Saul Kripke: Naming, Necessity, and Possible Worlds

1. Biography and Intellectual Context

Saul Aaron Kripke (1940–2022) was an American philosopher and logician who reshaped analytic philosophy in the late 20th century. A prodigy, he wrote his completeness theorem for modal logic at age 17 while still in high school, and published it in 1959.

Kripke taught at Harvard, Rockefeller, and Princeton. His three 1970 Princeton lectures, published as Naming and Necessity (1980), overturned the dominant Frege-Russell descriptivist theory of reference that had held for 70 years.

Core Innovations

Rigid designation and the causal-historical theory of reference
Kripke semantics for modal logic — possible worlds with accessibility relations
The necessary a posteriori and contingent a priori
Kripkenstein — the rule-following paradox
Fixed-point theory of truth for languages containing their own truth predicate

Context: Kripke worked against logical positivism and Quine's skepticism about modality. By giving modal notions a rigorous set-theoretic semantics, he legitimized talk of necessity, essence, and possible worlds — reviving metaphysics.

2. Naming and Necessity (1980)

2.1 Critique of Descriptivism

Frege and Russell held that a proper name 'a' is synonymous with a definite description: 'Aristotle' = 'the teacher of Alexander' or a cluster of descriptions. The description gives the sense, which determines reference.

Kripke offers three arguments:

Modal Argument

If 'Aristotle' meant 'teacher of Alexander', then "Aristotle taught Alexander" would be necessary. But it's contingent — Aristotle could have died young. Names are rigid; descriptions are usually non-rigid.

Epistemic Argument

We can use 'Feynman' to refer without knowing any uniquely identifying description. Reference does not require knowledge that picks out the referent.

Semantic Argument

Gödel-Schmidt case: Suppose Gödel stole the proof from Schmidt. We still refer to Gödel, not Schmidt, when we say 'Gödel'. So reference isn't fixed by the associated description.

2.2 Rigid Designators

A rigid designator refers to the same object in every possible world in which that object exists. Proper names ('Richard Nixon'), natural kind terms ('water', 'gold', 'tiger'), and some theoretical terms ('H2O') are rigid.

A definite description like 'the President of the US in 1970' is non-rigid: it refers to Nixon in the actual world, but to Humphrey in worlds where Humphrey won in 1968.

∀w₁∀w₂ ( w₁ ⊨ 'a' = o ∧ w₂ ⊨ ∃x(x=o) → w₂ ⊨ 'a' = o ) // rigidity

2.3 Causal-Historical Theory

Reference is fixed not by description but by:

Initial baptism: by ostension ("that baby will be called 'Nixon'") or by description used to fix reference (not give meaning)
Chain of communication: later users receive the name with intention to use it with same reference

This explains how we can refer to Aristotle despite massive ignorance and false beliefs.

2.4 Necessary A Posteriori and Contingent A Priori

Kripke breaks Kant's equation: necessity ≠ a priori, contingency ≠ a posteriori.

Category	Example	Explanation
Necessary a posteriori	Hesperus = Phosphorus	Both names rigidly designate Venus. Identity, if true, is necessary. Discovered by astronomy, so a posteriori.
Necessary a posteriori	Water = H₂O	Natural kinds have essential microstructure. Discovered empirically.
Contingent a priori	Stick S is 1 meter	At t₀ in Paris, define 'meter' via stick S. Known a priori by stipulation, but S could have been longer.

3. Interactive Demo: Rigid Designator Checker

Select a term. See what it refers to across three possible worlds.

Term:

World 2: Aristotle dies at 14, never teaches. Theophrastus teaches Alexander. World 3: Plato teaches Alexander.

4. Modal Logic and Kripke Semantics

Kripke's semantics (1959) gives truth-conditions for modal operators using possible worlds.

A Kripke frame is a pair ⟨W, R⟩ where W is a non-empty set of worlds, R ⊆ W×W is an accessibility relation. A model adds valuation V: Prop → ℘(W).

M, w ⊨ □φ iff ∀v (wRv → M, v ⊨ φ)
M, w ⊨ ◇φ iff ∃v (wRv ∧ M, v ⊨ φ)

Properties of R validate axioms: reflexive R → T (□p→p); transitive → 4 (□p→□□p); Euclidean → 5 (◇p→□◇p). S5 uses universal R.

Interactive Model Checker (3 worlds)

Accessibility R

w1 valuation

p q

w2 / w3

p q

Formula:

Edit R and valuations to test correspondence theory. Try making R reflexive for T axiom.

5. Wittgenstein on Rules and Private Language

In Wittgenstein on Rules and Private Language (1982), Kripke presents the "Kripkenstein" paradox.

The Skeptical Paradox

Consider addition. You've only added numbers <57. What fact makes it true that you meant PLUS not QUUS, where:

x quus y = x+y if x,y <57; else 5

Your past behavior, mental images, dispositions, instructions — all are finite and compatible with the quus hypothesis. There seems to be no fact that determines meaning.

Skeptical Solution

Kripke argues Wittgenstein offers not a straight solution (finding the fact) but a skeptical solution: meaning statements are not true in virtue of facts, but are justified by communal practice. To say "Jones means plus" is to say Jones can be relied upon to answer as we do, and will be corrected by the community if deviant.

Consequence: a private language is impossible because there is no distinction between seeming correct and being correct without a community standard.

6. Theory of Truth

Tarski showed that a classical language cannot contain its own truth predicate without paradox. Kripke (1975) uses a three-valued logic (Strong Kleene: True, False, Undefined) to build a language that can talk about its own truth.

Construction: Start with base language L₀ without 'True'. Build hierarchy:

S₀ = ∅
S_{α+1} = { φ | φ true under valuation with T-extension = S_α }
At limit λ: S_λ = ∪_{α<λ} S_α
Fixed point: S = S_{α} where no new sentences added

Grounded sentences (like "Snow is white" is true) eventually get value True. Ungrounded sentences like the Liar ("This sentence is false") never stabilize — they remain Undefined. This is the minimal fixed point.

Interactive Fixed-Point Builder

Stage: 0

Sentence	Content	Value

7. Philosophy of Mind: Argument Against Identity Theory

Materialist identity theory: pain = C-fiber firing.

Kripke's argument (Naming and Necessity, Lecture III):

If 'pain' and 'C-fiber firing' are rigid designators (they pick out the same phenomenon in all worlds), then if pain = C-fiber firing is true, it is necessarily true. (Necessity of identity: a=b → □a=b)
'Pain' is rigid via its essential phenomenological quality — pain is necessarily felt as pain.
But we can conceive of pain without C-fibers (in molluscs, Martians) and C-fibers without pain (unconscious stimulation). The identity appears contingent.
Therefore, pain ≠ C-fiber firing.

This is not a dualist proof, but it shows type-identity is incompatible with Kripke's semantic framework. It forces materialists toward functionalism or eliminativism.

8. Interactive Simulations & GNU Octave Code

Causal Chain Reference Propagation

Click to propagate a name through a community. Reference is preserved despite false beliefs.

Octave: Causal Chain

% simulate Kripke causal chain
n = 20; % agents
A = zeros(n); % adjacency
for i=1:n-1, A(i,i+1)=1; end % linear chain
ref = zeros(n,1); ref(1)=1; % baptism at node 1
for t=1:5
  ref = sign(A'*ref + ref); % propagate
end
disp(find(ref)) % who has reference

Octave: Fixed-Point Truth

% Kleene 3-valued: 1=T, -1=F, 0=U
function v = step(v, base)
  v_new = v;
  v_new(1) = base;           % S1 grounded
  v_new(2) = v(1);           % S2 = True(S1)
  v_new(3) = -v(3);          % Liar S3 = not S3
  if v_new(3)==v_new(3), v_new(3)=0; end % stays U
  v = v_new;
end
v = [0;0;0]; for i=1:10, v=step(v,1); end
disp(v) % [1;1;0] minimal fixed point

Octave: Modal Evaluator

% Kripke model evaluator
W = 1:3; R = [0 1 1; 0 1 0; 1 0 0];
V.p = [1 0 1]'; V.q = [0 1 1]';
box = @(phi) all(V.p(R(i,:)==1)); % simplified
% evaluate □p at world 1
i=1; all(V.p(find(R(i,:))))

Octave: Plot Accessibility

R = [0 1 1; 0 1 0; 1 0 0];
gplot(R, [0 1; 1 0.2; 1 1.8], '-o')
text([0 1 1],[1 0.2 1.8],{'w1','w2','w3'})
title('Kripke Frame ');