempty set Dw of ℑ called the inner domain of w,
4. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:
i. If σ is a name, then v(σ) is a member of ℑ.
ii. If σ is a zero-place predicate (sentence letter), v(σ, w) is one (but not both) of the values T or F. |
iii. If σ is a one-place predicate, v(σ, w) is a set of members of ℑ.
iv. If σ is an n-place predicate (n>1), v(σ, w) is a set of ordered n-tuples of members of ℑ.
(Nolt 314-315)
Valuation Rules for Leibnizian-based Meinongian Modal Predicate Free Logic
Given any Leibnizian valuation v, for any world w in Wv:
1. If Φ is a one-place predicate and α is a name, then
v(Φα, w) = T iff v(α) ∈ v(Φ, w);
v(Φα, w) = F iff v(α) ∉ v(Φ, w).
2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names, then
v(Φα1, ... , αn, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = F iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
3. If α and β are names, then
v(α = β, w) = T iff v(α) = v (β);
v(α = β, w) = F iff v(α) ≠ v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = T iff v(Φ, w) ≠ T;
v(~Φ, w) = F iff v(Φ, w) = T.
5 .
v(Φ & Ψ, w) = T iff both v(Φ, w) = T and v(Ψ, w) = T;
v(Φ & Ψ, w) = F iff either v(Φ, w) ≠ T or v(Ψ, w) ≠ T, or both.
6 .
v(Φ ∨ Ψ, w) = T iff either v(Φ, w) = T or v(Ψ, w) = T, or both;v(Φ ∨ Ψ, w) = F iff both v(Φ, w) ≠ T and v(Ψ, w) ≠ T.
7.
v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;
v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.
8 .
v(Φ ↔ Ψ, w) = T iff either v(Φ, w) = T and v(Ψ, w) = T, or v(Φ, w) ≠ T and v(Ψ, w) ≠ T;
v(Φ ↔ Ψ, w) = F iff either v(Φ, w) = T and v(Ψ, w) ≠ T, or v(Φ, w) ≠ T and v(Ψ, w) = T.
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
9 .
v(∀βΦ, w) = T iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T;
v(∀βΦ, w) = F iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) ≠ T;
10 .
v(∃βΦ, w) = T iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) = T;
v(∃βΦ, w) = F iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ T; |
11 .
v(□Φ, w) = T iff for all worlds u in Wv, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u in Wv, v(Φ, u) ≠ T;
12 .
v(◊Φ, w) = T iff for some world u in Wv, v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u in Wv, v(Φ, u) ≠ T.
(with the beta symbol following quantifiers shown as non-subscript, following the presentation in this section)
‘∃x x = α’, meaning ‘α exists’, can be notated as ‘E!α’.
Multivalued Logics
There are a number of reasons to be dissatisfied with the bivalence of classical logic, that is, with the limitation to just two values, true (T) and false (F). Among these reasons are: 1) some sentences are unintelligible and would thus seem to be neither T nor F, 2) some sentences have terms that fail to refer to an object, which might make us want to say they are neither T nor F, 3) semantic paradoxes, such as the liar’s paradox, would seem to have neither a T nor a F value, 4) there are metaphysical issues that could compel us to consider certain situations as being neither T nor F, as for example when making statements now about an unknowable future, 5) certain practical concerns, as for example in computer database creation, where designating certain formulas as neither T nor F is more useful, and 6) the vagueness of situations can make clear-cut propositions about their real status difficult on account of ambiguities, and hence we would not want such statements to be entirely T or F. Thus we have cause for devising logics with three or more values. Three such multivalued semantics are Bochvar’s, Kleene’s, and Łukasiewicz’s. In all three, a third value, namely indeterminate or ‘I’, is added, but how propositions containing indeterminate values are evaluated varies in each system.
Truth Tables for Bochvar’s Three-Valued Semantics
(Nolt 408)
Truth Tables for Kleene’s Three-Valued Semantics
(Nolt 412)
Truth Tables for Łukasiewicz’s Three-Valued Semantics
(Nolt 413)
Each will make certain classical tautologies valid or invalid, depending on how they evaluate in each semantic. And each one in some way may make certain counter-intuitive tautologies in classical logic become invalid, and certain intuitive classical tautologies become invalid. In each semantics, an operator for ‘it is true that’, symbolized T, may be used for establishing tautologies, since it converts values to just T or F. (Without this operator, classical tautologies in these three-valued semantics can be valued as indeterminate. Thus not all valuations would make it true, and hence they would no longer be tautologous). Its truth table is the following:
15.3
Supervaluations
[Basic technique of supervaluation]
A three-valued semantics was one way to deal with a number of situations where bivalence was unsatisfactory. A third value, I or indeterminate, was used. But instead of that third value, we can keep just T or F, although certain formulas can be assigned neither of these two values. These cases of no value are called truth value gaps. To evaluate the truth value of formulas with component truth value gaps, we can use a technique invented by Bas van Fraassen called supervaluation. First we begin by making a truth table and putting in the values we know, but we place gaps where there is no value. This is called a partial valuation. Next, we fill those gaps with T or F such that every possible combination of values is given. These are called classical completions. Finally, we make a supervaluation of these classical completions in the following way. If all the classical completions compute the formula as true, then the supervaluation is true. If all the classical completions compute the formula as false, then the supervaluation is false. And if not all the classical completions are either entirely T or F, then the supervaluation does not assign a value to the formula. Here is the formal definition of supervaluation:
DEFINITION A supervaluational model of a formula or set of formulas consists of
1. a partial valuation S, which assigns to each sentence letter of that formula or set of formulas the value T, or the value F, or no value. We use the notation ‘S(Φ)’ to denote the value (if any) assigned to Φ by S.
2. A supervaluation VS of S that assigns truth values VS (Φ) to formulas Φ according to these rules:
VS (Φ) = T iff for all classical completions S′ of S, S′(Φ) = T;
VS (Φ) = F iff for all classical completions S′ of S, S′(Φ) = F;
VS assigns no truth value to Φ otherwise.
(Nolt 415-416, boldface his)
Chapter 16:
Radically Nonclassical Logics
Infinite Valued and Fuzzy Logics
There are vague predicates or concepts which can potentially lead to counter-intuitive inferences. Consider for example the vague predicate in this inference:
A global population of 1,000,000,000 is sustainable.
If a global population of 1,000,000,000 is sustainable, so is a global population of 1,000,000,001.
∴ A global population of 1,000,000,001 is sustainable.
(Nolt 420)
Suppose we reiterate the premises, each time building from the prior conclusion’s numerical value, and adding one more in the process. After a while, the population number will get very large, and the conclusion will no longer be true (it will not satisfy the predicate any more). An infinite-valued semantics can deal with these situations. It allows us to assign partial values to propositions, such that instead of true and false we have 0, 1, and all the decimal values between. In our example, each iteration would receive slightly less of a truth value, and so eventually the reiterations would arrive at 0, corresponding to our intuition that their truth value should decrease as the population number increases. A common valuation scheme is:
1. V(¬Φ) = 1 – V(Φ)
2. V(Φ & Ψ) = min(V(Φ), V(Ψ))
3. V(Φ ∨ Ψ) = max(V(Φ), V(Ψ))
4. V(Φ → Ψ) = 1 + min(V(Φ), V(Ψ)) - V(Φ)
5. V(Φ ↔ Ψ) = 1 + min(V(Φ), V(Ψ)) - max((V(Φ), V(Ψ))
Consider a predicate like “is red”. Suppose we add the following argument to get “fresh blood is red”. This is clearly true. But what about “a sunset is red”? This is partly true. That means the set of things that belong to the predicate “is red” has items whose membership is not entire. They have a certain degree of membership, and that degree matches the truth value of the proposition predicating them. In other words, if “a sunset is red” has the truth value 0.2, that means sunsets only have a membership degree of 0.2 in the set of red things. These are fuzzy sets (that is, sets whose membership is a matter of degree), and they were invented by Lofti Zadeh. He also applied fuzzy sets to the logical values that can be assigned, such that a range of values would be assigned and certain values in that range are themselves assigned a partial value for their degree of membership in that range of truth values. Such a semantics based on fuzzy truth values is called a fuzzy logic. While such fuzzy systems involve perhaps too much complexity and arbitrarity, they have proven useful for artificial intelligence programming, and they have also appealed to people who are wary of too much logical or conceptual precision.
16.2
Intuitionistic Logics
[Basic set-up of Kripkean intuitionistic semantics]
Many logics base their notion or evaluation of truth on the correspondence of thoughts with the facts of a real world. Intuitionistic logic, however, holds that the “truth” or “falsity” of a statement is determined by whether or not the available data warrant us to assert it. We call this warrant, proof, or evidence confirmation. To assert a proposition means we have warrant to confirm it. And to assert its negation means we have warrant to say that it is not the case. In our formalized intuitionistic semantics, we assign to propositions one of two values: {1} confirmed, ‘C’, when it is warranted to assert it, and {2} unconfirmed, ‘U’, when it is not warranted to assert it. But there are two sorts of being unconfirmed for a proposition, namely, being either refuted (that is, confirming its negation) or being neither confirmed nor refuted (that is, the proposition is neither confirmed while its negation is not confirmed). One example of this “neither” status is the following. Consider the proposition P that there are seven consecutive sevens somewhere in the decimal expansion of the number π. Now suppose that no computer or human working on determining as many decimals of π as possible, after whatever extent of time, have discovered seven consecutive sevens. That means we do not have grounds to confirm P. But we also have no proof that there are not seven consecutive sevens, so we cannot confirm ~P. Another example has to do with historical statements which cannot be confirmed or denied, like “Napoleon ate breakfast on September 9, 1807.” Now, if we can neither confirm P nor confirm ~P, this means we cannot confirm P∨~P (which would be understood in intuitionistic logic to mean “‘P’ is either confirmed or refuted”). Thus the law of excluded middle does not hold in intuitionistic logic. We also cannot make double negation inferences. In other words, the intuitionistic value of ~~P is not necessarily the same as P. ~~P means that it is refuted that P is refuted. (But consider again the π example above. Here we can say that it is refuted that P is refuted, or ~~P, because we can know that no one has refuted it yet. So ~~P = C. But we also cannot confirm P, meaning that P = U. So we cannot derive P from ~~P in this instance by simply changing the value twice. Also it is possible to modify the example such that P = U and ~~P = U. The computer may have calculated the seven consecutive sevens, but on account of some technical problem, we cannot ever obtain that information from the computer’s processes. In other words, not only does negation not work by doubly switching the values, it also does not produce a consistent pattern of valuation in all cases). Now, in three-valued logics, we can always derive P from ~~P. So this is one way that distinguishes intuitionistic logic from three-valued (and also classical) logics. This in part leads us to regard the “neither” value (neither confirmed nor denied) as not being a third value but rather as a subspecies of non-confirmation. To more formally evaluate propositions in this bivalent way in intuitionistic logic, we need to consider the following concepts. One is “state of evidence” or “evidential state”, which is “the total evidence available to a person or culture at a given time” (431) and which are structurally similar to possible worlds in Kripkean model semantics. On the basis of evidential states, propositions are determined to be either confirmed or unconfirmed. A first evidential state is said to be consistent with (or have access from) a second one if all the confirmation assignments of the first are given as well in the second, even though the second may have additional confirmations not in the first. In intuitionistic semantics, it is stipulated that when something is confirmed in a first evidential state, it is confirmed in all others that the first is consistent with (those from which the first is accessible). This is the epistemic necessity of confirmation.
A Kripkean model in intuitionistic predicate logic is defined in the following way:
DEFINITION A Kripkean valuation or Kripkean model v for a formula or set of formulas of intuitionistic predicate logic consists of the following:
1. A nonempty set Wv of objects, called the worlds or evidential states of v.
2. A reflexive, transitive relation ℛ, consisting of a set of pairs of worlds from Wv.
3. For each world w in Wv a nonempty set Dw of objects, called the domain of w, such that for any worlds x and y, if xℛy, then Dx is a subset of Dy.
4. An assignment to each name α in that formula or set of formulas of an extension v(α) that is a member of the domain of at least one world.
|
5. An assignments to each predicate Φ and world w in that formula or set of formulas, an extension v(Φ, w) such that
i. If Φ is a zero-place predicate, v(Φ, w) is one (but not both) of the values C or U such that if v(Φ, w) = C, then for all u such that wℛu, v(Φ, u) = C.
ii. If Φ is a one-place predicate, v(Φ, w) is a set of members of Dw such that if d ∈ v(Φ, w), then for all u such that wℛu, d ∈ v(Φ, u).
iii. If Φ is an n-place predicate (n>1), v(Φ, w) is a set of ordered n-tuples of members of Dw such that if <d1, ..., dn> ∈ v(Φ, w), then for all u such that wℛu, <d1, ..., dn> ∈ v(Φ, u).
(432-433)
The following are the confirmation conditions for complex formulas in intuitionistic logic.
Let Φ and Ψ be any formulas and v any valuation of Φ and Ψ whose accessibility relation is ℛ. Then
1.
If Φ is a one-place predicate and α is a name, then v(Φα, w) = C iff v(α) ∈ v(Φ, w);
v(Φα, w) = U iff v(α) ∉ v(Φ, w).
2.
If Φ is an n-place predicate (n>1) and α1 ... , αn are names, then
v(Φα1, ... , αn, w) = C iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = U iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
3.
If α and β are names, then v(α = β, w) = C iff v(α) = v (β);
v(α = β, w) = U iff v(α) ≠ v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = C iff for all u such that wℛu, v(Φ, u) ≠ C.
v(~Φ, w) = U iff for some u such that wℛu, v(Φ, u) = C.
5.
v(Φ & Ψ, w) = C iff both v(Φ, w) = C and v(Ψ, w) = C.
v(Φ & Ψ, w) = U iff either v(Φ, w) ≠ C or v(Ψ, w) ≠ C, or both.
6.
v(Φ ∨ Ψ, w) = C iff either v(Φ, w) = C or v(Ψ, w) = C, or both.v(Φ ∨ Ψ, w) = U iff both v(Φ, w) ≠ C and v(Ψ, w) ≠ C.
7.
v(Φ → Ψ, w) = C iff for all u such that wℛu, v(Φ, u) ≠ C or v(Ψ, u) = C, or both.
v(Φ → Ψ, w) = U iff for some u such that wℛu, v(Φ, u) = C and v(Ψ, u) ≠ C.
8.
v(Φ ↔ Ψ, w) = C iff for all u such that wℛu, v(Φ, u) = v(Ψ, u).
v(Φ ↔ Ψ, w) = C iff for some u such that wℛu, v(Φ, u) ≠ v(Ψ, u).
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
9.
v(∀βΦ, w) = C iff for all worlds u such that wℛu and for all potential names α of all objects d in Du, v(α,d) (Φα/β , u) = C;
|
v(∀βΦ, w) = U iff for some world u such that wℛu, and some potential name α of some object d in Du, v(α,d) (Φα/β , u) ≠ C.
10.
v(∃βΦ, w) = C iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) = C;
v(∃βΦ, w) = U iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ C.
(Nolt 433-434)
Negation means refutation, which is more than non-confirmation in one evidential state. It means that there are no evidential states accessible from the first where the unrefuted formulation is confirmed. It is similar to impossibility in modal logic. Along with negation, the conditional, biconditional, and universal quantification require comparison with other evidential states accessible to the first. But conjunction, disjunction, and existential quantification need only the evidential state in question. A formula is intuitionistically valid if all evidential states confirm it. It is intuitionistically inconsistent if there is at least one evidential state that does not. A sequent is intuitionistically valid if there are no evidential states where the premises are confirmed while the conclusion is not.
16.3
Relevance Logics
[basic set-up of dialethic, relevant logic semantics]
In many logics, the following sequent is valid: P, ~P ⊢ Q. For, no matter which values we assign the formulas, none will make the premises true and the conclusion not true, as all the premises can never be made true anyway. This raises the concern that you can validly draw an inference that is completely irrelevant to the premises, which goes against our intuitions regarding how an inference from premises should work. (There are a number of irrelevant sequents that do not involve deriving irrelevant conclusion from contradictions, for example: Abe Lincoln was truthful. ∴ Nothing is both alive and not alive; Ta ⊢ ~∃x(Ax & ~Ax). Another is: P ⊢ Q → Q.) Now, inconsistencies can appear in many humanly constructed realms, like in statutory law, games, fictions, and even semantics as with the liar paradox. There are logics designed to prevent such irrelevant sequents, called relevance or relevant logics. Most relevant logics hold the following three things: {1} inconsistent premises do not imply every propositions but only those relevant to them; {2} valid formulas only follow validly from related premises; and {3} there is a sort of conditional that can only be true when its antecedent and consequent are relevant to one another. Paraconsistent logics do not allow for all propositions to be derived from inconsistent premises. Relevance logics do allow certain conclusions to be drawn from contradictory premises, but only relevant ones. Since not all propositions can be derived from contradictions in relevance logics, they can be considered paraconsistent. When a sequent lacks a counterexample, it is valid, and thus in classical logic P, ~P ⊢ Q is valid. (That is to say, no possible value assignment, when we are restricted to just true and just false, can make the premises true and the conclusion not true.) There are two sorts of strategies for ensuring that irrelevant sequents have counterexamples. {1} We add criteria of relevance to the definition of validity, or {2} we keep this definition of validity, but we expand our notion of counterexample to include ones that invalidate irrelevant sequents. The semantics we outline here, a “dialethic” logic, takes the second approach. It rejects bivalence and has the following four value assignments: {1} just true, {2} just false, {3} both true and false, and {4} neither true nor false. For technical reasons we exclude the conditional operator, and we think of the values in terms of sets: {T}, {F}, {T, F} or { }. The valuation rules and truth tables are:
1.
T ∈ v(~Φ) iff F ∈ v(Φ).
F ∈ v(~Φ) iff T ∈ v(Φ).
2.
T ∈ v(Φ & Ψ) iff T ∈ v(Φ) and T ∈ v(Ψ).
F ∈ v(Φ & Ψ) iff F ∈ v(Φ) or F ∈ v(Ψ), or both.
3.
T ∈ v(Φ ∨ Ψ) iff T ∈ v(Φ) or T ∈ v(Ψ), or both.
F ∈ v(Φ ∨ Ψ) iff F ∈ v(Φ) and F ∈ v(Ψ).
Truth Table for Negation
Truth Table for Conjunction
Truth Table for Disjunction
As we can see, the value assignments follow classical logic as much as possible. (When a formula’s component value is neither, but that value would not change a classical valuation anyway, then it takes the classical valuation. When part or all of a formula has both values, we determine all the possible classical valuations taking each singular value independently and include all the results in the value-set for the whole complex formula.) What is important in our validity evaluations of sequents is that the premises be at least true (even if they are also false) and the conclusion be not at all true (so if the conclusion is both true and false, then we cannot say it is not true. But if it has neither value or is just false, then we can say it is not true.) Consider the evaluation for P, ~P ⊢ Q:
As we can see, in the 10th and 12th lines, the premises are (at least true) and the conclusion is not (at all) true. (In line 13, for example, the premises are at least true, and the conclusion is at least false, but the conclusion is also at least true and thus not not-true. So that line’s valuation, v(P)={T,F} and v(Q)={T,F}, does not create a counterexample.) This dialethicist semantics does not overapply and create counterexamples for sequents that our intuition tells us should be valid; so for example it does not invalidate: ‘P & Q ⊢ P’. However, from contradictions we can derive relevant consequences, as in: ‘P & ~P ⊢ P’. One unfortunate exception, however, is that disjunctive syllogism, ‘P ∨ Q, ~P ⊢ Q’, is invalid in this dialethic semantics.
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
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