Jaakko Hintikka IF LOGIC MEETS PARACONSISTENT LOGIC

Jaakko Hintikka IF LOGIC MEETS PARACONSISTENT LOGIC 1. The uniqueness of IF logic My title might at first seem distinctly unpromising. Why should anyone think that one particular alternative logic could be relevant to another one? The most important part of a response to this question is to remind the reader of the fact that independence friendly (IF) logic is not an alternative or nonclassical logic. (See here especially Hintikka, There is only one logic, forthcoming.) It is not calculated to capture some particular kind of reasoning that cannot be handled in the classical logic that should rather be called the received or conventional logic. No particular epithet should be applied to it. IF logic is not an alternative to our generally used basic logic, the received first-order logic, aka quantification theory or predicate calculus. It replaces this basic logic in that it is identical with this classical first-order logic except that certain important flaws of the received first-order logic have been corrected. But what are those flaws and how can they be corrected? To answer these questions is to explain the basic ideas of IF logic. Since this logic is not as well known as it should be, such explanation is needed in any case. I will provide three different but not unrelated motivations for IF logic. 2. Quantifiers express dependence One motivation comes from a closer look at the semantical function of quantifiers. This function is often taken to be exhausted by the ranging of quantifiers over a class of values. Frege even proposed construing the existential and the universal quantifiers as C:\IF logic meets paraconsistent logic.090508.doc. 10/24/2008

higher-order predicates of lower-order predicates, expressing their nonemptyness and exceptionlessness respectively. The same view of quantifiers underlies the entire theory of so-called generalized quantifiers which is therefore subject to the same limitations as the ungeneralized first-order logic. This idea nevertheless captures only a part of the semantical job description of quantifiers. Their other important function is to express through the formal (syntactical) dependence or independence of a quantifier (Q 2 y) on another one (say (Q 1 x)) the dependence or independence of the variable y bound to the former of the variable x bound to the latter. Now much of the scientific enterprise consists in establishing dependencies and independencies of variables on each other. Hence any satisfactory logical language ought to be able to express all logically possible patterns of dependence and independence between variables. A language using the received first-order logic does not serve this purpose fully. The reason for this failure is the way in which the formal dependence of a quantifier (Q 2 y) on another one, say (Q 2 x) is expressed in it. As everybody who understands the notation of first-order logic knows, this dependence is expressed by the fact that the scope of (Q 2 y) is included in the scope of (Q 1 x), as in (1) (Q 1 x)( (Q 2 y)(-----) ) But this inclusion relation is of a rather special kind. It is among other things antisymmetric and transitive. Hence only some possible patterns of dependence and independence can be expressed by means of the received first-order logic. In view of the 2

importance of the dependence relations between variables this is a highly significant failure. This shortcoming of the received classical first-order logic is eliminated in IF logic. This can be done in different ways. We could merely relax the conventions governing the scope relation, without any new notation. A more practicable way is to introduce a slash notation that exempts a quantifier (Q 2 y) occurring within the formal scope of (Q 1 x) from its dependence on (Q 1 y) by writing it as (Q 2 y/x). The result is IF first-order logic. It could also be formulated without any new notation, merely by relaxing the conventions governing the (formal) scope of quantifiers, that is, governing the use of parentheses. Even though such a formulation is too clumsy to be feasible in practice, it illustrates the fact that IF logic is nothing but the received classical firstorder logic extended so as to make it more flexible. 3. Some characteristics of IF logic Moreover, some important propositions turn out t be logical truths of IF logic even though they are not logical truths of the received first-order logic. Among them there are the axiom of choice and König s lemma. Likewise, a number of crucially important concepts can be defined by means of IF logic without resorting to higher-order quantification. They include equicardinality, infinity, topological continuity and truth in a sufficiently rich first-order language in terms of the same language. In the received logic, they can be expressed only by quantifying over higher-order entities. Needless to say, the dependence relations expressible in IF logic (or in the received first-order logic) can also be expressed on the second-order level by asserting 3

the existence of the functions that mediate the dependencies in question. In technical logic, they are known as Skolem functions. They play an important role in the theory (and applications) of first-order logic. For one thing they are the functions whose existence is affirmed by the so-called axiom of choice. Thus to assert a quantificational sentence is to assert the existence of a full set of its Skolem functions This does not take us from the scope of IF first-order logic, for this logic is equivalent with the Σ 1 1 (sigma one-one) fragment of second-order logic in which the existence of Skolem functions can be explicitly expressed. One of the main facts about IF logic is that although all its semantical rules are in a suitable formulation precisely the old classical ones, the negation ~ defined by these rules is a strong dual negation not obeying the law of excluded middle. This is corrected by adding to IF logic a sentence-initial contradictory negation. The result is called extended IF logic (EIF logic), and it can be considered as the true basic logic. By IF logic I will in the following refer as much to EIF logic as to the original independencefriendly logic. 4. IF logic and game-theoretical semantics As was indicated, IF logic (and EIF logic) can be motivated in other ways, too. One intuitive and philosophically interesting way is to formulate first the semantics of the received first-order logic in terms of suitable games in the sense of the mathematical theory of games. Each interpreted first-order sentence S defines such game g(s). They are called semantical games, and they can be thought of as games in which one player ( the verifier ) defends the truth of S against an opponent ( the falsifier ). This is 4

philosophically interesting because of the precise form of the definition of truth in them. Pace Dummett, truth does not mean a win in a semantical game. It means the existence of a winning strategy for the verifier. This is equivalent with the earlier characterization of truth for the Skolem functions of S are nothing but ingredients of winning strategies for the verifier. From this game-theoretical semantics (GTS) for the traditional first-order logic we obtain a semantic for IF logic simply (and naturally) by allowing semantical games to be games with imperfect information. Since it is natural to define falsity as to existence of a winning strategy for the falsifier, the law of excluded middle is tantamount to the determinateness of semantical games. This determinateness fails in IF logic as a matter of (game-theoretical) course which shows that the failure of tertium non datur in IF logic should not be any surprise at all. The upshot is that not all IF first-order sentences are either true or false: Some have an indefinite truth-value. It turns out to be possible to assign probabilities to them in an eminently natural way. 5. IF logic and the definability of truth A third transcendental deduction of IF logic turns on an analysis of the notion of truth for quantificational sentences. When is such a sentence S true? One s first impulse and a sound one is to say that S is true when the witness individuals exist that show (in the sense of displaying) its truth. 5

A sentence of the form ( x)f[x] is thus true if and only if a witness individual b exists that makes F[b] true. A sentence of the form ( x)( y)f[x,y] is true if for an individual there exists a witness individual b which makes F[a,b] true and so on. The latter example shows that witness individuals may depend on other individuals. Hence the existence of all the truth-displaying witness individuals means the existence of all the functions that provide us with them as functions of other individuals. But a moment s reflection shows that these functions are nothing but the Skolem functions that we have already encountered repeatedly. Hence we arrive in this way at the same semantics as was reached by the first two ways of approaching the semantics of IF logic (as well as of classical logic). Now the existence of Skolem functions for an IF sentence S turns out to be expressible by another IF sentence T(S). I will not prove this result here, but I will illustrate the procedure of reaching T(S) from S. This procedure is illustrated by the equivalence of a sentence of the form (2) ( f)( x)f[a(f(x)] (where A(f(x) is the only context in which f occurs in F) with (3) ( x 1 )( x 2 )( y 1 / x 2 )( y 2 / x 1 ) (((x 1 =x 2 ) (y 1 =y 2 )) F[A(y 1 )] In order to see this, we can translate (3) into a second-order form, which obviously is (4) ( f 1 )( f 2 )( x 1 )( x 2 )((y 1 =x 2 ) f 1 (x 1 )=f 2 (x 2 )) & F[A(f 1 (x 1 )] 6

Here the first conjunct says that f 1 and f 2 are the same function. By identifying them in (3) we get back to (2). Other kinds of context in which f can occur in ( f)f(f) require a different but equally feasible treatment. The upshot is the definability of a truth-predicate for a suitable IF first-order language in the very same language. This puts the entire philosophical discussion of different so-called theories of truth to a new light. In logical theory, it opens the extremely important possibility of doing (at least some) of first-order logic by means of the same logic. 6. A link between IF logic and paraconsistent logic What relevance does such an IF logic have for paraconsistent logics? I will not try to answer this question fully here. Instead, I will show how IF logic suggests a large number of questions concerning paraconsistent logic. Their answers will eventually show what the total impact IF logic on the theory of paraconsistent logics will be The two have a different initial motivation, and they may look incommensurable. This prima facie incommensurability can be largely overcome in a simple way suggested by the failure of tertium non datur in IF logic. It makes possible a comparison which in the first place involves only a change in terminology. follows Let us assume that we decide to change the terms for the basic truth-values, as 7

OLD LOCUTION NEW LOCUTION true or indefinite false or indefinite true false indefinite tertium not datur true false true but not false false but not true true and false law of contradiction When a semantically characterized IF logic is expressed in the new terminology, we obtain an apparently new logic that reads in many ways just like a paraconsistent logic. To honor the host town of the meeting that inspired this paper, perhaps we may call this logic paratyconsistent logic. Its prima facie status as a paraconsistent logic prompts a large number of questions many of which point to possible directions of future research or logical theory. Here the more technical questions are considered first. So far, paratyconsistent logic has been characterized only semantically. What would it mean to axiomatize it? It would mean giving a recursive enumeration of all formulas that are true in paratyconsistent sense which corresponds to not false in IF logic terms. This can be done, for IF logic has a complete disproof procedure. Hence we can ask: What does a complete axiomatization of paratyconsistent logic look like? How is it related to axiomatic systems of paraconsistent logic? This possibility of a complete axiomatization of para(ty) consistent logic might nevertheless easily create a seriously mistaken impression. Such an axiomatization means simply a recursive enumeration of logically true formulas. In IF logic, there exists 8

a similar axiomatization of logically false (inconsistent) sentences, which is why in a paraty consistent logic logical truths allow a complete axiomatization (i.e. recursive enumeration). But inconsistent formulas are not any longer recursively enumerable in paraty consistent logic. In EIF logic the fragment corresponding to Σ 1 1 second-order logic allows for a complete axiomatization of logical inconsistencies but not of logical truths, 1 whereas its Π fragment allows for a recursive enumeration of logical truths but not of logical inconsistencies. 1 What all this shows methodologically is that the usual approach to different logics by means of construing them as logical systems of axioms and (recursive) rules of inference is not capable of doing justice to IF logic or to paratyconsistent logic. Hence there is no reason to expect that it can do justice to rightly understood paraconsistent logics, either. Formal axiomatization has to be supplemented by semantical methods. This is an important methodological suggestion of our comparison between IF logic and paraconsistent logics. 7. Challenges to and opportunities for paraconsistent logici If IF logic and paraconsistent logic are at bottom identical or can be made identical by developing a suitable new paraconsistent system, then all the achievements of IF logic should be possible to reproduce in paraconsistent logic. We should be able to define the same mathematical concepts by means of paraconsistent logic as can be defined with the help of IF logic although they cannot be defined in conventional first-order logic, including equicardinaltity and infinity. Paraconsistent logic should turn the axiom of choice into a logical truth, as in IF logic. A merger of paratyconsistent logic and 9

paraconsistent logic should also make it possible to formulate some kind of truth condition for a paraconsistent first-order language in the same language. What could it look like? These question are intertwined with yet others. Especially basic are questions concerning negation. What is the natural treatment of negation in paratyconsistent logic? Should the negation that is in fact used in paraconsistent logics be interpreted as the contradictory negation or as some kind of stronger (dual) negation? Some of the problems listed above can only be solved by introducing a second negation into one s logic. How can this be done in the context of paraconsistent logic? Other questions concerning the semantical basis of paraconsistent logics. In order to use it as its own metalogic, the concept of truth for a paraconsistent language must be expressible by means of paraconsistent logic. The analogous truth condition for an IF first-order language can be expressed in the same IF language. But there is a price to be paid for this achievement. The price is that the semantics of IF first-order logic is unavoidably noncompositional. Now to the best of my knowledge noncompositional semantics for paraconsistent logic has not been seriously contemplated. Hence something has to be fundamentally changed if in semantical treatments of paraconsistent logic paraconsistent logicians want to use the bridge provided by paratyconsistent logic for the purpose of reaching a self-applicable truth condition and for the wider purpose of enabling a paraconsistent language to serve as its own metalanguage. In a general theoretical perspective having to give up compositionality is likely to be the deepest methodological change needed for paraconsistent logicans to make use of the kinship of paraconsistent logic with IF logic. 10

These questions and suggestions mean challenges to theorists of paraconsistent logic. Since I cannot speak for them, I must leave actual attempts to follow up these leads to paraconsistent logicians. 8. On the interpretation of paraconsistent logics This does not exhaust the questions that can be raised here. The most important aspect of the comparison between IF logic and paraconsistent logic concerns their interpretation. In one perspective, the impact of IF logic looks like the best thing that could have happened to paraconsistent logic. One of its main weak spots has been its interpretation. We seem to have a reasonable pretheoretical idea of what it means for a proposition to be true and of what it means for it to be false. But what on earth can it mean for a proposition to be true and false? If Cain tells in a cross-examination that it is both true and false that he killed Abel, he might be cited for contempt of the court. The existing discussions of paraconsistent logics do not yield a satisfactory unique account of the concrete down-to-earth meaning of self-contradictory but yet not disprovable propositions. Here the semantics of IF logic, transposed into a paratyconsistent logic, seems to be the answer to paraconsistent semanticist s prayers. The problem of interpreting propositions of paraconsistent logic that are both true and false corresponds to the problem of interpreting the propositions in an IF language that have the truth-value indefinite. And as was pointed out above, the ascription of such a truth-value to a proposition amounts to an assertion of a certain objective fact about the world. It is a mere prejudice that the only way of conveying information about the world is to assert 11

the truth of a sentence. Game-theoretical semantics shows convincingly that this is not the only way. I can equally well inform you about the reality by confiding to you that a certain sentence is false or that it has an indefinite truth-value. Thus here IF logic and its game theoretical semantics seem to provide paraconsistent logic the concrete semantical interpretation it has been missing. If this interpretation (for brevity, I will call it the game-theoretical or GT interpretation) is accepted and put to work, we can expect all the nice results achieved by means of IF logic to be available also to paraconsistant logicians. What else could a logician hope for here? 9. Paraconsistent logic and epistemology A GT interpretation of paraconsistent logic along these lines should thus satisfy all logicians. However, it does not by itself satisfy a critical philosopher of logic. Such an interpretation is objective in that it does not involve any epistemic element. It relies entirely on the meaning relations (rules for semantical games) that are independent of any particular use of language and logic. This objectivity is a great merit of the GT interpretation. But it does not seem to be possible to reconcile with the more or less official motivation of the entire idea of paraconsistent reasoning. The most interesting and most convincing kinds of motivation of paraconsistent logic have been epistemic, not purely logico-semantical. Paraconsistent logic has been supposed to provide us with means of coping with situations in which the different items of information we have received do not agree with each other. The importance of paraconsistent logic is 12

supposed to derive from the importance and frequency of such informational conflicts in our epistemic life. This motivation now seems to be entirely lost, which would might seem to leave paraconsistent logic philosophically high and dry. One possible response here is to suggest that the correlation between the semantics for IF logic and the semantics for paraconsistent logic explained above can apparently provide an epistemic pragmatics for paraconsistent logic and hence vindicate its rationale. One of the most striking recent applications of logical ideas to epistemology is the discovery of a near-identity of the optimal strategies of logical deduction and the optimal strategies of interrogative inquiry, at least in contexts of pure discovery. This strategic significance of logic for inquiry in general can presumably be preserved in the transition from IF logic to paraconsistent logic. In this correlation, the gradual elimination of de facto contradictions which paraconsistent logics are supposed to facilitate corresponds to a gradual elimination of truth-value gaps. This is indeed an eminently viable way of (albeit not the only one) of conceptualizing our epistemological enterprise. It does in fact provide a coherent and realistic account of the epistemological services that paraconsistent logic can provide. This perspective hence provides us with a viable pragmatic and epistemological raison d etre of paraconsistent logic. However, this vindication does not agree fully with the official epistemological motivation of paraconsistent logics. What the gradual filling of truth value gaps means epistemologically is the reconciliation of apparently discrepant items of information or their integration with a coherent total account of the reality. This happens through the acquisition of further information. The most important 13

function of logic in epistemology is to provide strategies for this search for new information. In so far as the paratyconsistent interpretation provides an account of how paraconsistent logics can help this reconciliation process, it vindicates the paraconsistency idea epistemologically. However, in order for them to serve this purpose, the strategic aspects of paraconsistent logics have to be developed more fully than before. Furthermore, and most significantly, this reconciliation process does not involve the outright (or temporary) rejection of any items of information. About the conditions of such rejection, paraconsistent logics do not tell anything non-trivial. But this is not a flaw of paraconsistent logics. It is a fact of a logician s life. No logic in a genuine sense of the term can tell us which putative information to reject or not to reject in a context of actual material inquiry. The reason is that the advisability of rejection cannot be judged fully on the basis of information so far reached in an empirical inquiry. It depends on the total reality itself including its so far unknown parts and aspects. What they are can only be guessed at, not inferred by any logic. Hence paraconsistent logic can serve legitimate epistemological purposes, but not as a logic of belief revision in a sense that would include the outright rejection of any items of prima facie information. If this implies a change in the current philosophical motivation of paraconsistent logic, than that change is unavoidable. Perhaps there is also something of a logical confusion threatening here. Perhaps the illusion that paraconsistent logics could tell us something about the rejection of 14

information is due to an uncertainty about the meaning of negation in paraconsistent logic. It seems that some logicians and philosophers think of it as being basically the contradictory negation. Yet what seems to be the only really viable semantics for paraconsistent logic involves inevitably also the dual (strong) negation, explicitly or implicitly. 10. Different Games One suggestion that may clarify the motivation of paraconsistent logic is to emphasize distinctions between different epistemologically and semantically relevant activities, alias games. The semantical games explained in sec. 4 above in terms of which truth can be defined are not games of truth-seeking. The truth of a sentence S means the existence of a winning strategy, for the verifier in the semantical game G(S) correlated with S. It does not mean to win the semantical game G(S). Hence to try to find the truth of S is to try to find such a winning strategy. Such enterprises are essentially and deeply different from semantical games. I have urged that they should be conceptualized as questioning games. Such interrogative games involve deduction as one component, but they are neither semantical games nor games of formal proofs of logical truth. It seems to me eminently salutary in general to distinguish these three types of games from each other. For instance, establishing the logical truth (truth in all interpretations) of S through a formal proof is something categorically different from establishing the actual truth of S (i.e. truth in the actual world). In particular, the problem of coping with contradictory information belongs squarely to interrogative games, not to semantical games or to the formal games of theorem-proving. Hence paraconsistent 15

logicians might have to reconsider how their ostensive arm of dealing with contradictory information might best be realized. Whether or not a deductive logic can be paraconsistent is one problem. A different question which is not even controversial is that the logic of interrogative games should be paraconsistent in the sense of admitting situations where one s prima facie information involves contradictions. What is significant about the interrogative games that codify empirical reasoning is that contradictions are in them dealt with by means of strategic rules and not by means of move-by-move rules like rules of inference in deductive logic. Maybe the basic idea of paraconsistency should be developed in the context of a strategy theory for interrogative games. This theory is the true logic of belief revision. 16

Literature For IF logic consult Jaakko Hintikka, Principles of Mathematics Revisited, Cambridge U.P., 1996, or its French translation (2007). Gabriel Sandu has a treatise on IF logic forthcoming. The interrogative model of inquiry and its logical foundations are studied in Jaaakko Hintikka, Socratic Epistemology, Cambridge U.P., 2007, and in the essays collected in Jaakko Hintikka Inquiry as Inquiry (Selected Papers, vol. 5), Kluwer Academic (Springer) Dordrecht, 1999. 17

IF LOGIC MEETS PARCONSISTENT LOGIC Relevance of IF logic to work in paraconsistent logic: (1) Possibly (ideally?) translation (2) The semantics of IF logic provides an interpretation of paraconsistent logic. In particular, it shows the interpretation of negation (3) It suggests that in paraconsistent logic, too, we need two negations. (4) The interpretation is completely logico-combinatorial. Is that compatible with the intended applications of paraconsistent logic? (5) & (6) Text Missing (7) IF logic is not axiomatizable. Hence if paraconsistent logic is aimed at completeness and if it is equivalent to IF logic, it is not axiomatizable either. It cannot be characterized in terms of some one logical system in the usual sense that relies only on recursive enumeration. (8) This is reinforced by the fact that IF logic is not compositional. Compositionality is a matter of semantics (interpretation). (9) Problems of dependency in combined logics. (10) Suggestion: Leave the epistemic task to interrogative games or else construe paraconsistent logic as the logic of questioning (alternative interpretation). 18