Download Truth and proof

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 xy 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
xy (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=: (Mx x ≤ M)  (Ny 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.
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GBM: example 2
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A=: (Mx x≤M)  (Ny 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 (Mx x≤M)  (Ny N<y)
(indoors)
• Formal proof in PA
• Winning strategy in dialogical logic.
Proof games in dialogical logic
1.
2.
3.
4.
5.
6.
7.
Proponent: (p  p)
Opponent: ?
Proponent: p
Opponent: p
Proponent cannot defend p
Proponent goes back and answers (2): p
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
•
1.
2.
3.
The general form of the argument:
PA proves ConPA  G
Prove ConPA in a metasystem
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.