Saul Kripke

Before Kripke, modal logic — the logic of □ ("necessarily") and ◇ ("possibly") — was studied almost entirely through axioms. C. I. Lewis and others had built several competing proof systems, labeled S1 through S5, but there was no agreed account of what these symbols actually meant, and no way to check whether a given system was even consistent, beyond syntactic proof-hunting.

Kripke's fix, worked out while he was still a teenager and refined over the next few years, was to treat necessity as truth in every world that is accessible from the one you're in, and possibility as truth in at least one accessible world. A frame is just a set of possible worlds together with an accessibility relation between them; a model adds an assignment of truth values to sentence letters at each world.

What makes this powerful is that the disputed proof systems turn out to correspond exactly to structural properties of the accessibility relation. Require the relation to be reflexive — every world can see itself — and you get system T. Add transitivity and you get S4. Make it an equivalence relation (reflexive, symmetric, and transitive, so that within a cluster every world sees every other) and you get S5, the system usually used for logical or metaphysical necessity. The competing axiom systems weren't rival guesses about the same thing; they were descriptions of different kinds of accessibility.

Try it

The diagram builds a small frame: five worlds, one designated as actual (gold), connected by a fixed set of arrows. Toggle the checkboxes to add reflexivity, symmetry, or transitivity to the relation, and toggle where a proposition P holds. Watch which modal system the relation now matches, and whether □P and ◇P come out true at the actual world.

Note the case where the actual world has no accessible worlds at all: □P then comes out true automatically — not because P is guaranteed, but because there's nothing left to falsify it against. This "vacuous" truth is a real feature of the semantics, not a glitch.

The dominant view Kripke was arguing against, descended from Frege and Russell and sharpened by Searle, held that a proper name is shorthand for a description: "Aristotle" means something like "the student of Plato who taught Alexander," or perhaps a cluster of such descriptions. Kripke attacked this from three directions at once.

The modal argument: it's perfectly coherent to say Aristotle might never have taught anyone. If the name just meant that description, the sentence would be a flat contradiction — but it isn't one. The epistemic argument: which description you attach to a name is something you could be wrong about, or never learn, while still using the name to refer correctly — most competent speakers of "Einstein" couldn't state anything uniquely identifying beyond "physicist," yet they refer to the man just fine. The semantic argument: the identifying description people do associate with a name is sometimes false of its bearer (Einstein did not invent the atomic bomb), and reference survives the error.

Rigid designators

Kripke's positive proposal: a name is a rigid designator — it picks out the same individual in every possible world in which that individual exists, and nothing else in any world. Most definite descriptions are not rigid: "the inventor of bifocals" picks out Franklin in the actual world, but would pick out someone else in a world where somebody else got there first.

How reference gets fixed

In place of descriptivism, Kripke sketches a causal-historical chain: a name is pinned to an object in an initial act — a baptism, an ostension, sometimes a description used just to fix the reference rather than give the meaning — and then passed from speaker to speaker through a chain of communication, each link intending to use the name with the same reference as the person they got it from. You can refer to Aristotle correctly while knowing almost nothing about him, because you're plugged into that chain.

Necessary a posteriori, contingent a priori

Because names and certain natural-kind terms are rigid, true identity statements built from them — "Hesperus is Phosphorus," "water is H₂O" — are necessarily true: there's no possible world where the very object is not identical to itself under another name. Yet these identities were discovered by observation, not derived from armchair reasoning, so they're knowable only a posteriori. This came as news: philosophers had mostly assumed "necessary" and "a priori" were the same divide seen from two sides. Kripke also points to the mirror-image case — a stipulated definition, like fixing "one meter" as the length of a particular bar at a particular time, gives you something knowable a priori (true by the stipulation itself) yet only contingently true (the bar could have been longer or shorter under different conditions).

Kripke also floats a further, more tentative essentialist claim: that an individual's origin is necessary to it — this desk couldn't have been made from an entirely different hunk of wood, on his view, and a person couldn't have come from different biological parents. He presents this as a suggestion rather than a settled result, and it remains one of the more contested parts of the lectures.

Kripke reads a skeptical puzzle out of Wittgenstein's remarks on rule-following — one he's careful to flag as possibly not Wittgenstein's own argument in every detail, worth considering on its own merits regardless of the exegesis, which is why the reading is often nicknamed "Kripkenstein" rather than just attributed straight to Wittgenstein.

The setup: suppose you've used the "+" sign many times, always on numbers smaller than 57, and always correctly computing their sum. Today you're asked for 68 + 57, a computation you've never actually performed. You answer 125. But a skeptic asks: how do you know you meant the plus function all along, rather than a bent function — call it quus — that agrees with plus for every pair you've actually tried, but is defined to return 5 for any pair involving a number 57 or larger?

Kripke argues this is a skeptical problem far more radical than Hume on induction or causation: it's not just that we can't be certain what will happen next, it's that there seems to be no fact at all — not even in principle — that constitutes your having meant addition rather than quaddition. And the same argument generalizes to any word or rule whatsoever.

The "skeptical solution"

Rather than hunting for the missing fact (a "straight" solution), Kripke reconstructs what he takes to be Wittgenstein's move: a skeptical solution, which concedes there's no such fact but argues our talk of meaning doesn't need one. What justifies saying you mean plus isn't a private fact about you; it's your standing within a community practice that checks your answers against others' and corrects departures. An isolated individual, on this view, has no meaning-facts about their own rule-following at all — meaning is essentially a social phenomenon, not a private mental one. This is also Kripke's route into Wittgenstein's private-language argument: a language only one person could in principle check is not a language that can have right and wrong uses at all.

Tarski had shown that a language rich enough to contain its own unrestricted truth predicate falls into contradiction — the Liar sentence, "this sentence is false," is neither consistently true nor consistently false. His fix was a hierarchy: an object language gets its truth predicate only from a distinct, richer metalanguage, which needs its own metalanguage to talk about its truth, and so on. Workable for formal languages, but a poor match for English, which seems to have exactly one word, "true," that applies quite happily to English sentences, including sentences about truth itself.

Kripke's alternative lets a single language have a partially defined truth predicate that can apply to its own sentences, using Kleene's three-valued logic (true / false / neither) and building the predicate up in stages rather than assuming it's fixed all at once.

A sentence that acquires a truth value at some stage is called grounded. Ordinary sentences, even ones about truth, are grounded almost immediately: "'snow is white' is true" is settled the moment "snow is white" is. But some self-referential sentences never receive a value at any stage, however far the construction runs — they stay ungrounded, meaningful in the sense of being well-formed, but permanently without a truth value. The Liar sentence is the notorious case: assigning it a value at any stage would only unravel at the next. The "truth-teller" ("this sentence is true") is a milder cousin — not paradoxical, just permanently undetermined either way.

The payoff is that a single, self-applicable truth predicate is coherent after all — you don't need Tarski's infinite tower of languages, only a tolerance for some sentences never settling into true or false. Kripke's construction became a standard reference point for later work on truth, vagueness, and paradox.

Possible-worlds semantics is now the default vocabulary for modal logic, and spread far beyond it — into the semantics of conditionals, tense, epistemic logic, and eventually into linguistics' treatment of modality in natural language. Naming and Necessity is widely credited with reviving serious metaphysics — talk of essential properties and de re necessity had gone out of fashion under the influence of Quine's skepticism about modality, and Kripke's arguments helped make it respectable again. It also created a headache for identity theories in philosophy of mind: if "pain is C-fiber firing" is meant as a genuine identity between rigid designators, it should be necessary if true, yet it intuitively seems like pain without the physical correlate is at least conceivable — a tension Kripke pressed hard against type-identity materialism.

The rule-following considerations reshaped debates about meaning, mental content, and dispositional theories of concepts, and are still argued over by philosophers who read Wittgenstein very differently from Kripke's reconstruction. The theory of truth became a standard reference point for formal work on paradox, alongside later revision-theoretic and paracomplete approaches it helped provoke. Kripke is also known for a companion puzzle about belief (the "Pierre puzzle," 1979), which uses the same rigid-reference machinery to unsettle standard treatments of belief reports.

Kripke taught at Princeton from 1968 and later at the CUNY Graduate Center, where the Saul Kripke Center preserves his papers and unpublished lectures — a substantial amount of his work reached print only as recordings and notes, transcribed and edited long after the fact. He received the Schock Prize in Logic and Philosophy in 2001, an honor sometimes described as philosophy's closest analogue to a Nobel. He died in 2022.

All of the above concerns Kripke's philosophical reception; where his arguments are contested — the reading of Wittgenstein, the necessity of origin, how far causal reference extends to natural-kind terms — that's noted in the sections above rather than glossed over, since these remain live disputes rather than settled consensus.