Saul Kripke

Modal logic

Two of Kripke's earlier works, "A Completeness Theorem in Modal Logic" (1959) ^[21] and "Semantical Considerations on Modal Logic" (1963), the former written when he was a teenager, were on modal logic. The most familiar logics in the modal family are constructed from a weak logic called K, named after Kripke. Kripke introduced the now-standard Kripke semantics (also known as relational semantics or frame semantics) for modal logics. Kripke semantics is a formal semantics for non-classical logic systems. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The discovery of Kripke semantics was a breakthrough in the making of non-classical logics, because the model theory of such logics was absent before Kripke.

A Kripke frame or modal frame is a pair ${\displaystyle \langle W,R\rangle }$, where W is a non-empty set, and R is a binary relation on W. Elements of W are called nodes or worlds, and R is known as the accessibility relation. Depending on the properties of the accessibility relation (transitivity, reflexivity, etc.), the corresponding frame is described, by extension, as being transitive, reflexive, etc.

A Kripke model is a triple ${\displaystyle \langle W,R,\Vdash \rangle }$, where ${\displaystyle \langle W,R\rangle }$ is a Kripke frame, and ${\displaystyle \Vdash }$ is a relation between nodes of W and modal formulas, such that:

• ${\displaystyle w\Vdash \neg A}$ if and only if ${\displaystyle w\nVdash A}$,
• ${\displaystyle w\Vdash A\to B}$ if and only if ${\displaystyle w\nVdash A}$ or ${\displaystyle w\Vdash B}$,
• ${\displaystyle w\Vdash \Box A}$ if and only if ${\displaystyle \forall u\,(w\;R\;u}$ implies ${\displaystyle u\Vdash A)}$.

We read ${\displaystyle w\Vdash A}$ as "w satisfies A", "A is satisfied in w", or "w forces A". The relation ${\displaystyle \Vdash }$ is called the satisfaction relation, evaluation, or forcing relation. The satisfaction relation is uniquely determined by its value on propositional variables.

A formula A is valid in:

• a model ${\displaystyle \langle W,R,\Vdash \rangle }$, if ${\displaystyle w\Vdash A}$ for all w ∈ W,
• a frame ${\displaystyle \langle W,R\rangle }$, if it is valid in ${\displaystyle \langle W,R,\Vdash \rangle }$ for all possible choices of ${\displaystyle \Vdash }$,
• a class C of frames or models, if it is valid in every member of C.

We define Thm(C) to be the set of all formulas that are valid in C. Conversely, if X is a set of formulas, let Mod(X) be the class of all frames which validate every formula from X.

A modal logic (i.e., a set of formulas) L is sound with respect to a class of frames C, if L ⊆ Thm(C). L is complete with respect to C if L ⊇ Thm(C).

Semantics is useful for investigating a logic (i.e., a derivation system) only if the semantical entailment relation reflects its syntactical counterpart, the consequence relation (derivability). It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and for them, to determine which class it is.

For any class C of Kripke frames, Thm(C) is a normal modal logic (in particular, theorems of the minimal normal modal logic, K, are valid in every Kripke model). However, the converse does not hold generally. There are Kripke incomplete normal modal logics, which is unproblematic, because most of the modal systems studied are complete of classes of frames described by simple conditions.

A normal modal logic L corresponds to a class of frames C, if C = Mod(L). In other words, C is the largest class of frames such that L is sound wrt C. It follows that L is Kripke complete if and only if it is complete of its corresponding class.

Consider the schema T : ${\displaystyle \Box A\to A}$. T is valid in any reflexive frame ${\displaystyle \langle W,R\rangle }$: if ${\displaystyle w\Vdash \Box A}$, then ${\displaystyle w\Vdash A}$ since w R w. On the other hand, a frame which validates T has to be reflexive: fix w ∈ W, and define satisfaction of a propositional variable p as follows: ${\displaystyle u\Vdash p}$ if and only if w R u. Then ${\displaystyle w\Vdash \Box p}$, thus ${\displaystyle w\Vdash p}$ by T, which means w R w using the definition of ${\displaystyle \Vdash }$. T corresponds to the class of reflexive Kripke frames.

It is often much easier to characterize the corresponding class of L than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose L₁ ⊆ L₂ are normal modal logics that correspond to the same class of frames, but L₁ does not prove all theorems of L₂. Then L₁ is Kripke incomplete. For example, the schema ${\displaystyle \Box (A\equiv \Box A)\to \Box A}$ generates an incomplete logic, as it corresponds to the same class of frames as GL (viz. transitive and converse well-founded frames), but does not prove the GL-tautology ${\displaystyle \Box A\to \Box \Box A}$.

Intuitionistic logic

Kripke semantics for intuitionistic logic follows the same principles as the semantics of modal logic, but uses a different definition of satisfaction.

An intuitionistic Kripke model is a triple ${\displaystyle \langle W,\leq ,\Vdash \rangle }$, where ${\displaystyle \langle W,\leq \rangle }$ is a partially ordered Kripke frame, and ${\displaystyle \Vdash }$ satisfies the following conditions:

• if p is a propositional variable, ${\displaystyle w\leq u}$, and ${\displaystyle w\Vdash p}$, then ${\displaystyle u\Vdash p}$ (persistency condition),
• ${\displaystyle w\Vdash A\land B}$ if and only if ${\displaystyle w\Vdash A}$ and ${\displaystyle w\Vdash B}$,
• ${\displaystyle w\Vdash A\lor B}$ if and only if ${\displaystyle w\Vdash A}$ or ${\displaystyle w\Vdash B}$,
• ${\displaystyle w\Vdash A\to B}$ if and only if for all ${\displaystyle u\geq w}$, ${\displaystyle u\Vdash A}$ implies ${\displaystyle u\Vdash B}$,
• not ${\displaystyle w\Vdash \bot }$.

Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the Finite Model Property.

Intuitionistic first-order logic

Let L be a first-order language. A Kripke model of L is a triple ${\displaystyle \langle W,\leq ,\{M_{w}\}_{w\in W}\rangle }$, where ${\displaystyle \langle W,\leq \rangle }$ is an intuitionistic Kripke frame, M_w is a (classical) L-structure for each node w ∈ W, and the following compatibility conditions hold whenever u ≤ v:

• the domain of M_u is included in the domain of M_v,
• realizations of function symbols in M_u and M_v agree on elements of M_u,
• for each n-ary predicate P and elements a₁,...,a_n ∈ M_u: if P(a₁,...,a_n) holds in M_u, then it holds in M_v.

Given an evaluation e of variables by elements of M_w, we define the satisfaction relation ${\displaystyle w\Vdash A[e]}$:

• ${\displaystyle w\Vdash P(t_{1},\dots ,t_{n})[e]}$ if and only if ${\displaystyle P(t_{1}[e],\dots ,t_{n}[e])}$ holds in M_w,
• ${\displaystyle w\Vdash (A\land B)[e]}$ if and only if ${\displaystyle w\Vdash A[e]}$ and ${\displaystyle w\Vdash B[e]}$,
• ${\displaystyle w\Vdash (A\lor B)[e]}$ if and only if ${\displaystyle w\Vdash A[e]}$ or ${\displaystyle w\Vdash B[e]}$,
• ${\displaystyle w\Vdash (A\to B)[e]}$ if and only if for all ${\displaystyle u\geq w}$, ${\displaystyle u\Vdash A[e]}$ implies ${\displaystyle u\Vdash B[e]}$,
• not ${\displaystyle w\Vdash \bot [e]}$,
• ${\displaystyle w\Vdash (\exists x\,A)[e]}$ if and only if there exists an ${\displaystyle a\in M_{w}}$ such that ${\displaystyle w\Vdash A[e(x\to a)]}$,
• ${\displaystyle w\Vdash (\forall x\,A)[e]}$ if and only if for every ${\displaystyle u\geq w}$ and every ${\displaystyle a\in M_{u}}$, ${\displaystyle u\Vdash A[e(x\to a)]}$.

Here e(x→a) is the evaluation which gives x the value a, and otherwise agrees with e.

Naming and Necessity

The three lectures that form Naming and Necessity constitute an attack on the descriptivist theory of names. Kripke attributes variants of descriptivist theories to Frege, Russell, Wittgenstein, and John Searle, among others. According to descriptivist theories, proper names either are synonymous with descriptions, or have their reference determined by virtue of the name's being associated with a description or cluster of descriptions that an object uniquely satisfies. Kripke rejects both these kinds of descriptivism. He gives several examples purporting to render descriptivism implausible as a theory of how names get their references determined (e.g., surely Aristotle could have died at age two and so not satisfied any of the descriptions we associate with his name, but it would seem wrong to deny that he was still Aristotle).

As an alternative, Kripke outlined a causal theory of reference, according to which a name refers to an object by virtue of a causal connection with the object as mediated through communities of speakers. He points out that proper names, in contrast to most descriptions, are rigid designators: that is, a proper name refers to the named object in every possible world in which the object exists, while most descriptions designate different objects in different possible worlds. For example, "Richard Nixon" refers to the same person in every possible world in which Nixon exists, while "the person who won the United States presidential election of 1968" could refer to Nixon, Humphrey, or others in different possible worlds.

Kripke also raised the prospect of a posteriori necessities—facts that are necessarily true, though they can be known only through empirical investigation. Examples include "Hesperus is Phosphorus", "Cicero is Tully", "Water is H₂O", and other identity claims where two names refer to the same object. According to Kripke, the Kantian distinctions between analytic and synthetic, a priori and a posteriori, and contingent and necessary do not map onto one another. Rather, analytic/synthetic is a semantic distinction, a priori/a posteriori is an epistemic distinction, and contingent/necessary is a metaphysical distinction.

Finally, Kripke gave an argument against identity materialism in the philosophy of mind, the view that every mental particular is identical with some physical particular. Kripke argued that the only way to defend this identity is as an a posteriori necessary identity, but that such an identity—e.g., that pain is C-fibers firing—could not be necessary, given the (clearly conceivable) possibility that pain could be separate from the firing of C-fibers, or the firing of C-fibers be separate from pain. (Similar arguments have since been made by David Chalmers.^[24]) In any event, the psychophysical identity theorist, according to Kripke, incurs a dialectical obligation to explain the apparent logical possibility of these circumstances, since according to such theorists they should be impossible.

Kripke delivered the John Locke Lectures in philosophy at Oxford in 1973. Titled Reference and Existence, they were in many respects a continuation of Naming and Necessity, and deal with the subjects of fictional names and perceptual error. In 2013 Oxford University Press published the lectures as a book, also titled Reference and Existence.

In a 1995 paper, philosopher Quentin Smith argued that key concepts in Kripke's new theory of reference originated in the work of Ruth Barcan Marcus more than a decade earlier.^[25] Smith identified six significant ideas in the New Theory that he claimed Marcus had developed: (1) that proper names are direct references that do not consist of contained definitions; (2) that while one can single out a single thing by a description, this description is not equivalent to a proper name of this thing; (3) the modal argument that proper names are directly referential, and not disguised descriptions; (4) a formal modal logic proof of the necessity of identity; (5) the concept of a rigid designator, though Kripke coined that term; and (6) a posteriori identity. Smith argued that Kripke failed to understand Marcus's theory at the time but later adopted many of its key conceptual themes in his New Theory of Reference.

Other scholars have subsequently offered detailed responses arguing that no plagiarism occurred.^[26][27]

Wittgenstein

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First published in 1982, Kripke's Wittgenstein on Rules and Private Language contends that the central argument of Wittgenstein's Philosophical Investigations centers on a devastating rule-following paradox that undermines the possibility of our ever following rules in our use of language. Kripke writes that this paradox is "the most radical and original skeptical problem that philosophy has seen to date", and that Wittgenstein does not reject the argument that leads to the rule-following paradox, but accepts it and offers a "skeptical solution" to ameliorate the paradox's destructive effects.

Most commentators accept that Philosophical Investigations contains the rule-following paradox as Kripke presents it, but few have agreed with his attributing a skeptical solution to Wittgenstein. Kripke himself expresses doubts in Wittgenstein on Rules and Private Language as to whether Wittgenstein would endorse his interpretation of Philosophical Investigations. He says that the work should not be read as an attempt to give an accurate statement of Wittgenstein's views, but rather as an account of Wittgenstein's argument "as it struck Kripke, as it presented a problem for him".

The portmanteau "Kripkenstein" has been coined for Kripke's interpretation of Philosophical Investigations. Kripkenstein's main significance was a clear statement of a new kind of skepticism, dubbed "meaning skepticism": the idea that for isolated individuals there is no fact in virtue of which they mean one thing rather than another by the use of a word. Kripke's "skeptical solution" to meaning skepticism is to ground meaning in the behavior of a community.

Kripke's book generated a large secondary literature, divided between those who find his skeptical problem interesting and perceptive, and others, such as Gordon Baker, Peter Hacker, and Colin McGinn, who argue that his meaning skepticism is a pseudo-problem that stems from a confused, selective reading of Wittgenstein. Kripke's position has been defended against these and other attacks by the Cambridge philosopher Martin Kusch, and Wittgenstein scholar David G. Stern considers Kripke's book "the most influential and widely discussed" work on Wittgenstein since the 1980s.^[29]

Truth

In his 1975 article "Outline of a Theory of Truth", Kripke showed that a language can consistently contain its own truth predicate, something deemed impossible by Alfred Tarski, a pioneer in formal theories of truth. The approach involves letting truth be a partially defined property over the set of grammatically well-formed sentences in the language. Kripke showed how to do this recursively by starting from the set of expressions in a language that do not contain the truth predicate, and defining a truth predicate over just that segment: this action adds new sentences to the language, and truth is in turn defined for all of them. Unlike Tarski's approach, however, Kripke's lets "truth" be the union of all of these definition-stages; after a denumerable infinity of steps the language reaches a "fixed point" such that using Kripke's method to expand the truth-predicate does not change the language any further. Such a fixed point can then be taken as the basic form of a natural language containing its own truth predicate. But this predicate is undefined for any sentences that do not, so to speak, "bottom out" in simpler sentences not containing a truth predicate. That is, " 'Snow is white' is true" is well-defined, as is " ' "Snow is white" is true' is true," and so forth, but neither "This sentence is true" nor "This sentence is not true" receive truth-conditions; they are, in Kripke's terms, "ungrounded."

Gödel's first incompleteness theorem demonstrates that self-reference cannot be avoided naively, since propositions about seemingly unrelated objects (such as integers) can have an informal self-referential meaning, and this idea – manifested by the diagonal lemma – is the basis for Tarski's theorem that truth cannot be consistently defined. But Kripke's truth predicate does not give a truth value (true/false) to propositions such as the one built in Tarski's proof, since it is provable by induction that it is undefined at stage ${\displaystyle n}$ for every finite ${\displaystyle n}$.

Kripke's proposal is problematic in the sense that while the language contains a "truth" predicate of itself (at least a partial one), some of its sentences – such as the liar sentence ("this sentence is false") – have an undefined truth value, but the language does not contain its own "undefined" predicate. In fact it cannot, as that would create a new version of the liar paradox, the strengthened liar paradox ("this sentence is false or undefined"). Thus while the liar sentence is undefined in the language, the language cannot express that it is undefined.^[30]