LOGIC AND EXPERIENCE IN THE LIGHT OF DIALOGIC LOGIC

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1 Bulletin of the Section of Logic Volume 12/4 (1983), reedition 2008 [original edition, ] Witold Marciszewski LOGIC AND EXPERIENCE IN THE LIGHT OF DIALOGIC LOGIC 1. There seems to be something mysterious about alications of formal systems, including those of logic, to emirical reality 1. If logic is to be alied to emirical situations, like those described in an ordinary language, then it seems to some eole its statements cannot be necessary, or analytic, roositions. However, they are both alicable and necessary. This suosed uzzle constitutes a significant art of the roblem of hilosohical foundations of logic 2. To this mind of the resent writer, there is no mystery of alications, since any emirical, or even ostensive, redicate can be involved in certain meaning ostulates, e.g. No red is green, which are both emirical and necessary roositions; emirical as they involve ostensibly defined redicated; necessary, as it is enough to know their meaning to state, their validity. The case of logical theorems can be considered at the same footing, logical constants being treated as defined ostensively. Nevertheless, not an individual state of mind but the actual state of scholarly discussions makes something a roblem; in the resent state of foundational discussions on logic, the mystery of alications reserves its vitality, hence any new way of dealing with it roves welcome. 1 This aer was resented at the conference in Jab lonna in the version bearning the title: On the dialogical foundations of logic. 2 A clear exosition of the roblem of relations between logic and exerience has been given by K. Ajdukiewicz in his essay Logika a doświadczenie (1947), contained in the collection Jȩzyk i oznanie, vol. 2, PWN 1965, and also in his Pragmatic Logic, Reidel-PWN 1974,. 198 ff.

2 174 Witold Marciszewski 2.1. A new way is offered by dialogic logic as created by P. Lorenzen with collaboration of K. Lorenz 3. Dialogic logic is an inferential system using only indirect roofs. Proof stes are distributed into two dialogueing arts called roonent and oonent : the former states the roosition to be discussed and defends it, the latter attacks this roosition, each attack being followed by a move of the roonent which is either a defense or a counterattack. There are two kinds of rules in dialogic logic: (i) rules of inference, all of them being elimination rules, i.e. rescribing how to dro the main (in the formula in question) logical constant in the rocess of decomosing the initial roosition; (ii) structural rules, i.e. those which rescribe a sequence of layers moves, e.g. whether defence is to follow immediately after attack, or may be ostoned. The latter kind of rules is crucial for that urose of dialogical logic which consists in drawing a demarcation line between classical and intuitionistic logic. Elimination rules are related to another urose of dialogic logic which is to show roots of logic in the ordinary language as used in emirical situations and conversational activities. Within the limits of this aer only the elimination rules, as related to its subject, can be taken into consideration Here are the elimination rules of dialogic logic, each of them designated by the symbol whose elimination the rule deals with, together with the indication whether the rule in question is concerned with defense (D) or refutation (R), i.e. attack. (D.&) To defined A&B, rove A and rove B. D. ) To defined A B, either rove A or rove B (or both). (R. ) To refute A, rove A. (R. ) To refute A B, rove B. (D ) To defend A B, when A is roved, rove B. (D.U) To defend (Ux)A(x), rove any instance of A(x) as required by the oonent, e.g. A(b). 3 See P. Lorenzen and K. Lorenz: Dialogische Logic, Wissenschaftliche Buchsellschaft 1978, Darmstadt. See also relevant essays in: K. Lorenz (ed.), Konstruktionen versus Positionen: Beiträge zur Diskussion um die Konstruktive Wissenschaftstheorie, Band II Allgemeine Wissenschaftstheorie ; Walter de Gruyter 1979, Berlin New York. In these books there can be found many other bibliograhical references concerning dialogic logic.

3 Logic and Exerience in the Light of Dialogic Logic 175 (D.E) To defend (Ex)A(x), rove any instance of A(x) whatever, e.g. A(b). The roonent wins the game, i.e. succeeds in roving that a formula is a theorem of logic, when the oonent in his attemts at refutation necessarily turns inconsistent, either by assuming two contradictory statements (each in a searate move), or by assuming just this sentence which is required by the roonent to defined his osition 4. The oonent wins the game if he saves himself from the inconsistency, while all the ossibilities of defense have been exhausted by the roonent Before we discuss how the dialogical elimination rules are related to our everyday exerience, let us state the following eistemological rincile: the construction or the acquisition of new concet by means of ostensive definitions or analogous devices requires certain resuosed notions (the term resuosed is used here as less committing then the term a riori ). For instance, when we learn a new concet, i.e. the meaning of a redicate, by means of an ostensive definition, then what must be resuosed is the notion of a set which is necessary to gras the function 4 Here are examles of dialogical game-roofs of logical theorems, with O standing for the oonent and P for the roonent. ( ) ( ) ( ) ( ) (( q)&) q ( q)& 1? q 2? q q (( q& ) q ( q)& 1? q 2?? (refers to the disjunction above) (( q) ) ( q) q As to the last move, 0 has two otions: if he chooses, then he gets inconsistent by assuming both and (as above) ; if he chooses q, then he rovides P with the assertion required for the defense of the conditional in question.

4 176 Witold Marciszewski of the redicate as redicate: otherwise the examle object resented in the ostensive definition would not be regarded as a reresentative of redicate s extension (still more is resuosed in ostensive definitions, but let this examle suffice). Now, it is assumed in the hilosohy of dialogic logic (as understood by the resent author) that the meaning of a logical constant can be defined in certain situation contexts that oerate like ostensive definitions. If so, we are to describe how this is carried out, and to find resuosed notions. The notions we find are: that of obligation and that of roof, or demonstration, together resulting in a more involved notion: the obligation to rove Let us note that the whole of our social life is built uon a structural framework (a skeleton, so to say) of mutual obligations, or duties, and their converses called rights. Without such an obligation framework there would be amorhous collections of individuals instead of solid social structures. These observations are relevant for the roblem of foundations of logic since they show how fundamental the idea of obligation is for human thinking. Rewards and unishments as alied in the teaching of a articular obligation are intelligible for the erson being taught only when the general notion of obligation is rovided. The idea of obligation contributes to definitions of logical constance in the following way. First, let us note that it is ossible to combine an ostensive definition with a contextual one; such a combination aears when we define an oeration (which requires a contextual definition) in a situation in which the truth of oeration arguments is being ercetually ostended. Such a rocedure may be called: the definition by a situational context. Second, let us note that the fulfilling of the duty of roof by a erson x can be rewarded by a sign of aroval (not necessarily a verbal sign) of a erson y, and this sign is to mean that the sentence in question has got roven. That the arguments are roven is (in the situations like those here discussed) just what the ersons involved can see with their eyes; this fact is crucial for roblem logic and exerience, since it yields an emirical interretation of logical oerations as due to their emirical origin in situational contexts Let us discuss some examles. The logical constant ASSERTION, e.g. in the form... is true, can be defined in the following situational context. A erson x has the duty to erson y to rove A. If x manages

5 Logic and Exerience in the Light of Dialogic Logic 177 to ostend the state of affairs denoted by A, as observable for both arts, then y s aroval combined with an utterance like A is true, or A has got roven, comletes the situational definition of assertion (as a concet being introduced to the vocabulary of x). The oeration DENIAL can be situationally defined by the context in which someone denies A by the assertion of A (comare rule (R. )). Let a erson x ought to rove DISJUNCTION either A or B, where A and B are ercetual statements; x roves, say, A by ostending the state in question, and thereby he wins the aroval of a cometent erson y (e.g., a native seaker, or a teacher of logic); on another occasion, x roves the same disjunction by roducing B and wins y s aroval as well. This exerience becomes a aradigm of using the logical constant either... or, what for a amounts to entering into ossession of a rule like (D, ). Such a rocess of learning logical oeration (as exressed by logical constants) may be comleted by the situations in which the lack of success of roving is followed by the disaroval of a cometent erson. Thus we can obtain a ragmatic counterart of truth tables in which the truth of an argument would be reresented by the ercetion of an observable state of affairs, the lack of truth by the lack of required ercetion, while for the comound roosition the truth and its lack would be reresented by the aroval and the disaroval of a cometent erson, resectively. Another examle: in some everyday situations we learn how to refute a conditional A B by roducing the assertion of A together with the denial of B; e.g., when I do not agree that a dog bites if it is teased, I tease it and show that it fails to bite. Hence, whoever tries to refute a conditional has to start from roving its antecedent, and this is just what the rule (R, ) tells us. Therefore, the antecedent having been roved, the rule of defense, viz. (D. ) demands that the consequent be roved. 4. Thus, when defining logical constants by situational contexts, we fix their meaning with inference rules, the original situational contexts being aradigms of alication or rules. Theorems of logic are analytic roositions whose validity is totally deendent on the meaning of their logical constants; in this sense they are indeendent of exerience. However, the logical constants themselves have such a meaning that its emirical interretation derives from the very mode of defining them, viz. defining by situational contexts. Now, if somebody sees a uzzle of alication, it is u to him to state roositions about analyticity and emirical alicability

6 178 Witold Marciszewski of logic which would seem to be inconsistent with each other. Deartment of Logic Methodology and Philosohy of Science Warsaw University Bia lystok Branch