Between proof and truth

Gabriel Sandu

Univ. of Helsinki

Dummett’s verificationism

• Truth: an effective condition which establishes a statement as conclusive.

• In mathematics, it is a proof (deductive argument) or a computation

• The effectiveness of proofs and of computations guarantees their epistemic accessibility

Verificationism in arithmetics

• Two kinds of verification procedures

1. Verification procedures as proofs

2. Verification procedures as computations (unproblematic cases)

Computation in unproblematic cases

"that it is unproblematic to hold the principle of bivalence for recursive or decidable statements of arithmetic. This is because all the quantifiers are bounded. Hence the determination of a truth-value for such a statement involves at most a finite search among natural numbers together with decision making with respect to effective operations and relations on them."

Hintikka’s verificationism

• Verification procedures as winning strategies in semantical games

• Semantical games are played with sentences on a given model

• They can be learned and taught

• Truth in a model as the existence of a winning strategy in the semantical game

Epistemic accessibility

• Dummett’s verification procedures (proofs and computations) are epistemically accessible (effective)

• Hintikka’s verification procedures are not (always): there are winning strategies (functions) which are not computable (effective).

Hintikka’s proposal: playability of games

“For the basis of my argument was the requirement that the semantical games that are the foundations of our semantics and logic must be playable by actual human beings, at least in principle. This playability of our "language games" is one of the most characteristic features of the thought of both Wittgenstein and Dummett.” (Hintikka 1996)

Consequences of Hintikka’s proposal

• The notion of effective (computable) strategy is well defined only on the universe of natural numbers

• Even so, the requirement of playability makes sense only on structures where predicate and function symbols of the arithmetical language are interpreted by recursive relations and functions.

• If our model is not recursive, speaking of computable strategies does not make much sense, since in that case, even simple atomic formulas cannot be computed.

Two kinds of verificationism

• Verification procedures = proofs = effective winning strategies in dialogical games (Dummett, Lorenz and Lorenzen)

• Verification procedures = effective truth = effective winning strategies in semantical games played on recursive models (Hintikka)

Dummett and Hintikka

• What is the relation between Dummett’s verification procedures for arithmetic and Hintikka’s computable verification procedures in standard semantical games played on the standard model?

Dummett and Hintikka continued

• What is the relation between proof and computable truth?

• Does any proof yield a computable truth?

• Is it so that to any inference corresponds to a computation?

A new proposal: Coquand-Krivine

• Games with backwards moves (GBM)

• They are ”like” Hintikka’s semantical games

• They are ”like” dialogical games.

• On the standard model, winning strategies in GBM are computable.

Three verification procedures (arithmetics)

1. Computable truth: computable winning strategies in semantical games played on the standard structure (Hintikka)

2. Computable truth: computable winning strategies in the GBM associated with (1) (Coquand-Krivine)

3. Proof: effective winning strategies in dialogical games (Lorenz-Lorenzen)

Semantical game: example 1

• Game with xy x ≤ y) on the standard structure

1. V chooses 0

2. F chooses m

3. V has a win against any m chosen by F

GBM: example 1

• The corresponding GBM played with

xy (x ≤ y)

on the standard structure:

1.V chooses 1

2. F chooses m

3. If 1 > m, then V remakes her first choice, and chooses 0.

Semantical games: example 2

• A=: (Mx x ≤ M) (Ny N < y):• V chooses left• F chooses n₀ for N.• V chooses m₀ for y.• V has a winning strategy: choose

m₀ = n₀+1 • Hence A is true on the universe of natural

numbers.

GBM: example 2

• A=: (Mx x≤M) (Ny N<y)• V chooses right.• F chooses n₀ for N.• V changes her mind and chooses left• V chooses n₀ for M.• F chooses m₀ for x.• If m₀ ≤ n₀, then V wins. • Otherwise (i.e. m₀>n₀) Eloise prolongs the play:• She goes back to the position where she chose left but

decides now to choose m₀ as a value for y - hence winning the play.

Proof games

• Played with (Mx x≤M) (Ny N<y) (indoors)

• Formal proof in PA

• Winning strategy in dialogical logic.

Proof games in dialogical logic

1. Proponent: (p p)

2. Opponent: ?3. Proponent: p

4. Opponent: p

5. Proponent cannot defend p

6. Proponent goes back and answers (2): p

7. Proponent wins.

GBM are ”like” semantical games

• Coquand (1995), Bonnay (2006):

1. V has a w.s. in a semantical game iff V has a w.s. in the GBM associated with it.

2. F has a w.s. in a semantical game iff F has a w.s. in the GBM associated with it.

Epistemic accessibility (playability)

• Krivine (2003): Let A be a sentence. The following are equivalent (all are played on the standard structure):

1. V has a w.str. in the semantical game played with A.

2. V has a w.str. in the corresponding GBM

3. V has a computable w.str. in the corresponding GBM.

Dummett and Hintikka: Completeness

1. Do proofs in PA yield computable verifying winning strategies in standard semantical games played on the standard structure? No!

2. Do proofs in PA yield computable winning strategies in GBM played on natural numbers? Yes!

Proofs vs. computable truth

• A proof in PA of any sentence A yields a computable w.str. for V in the GBM played on the natural numbers.

• Thus GBM represent an interesting antirealist position between proof and truth.

Completeness result

• PA A there is a computable w.str. in GBM

• Is any computable w.str. in GBM given by a proof?

The Gödelian sentence G

• G is the arithmetical statement which says ”I am unprovable”.

• G is undecidable in PA but true

• How does one ”recognize” the truth of G?

Recognizing the truth of G

• Three answers, all involve a meta-level reasoning:

1. Tarski: Semantical proof in the metalanguage (Tarski-Ketland 1997).

2. Dummett-Tennant: proof in a meta-system (Tennant 2002).

3. Computable w.str. in GBM.

Tarski-Dummett-Tennant

• The general form of the argument:

1. PA proves ConPA G

2. Prove ConPA in a metasystem

3. Whence, the metasystem proves G

Tarski’s solution

• The metasystem is ”PA + Tarski’s theory of truth”

• The metasystem proves

(*) x(ProvPA(x) Tr(x))

• From (*) one can prove ConPA

• Tarskian theory of truth is not conservative over PA.

Dummett-Tennant: meta-proof

• The metasystem is PA plus reflection principles

ProvPA(<>)

for any primitive recursive .

• The metasystem proves ConPA.

Tennant’s answer: Sound proof suffices for truth

• Any proof in PA* of a sentence in the language of L, is a ground for asserting , even though might have no proof in the weaker system . All is needed, for the assertion of , is some proof of …In particular if it turns out that there is a proof of in some sound system PA*, then we are justified in asserting . For, from any philosophical perspective, sound proof suffices for truth.

Recognizing the truth of G in GBM

• There is a w. str. for V in the standard semantical game played with G on the standard model.

• Whence (Krivine) there is a computable w. str. for V in the GBM associated with it.

A possible objection

• Dummett: the argument is circular because it makes use of the standard model of PA

• Circularity: One cannot specify the standard model without using the natural numbers

Reply

• Reply: The requirement of playability of games implies that addition and multiplication must be computable.

• Hence, by Tennenbaum Theorem, it follows that any of the structure in which the game is played must be the standard one.

Truth-seeking games outside mathematics

• Three friends are in a café and one of them is giving the order: beer, wine and water.

• W: Who has wine?• Then he puts that glass.• W: Who has beer?• Then he puts the glass. • Finally he does not ask any more, but he

simply puts the remaining glass.

The logical structure of the argument

• Two questions, two answers and one inference.

• The final stage can be described by the valid propositional schema:

A B C, A, B C.

Logical theory of rational inquiry

• The major ingredients:

1. Questions,

2. Answers (observations, measurements, updates, announcements),

3. Inferences, and

4. Corrections (revisions).

The interrogative model of inquiry: truth-seeking games

• Two kinds of moves:

1.Logical moves: Deductive steps, a variant of the tableau-building rules

2.Interrogative steps: questions to an oracle

• The answers provide new information in the form of new premises from which further deductive inferences can be performed.

Dynamic epistemic logic

• Knowledge updates in a group of agents on the basis of incoming new information.

• Information come through several channels: observations, communicative acts (questions) and announcements, etc.

Dynamic epistemic logic continued

• The underlying logical system: multi-agent epistemic logic.

• The new star is [!P]B: After the (public) announcement that P, B is the case.

• Goal: higher-order knowledge, that is, how a new observation or communication leads to a change in the knowledge of the agents and leads eventually to common knowledge.

3 Kinds of updates

1. Games (cluelo, backward induction), 2. Traditional epistemic puzzles (muddy

children, the surprise examination, Fitch’s paradox),

3. Preference logics and social choice theory.

• The main concern is with knowledge, not beliefs, whence the emphasis on updates, not revisions.