Reflections on Dialogical Logic

Guido Del Din

DIALOGICAL LOGIC• With such expression we refer to an approach to logic introduced in the 60’s by the German logicians Paul Lorenzen and Kuno Lorenz.

• Nowadays, the most important center of research in Dialogical Logic is the University of Lille (S. Rahman, L. Keiff).

• Dialogical Tableaux are the distinctive graphical instrument of this approach to logic.

O P ?$

Pc1

?" / c2Pc2

$x (Px→"xPx) Pc1→"xPx

"xPxPc2→"xPx

Pc2

Example:Simplified dialogical tableau

for a FOL formula

Some general observations• 2-column table, model of a dispute between two speakers about a thesis (assertions-challenges-defenses).

• The dispute is interpreted as a 0-sum game. The two speakers play each move alternately.

• The thesis is the upper formula in the right column. Premises appear in the left column at the beginning.

• The player who states the thesis is the Proponent P and all his moves appear in the right column; the rival is the Opponent O (left column).

• Dialogical tableaux can split, producing branches: each branch represents a different course of the dialogue, determined by the choices of the players between alternative moves.

Why are dialogical tableaux noteworthy?

• They are a recent notational device that has been adopted in didactic textbooks of logic (above all, in the German area).

• They share features with other more common notational instruments, mixing such features in a peculiar way.

• They turn on a conceptual framework that may call into question the standard understanding of syntax and semantics.

The main notational devices in modern symbolic logic

• Logistic Systems (Hilbert-style axiomatic systems)

• Truth Tables (Wittgenstein)

• Calculi for Natural Deduction (Gentzen, Fitch)

• Semantic Tableaux (Beth)

Truth Tables and Semantic Tableaux are effective decision procedure.

Natural Deduction Systems aims to be a model of our argumentative behavior.

Dialogical tableaux are both:

Effective decision procedures for formulae of Propositional Logic and First Order Logic (criterion of validity: existence of a formal winning strategy for P)

Simplified models of a linguistic natural behavior: a dialogical dispute.

The working principles of dialogical tableaux

• Language• Rules1. Particle rules2. Structural rules3. Justification rules for

prime formulae• Dialogical criterion of validity

The Language of Dialogical Logic

Independently of the graphical representation in a 2-column table (dialogues may be also displayed in tree-like notations or in Fitch-style diagrams).

• Usual language for PL and FOL• Two metalogical labels O and P, standing for the players

• Numbers that specify the order of the moves• Two force symbols: “!” for the assertion (commitment to defend the formula)“?...” for the attack (the dots stand for indices that explicit the target of the attack)

Particle rules

• A particle rule is the abstract description of an argumentation form, that sets how a complex formula can be criticized and defended, according to its outmost logical form.

• They determine the dialogical meaning of each logical constant.

• They determine the single steps of the dialogue, namely, which moves can follow from a given dialogical precondition.

• In this sense, they fix the local semantics: they don’t say anything about the general organization of the dialogue.

UTTERED FORMULA ATTACK DEFENSE 

A ˄ B  

A ˅ B 

¬A 

A → B     

"x A(x)   

$x A(x)  

 ? against the left and then ? against the right conjunct, or vice versa ? against the entire disjunction 

A A     ?"/c

(constant “c” chosen by who carries out the attack) 

?$

 assertion of the attacked conjunct, that must be defended assertion of one of the two disjunct no defense 

B (the commitment to assert “B” is compelling once the rival has defended “A” against the possible counterattack) 

A(c)(replacing with “c” the free occurrences of the variable in “A”) 

A(c)(constant “c” chosen by who carries out the defense)

 

Conjunction

“?L” means that O chooses to attack the left conjunct; “?R”, the right conjunct. The split in the tableau shows that the choice of O generates two different courses of the dialogue. The force symbol “!” is omitted.

O P

?L ?R

?R ?L

A ˄ B A B

B A

Disjunction

Here P can choose which formula to defend: either the left or the right formula of the disjunction.

O P ?˅

A ˅ B A B

Negation

There is no proper attack against a negation: the reaction of O consists in asserting the negated formula. Moreover, there is no proper defense: the reaction of P is a counterattack.

O P A

¬A ?

O P

A

A → B

B ?

BDefense of A

Implication

Here P, in his second move, can choose either to attack “A” or to assert “B”. In the first case, if O succeeds in defending “A”, P is committed in his third move to assert “B”.

Universal quantifier

“?"/c” means that O chooses an individual constant “c” and then P is committed to defend the formula “A(c)”, resulting from the substitution in “A” of the bound variable “x” with “c”.

O P

?"/c "x A(x) A(c)

Existential quantifier

In this case, P has the right to choose the individual constant “c” for the substitution of the bound variable.

O P ?$

$x A(x) A(c)

(PR) Observations I

• These rules convey the meaning of the logical constants without reference to the truth-value of the formulae at stake.

• About the quantifiers: no appeal to (sequence of) objects, like in model-theoretic semantics, but appeal to constants.

• The dialogical approach is clearly distant from standard model-theoretic semantics.

(PR) Observations II• Blurred similarity with inference rules in natural deduction: both kinds of rules mirror argumentative forms and determine the single steps of a dialogue/proof.

• But dialogical PRs don’t show the inversion principle: the analogy can be seen only with elim-rules. This depends on the decompositional nature of dialogues (just like semantic tableaux)

• Can you find a dialogical particle rule for Prior’s connective “tonk”?

(PR) Observations III

• Particular status of “¬” and “→”: these are the only connectives which compel the Opponent to assert a formula

• for “¬”: no defense, but only possibility of counterattack

• for “→”: the challenger can choose between defense and counterattack

Structural rules• They define the general organization of the game: how to start and how to conduct the dialogue.

• They determine, above all, which player wins and in what conditions he wins.

• The aim of the structural rules is to provide a method of decision.

• Different sets of structural rules mirror different concepts of logical consequence, varying the allowed applications of the particle rules.

• In this sense, they fix the global semantics: they specify the context in which logical constants take on their meaning.

(SR-0) The expressions of a dialogue are numbered, and are alternatively asserted

by P and O. The thesis carries the number 0 and is asserted by P. All moves after the thesis obey particle and structural rules. The even moves (2,4,…) are made by P; the odd ones (1,3,…) are made by O.

(SR-1intuitionistic) Whenever player X is to play, he can attack any move of Y or defend against the last attack of Y, provided he has not already defended against it. A player may postpone a defense as long as there are attacks that can be put forth.

(SR-1classical) Whenever player X is to play, he can attack any move of Y or defend against any attack of Y (even the ones against which he has already defended). In other terms, players can play again earlier defenses.

(SR-2) There are three cases in which a dialogue will be extended in such a way that it will generate two distinct dialogue games (branching). These cases are when O defends a disjunction, O attacks a conjunction, or O reacts to an attack against a conditional. A dialogue can be seen as a set of dialogue

games. SR-2 is grounded in the fact that the splits generated by P’s choices are not pertinent when the validity of the thesis is at stake

(SR-3) Prime formulae (formulae without connectives or quantifiers) can be uttered for the first time only by O. The Proponent can utter an atomic formula only if the same formula was already uttered by O. Atomic formulae cannot be attacked. SR-3 assures the formality of the dialogue: the winning of P doesn’t depends on the meaning of the prime formulae.

(SR-4) A dialogue game is closed if and only if the same atomic formula appears in two subsequent positions, one uttered by O and the other by P. Otherwise the game is still open. P wins if and only if the game is closed. A

dialogue game is finished if and only if either it’s closed or there is no further move to make according to the rules. O wins if and only if the game is finished and open.

(SR-5) (A rule which assures that the course of the dialogue is finite, limiting the repetitions of the attacks; the formulation of this rule is a bit cumbersome and we will skip it here)

Justification rules for prime formulae

• They determine the “actions” that the player has to execute defending a prime formula he asserted.

• The favorite example: prime formulae of arithmetic. Arithmetic is interpreted as a game with “figures”. Justification rules are the construction rules according to which such “figures” are build. The defense of a prime formula consists in the construction of a figure.

• They fix the operational meaning of the prime formulae, the content semantics of the dialogue. The players have preliminarily agreed upon such meaning of the prime formulae

• As far as logical concepts are concerned, the implementation of (SR-3) permits to skip the issue.

Criterion of validityThe formula stating the thesis is said to be valid if and only if P, in accordance with the appropriate rules, can succeed in defending the thesis against all possible allowed criticism by O. P possesses a winning strategy for the thesis, so that he is sure to win every game of the dialogue, whatever O-moves are. (The dialogical tableau for the formula is closed)

Intuitionistic vs. Classical Logic• One of the most fruitful application of dialogical logic: to encompass in a unique framework different notions of logical validity.

• Differences between logical systems are conceived as difference in the sets of structural rules

• The difference between classical and intuitionistic logic is grasped by means of a liberalization: the constraints that limit how to implement particle rules are stricter in intuitionistic dialogues than in classical dialogues. Indeed, in the second case, a wider range of moves is available to the proponent: precisely, he can apply with more freedom the defenses, referring to precedent challenges of the rival.

(SR-1intuitionistic) Whenever player X is to play, he can attack any move of Y or defend against the last attack of Y, provided he has not already defended against it. A player may postpone a defense as long as there are attacks that can be put forth.

(SR-1classical) Whenever player X is to play, he can attack any move of Y or defend against any attack of Y (even the ones against which he has already defended). In other terms, players can play again earlier defenses.

The law of the excluded middleINTUITIONISTIC TABLEAU

Applying (SR-1intuitionistic), P cannot defend twice the same formula. In this way, the tableau results open, because after the third move (in which O asserts A reacting to the negation stated by P) there is no particle rule that P can implement

O P

?˅ A

A ˅ ¬A ¬A

The law of the excluded middleCLASSICAL TABLEAU

This tableau represents the course of a dialogue about “A ¬A” applying (SR-1classical). According to the set of dialogical rules for classical logic, the formula at stake is valid: the tableau is closed, because in the last move P defends for the second time the thesis, asserting the first disjunct, which corresponds to the formula that O has just stated.

O P

?˅ A

A ˅ ¬A ¬A A

The difference between classical and intuitionistic logicA PROOF-THEORETIC ACCOUNT

•Different systems of natural deduction with two different sets of inference and deduction rules. Different inference rules for ( A =def A→ )

INTUITIONISTIC: CLASSICAL: ( A) A

A

•If inference rules are constitutive of the meaning of the logical constants, then negation has two different meanings in classical and in intuitionistic logic.

The difference between classical and intuitionistic logic

A MODEL-THEORETIC ACCOUNT

• Two different conceptions of Truth (the concept of truth is the cornerstone of model theory)

• CLASSICAL LOGIC: realist notion of truth, or a similar conception, that assures the principle of bivalence.

• INTUITIONISTIC LOGIC: epistemic notion of truth: the truth of a sentence consists of our ability to verify it. To assert P is to have a proof of P, and to assert not-P is to have a refutation of P. A refutation of not-P is not ipso facto a proof of P. The principle of bivalence can be discussed.

The dialogical account of Int vs Cl, instead

• can’t be referred to differences in the meaning of logical constants (particle rules are the same in both logics),

• can’t be referred to differences in general philosophical conceptions,

• because it consists of the difference between two games (determined by two different structural rules).

Epistemic difference between the two games: the classical version allows P to revise her own former utterances, resorting to information subsequently conceded by O; the intuitionistic version reflects a situation in which two parties are speaking just one against the other, without possibility of self-correction.

Philosophical remarksSYNTAX – SEMANTICS - PRAGMATICS

• Proof-theoretic approach: preeminence of syntax

• Model-theoretic approach: preeminence of semantics

• Dialogical approach: preeminence of pragmatics, encompassing syntax and semantics?

An open line of research:

HOW INCOMPLETENESS GET MANIFESTED IN DIALOGUES?• Study of incomplete logics (e.g. Second

Order Logic)

• Study of incomplete theories? Let’s take an axiom theory in which

Gödel’s incompleteness theorem holds. The axioms are the premises (they appear in Opponent’s column). The Proponent states as thesis an undecidable formula, constructed according to Gödel’s instructions (Gödel’s sentence). Can we find a winning strategy for P?

O P

Q1, Q2, Q3, Q4, Q5, Q6, Q7

…

…

GQ

...

…

The case of Robinson arithmetic

Under intuitive interpretation, G states its own unprovability in Q (“G is true but unprovable”)

Does this tableau get closed? If that were the case, the dialogical approach would prove to be irreducible to the proof-theoretic one.