Logical Options: An Introduction to Classical and Alternative Logics

Logical Options
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Imagine two players that choose to either cooperate or cheat. If both cooperate, they both achieve a reward of 3 points, if they both cheat, they both get nothing, and if one cooperates and the other cheats, the cheater makes off with 5 points and the cooperator gets nothing. If both players are altruistic and motivated to maximize the sum of their rewards, they will both cooperate, as this is the best they can do together. However, they are both tempted to cheat to increase their own reward from 3 to 5. On the other hand, if they are rational, they may recognize that if they cheat their opponent threatens to cheat and leave them with nothing.

So cooperation is the best one can do given this threat. And if each thinks the other realizes this, they may be motivated to cooperate. An extended or iterated version of this game gives the players multiple moves, that is, repeated opportunities to play and collect rewards. See Grim et. In games like Chess, players take turns making their moves and their opponents can see the moves made. This illustrates the interest of games with imperfect information.

The application of games to logic has a long history. GTS has significantly stronger resources that standard Tarski-style semantics, as it can be used for example to explain how meaning evolves in a discourse a sequence of sentences. However, the work on games and modal logic to be described here is somewhat different. Instead of using games to analyze the semantics of a logic, the modal logics at issue are used to analyze games. The structure of games and their play is very rich, as it involves the nature of the game itself the allowed moves, and the rewards for the outcomes , the strategies which are sequences of moves through time , and the flow of information available to the players as the game progresses.

Therefore, the development of modal logic for games draws on features found in logics involving concepts like time, agency, preference, goals, knowledge, belief, and cooperation. To provide some hint at this variety, here is a limited description of some of the modal operators that turn up in the analysis of games and some of the things that can be expressed with them. The basic idea in the semantics is that a game consists of a set of players 1, 2, 3, …, and a set of W of game states.

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This collection of relations defines a tree whose branches define every possible sequence of moves in the game. The semantics also assigns truth-values to atoms that keep track of the payoffs. It is crucial to the analysis of games to have a way to express the information available to the players. One way to accomplish this is to borrow ideas from epistemic logic.

The technical side of the modal logics for games is challenging. The project of identifying systems of rules that are sound and complete for a language containing a large collection of operators may be guided by past research, but the interactions between the variety of accessibility relations leads to new concerns.

Furthermore, the computational complexity of various systems and their fragments is a large landscape largely unexplored. Game theoretic concepts can be applied in a surprising variety of ways — from checking an argument for validity to succeeding in the political arena. So there are strong motivations for formulating logics that can handle games. What is striking about this research is the power one obtains by weaving together logics of time, agency, knowledge, belief, and preference in a unified setting.

The lessons learned from that integration have value well beyond what they contribute to understanding games. One would simply add the standard or classical rules for quantifiers to the principles of whichever propositional modal logic one chooses. However, adding quantifiers to modal logic involves a number of difficulties.

Some of these are philosophical. For example, Quine has famously argued that quantifying into modal contexts is simply incoherent, a view that has spawned a gigantic literature. A second kind of complication is technical. There is a wide variety in the choices one can make in the semantics for quantified modal logic, and the proof that a system of rules is correct for a given choice can be difficult. The work of Corsi and Garson goes some way towards bringing unity to this terrain, and Johannesson introduces constraints that help reduce the number of options; nevertheless the situation still remains challenging.

Another complication is that some logicians believe that modality requires abandoning classical quantifier rules in favor of the weaker rules of free logic Garson The main points of disagreement concerning the quantifier rules can be traced back to decisions about how to handle the domain of quantification. The simplest alternative, the fixed-domain sometimes called the possibilist approach, assumes a single domain of quantification that contains all the possible objects.

On the other hand, the world-relative or actualist interpretation, assumes that the domain of quantification changes from world to world, and contains only the objects that actually exist in a given world. The fixed-domain approach requires no major adjustments to the classical machinery for the quantifiers. For an account of some interesting exceptions see Cresswell The fixed-domain interpretation has advantages of simplicity and familiarity, but it does not provide a direct account of the semantics of certain quantifier expressions of natural language.

However, it seems a fundamental feature of common ideas about modality that the existence of many things is contingent, and that different objects exist in different possible worlds. The defender of the fixed-domain interpretation may respond to these objections by insisting that on his her reading of the quantifiers, the domain of quantification contains all possible objects, not just the objects that happen to exist at a given world. Cresswell makes the interesting observation that world-relative quantification has limited expressive power relative to fixed-domain quantification.

Although this argues in favor of the classical approach to quantified modal logic, the translation tactic also amounts to something of a concession in favor of free logic, for the world-relative quantifiers so defined obey exactly the free logic rules. A more serious objection to fixed-domain quantification is that it strips the quantifier of a role which Quine recommended for it, namely to record robust ontological commitment. However, some work on actualism Menzel, tends to undermine this objection.

For example, Linsky and Zalta and Williamson, argue that the fixed-domain quantifier can be given an interpretation that is perfectly acceptable to actualists. Pavone even contends that on the haecceitist interpretation, which quantifies over individual essences, fixed domains are required. Actualists who employ possible worlds semantics routinely quantify over possible worlds in their semantical theory of language.

By populating the domain with abstract entities no more objectionable than possible worlds, actualists may vindicate the Barcan Formula and classical principles. Note however, that some actualists may respond that they need not be committed to the actuality of possible worlds so long as it is understood that quantifiers used in their theory of language lack strong ontological import.

Furthermore, Hayaki argues that quantifying over abstract entities is actually incompatible with any serious form of actualism. In any case, it is open to actualists and non actualists as well to investigate the logic of quantifiers with more robust domains, for example domains excluding possible worlds and other such abstract entities, and containing only the spatio-temporal particulars found in a given world.

For quantifiers of this kind, a world-relative domains are appropriate. Such considerations motivate interest in systems that acknowledge the context dependence of quantification by introducing world-relative domains. Here each possible world has its own domain of quantification the set of objects that actually exist in that world , and the domains vary from one world to the next.

When this decision is made, a difficulty arises for classical quantification theory. Then this theorem says that it is necessary that Saul Kripke exists, so that he is in the domain of every possible world. The whole motivation for the world-relative approach was to reflect the idea that objects in one world may fail to exist in another.

This seems incompatible with our ordinary practice of using terms to refer to things that only exist contingently. One response to this difficulty is simply to eliminate terms. Kripke gives an example of a system that uses the world-relative interpretation and preserves the classical rules. However, the costs are severe. First, his language is artificially impoverished, and second, the rules for the propositional modal logic must be weakened. Presuming that we would like a language that includes terms, and that classical rules are to be added to standard systems of propositional modal logic, a new problem arises.

The whole idea was that existence of objects is contingent so that there are accessible possible worlds where one of the things in our world fails to exist. Universal Generalization. The rule of Universial Generalization is modified in the same way. A final complication in the semantics for quantified modal logic is worth mentioning.

A term is non-rigid when it picks out different objects in different possible worlds. The semantical value of such a term can be given by what Carnap called an individual concept, a function that picks out the denotation of the term for each possible world.

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So these prerequisites are really just a minimal to bootstrap our study on logic. This is a great book on logic, beautifully written. Do note that this account applies equally well to two theories that rely on the same distinctions possibly combined with different data , as to two theories with different levels of abstraction and possibly the same data. Floridi summarises this epistemological insight as follows: Knowledge is not about getting the message from the world; [but] first and foremost about negotiating the right kind of communication with it. The whole motivation for the world-relative approach was to reflect the idea that objects in one world may fail to exist in another. This language is extended by using , , , for terms and as special symbols that contain a preceding question mark.

However, in a language that treats non rigid expressions as genuine terms, it turns out that neither the classical nor the free logic rules for the quantifiers are acceptable. The problem can not be resolved by weakening the rule of substitution for identity. A solution to this problem is to employ a more general treatment of the quantifiers, where the domain of quantification contains individual concepts rather than objects.

This more general interpretation provides a better match between the treatment of terms and the treatment of quantifiers and results in systems that are adequate for classical or free logic rules depending on whether the fixed domains or world-relative domains are chosen. Texts on modal logic with philosophers in mind include Hughes and Cresswell , , , Chellas , Fitting and Mendelsohn , Garson , Girle , and Humberstone Humberstone provides a superb guide to the literature on modal logics and their applications to philosophy. The bibliography of over a thousand entries provides an invaluable resource for all the major topics, including logics of tense, obligation, belief, knowledge, agency and nomic necessity.

Gabbay and Guenthner provides useful summary articles on major topics, while Blackburn et. What is Modal Logic?

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Logical Options: An Introduction to Classical and Alternative Logics [John L. Bell, David DeVidi, Graham Solomon] on lahocybu.ga *FREE* shipping on. An Introduction to Classical and Alternative Logics Logical Options introduces the extensions and alternatives to classical logic which are most discussed in.

Modal Logics 3. Deontic Logics 4. Temporal Logics 5.

Mathematical Logic

Conditional Logics 6. Possible Worlds Semantics 7. Modal Axioms and Conditions on Frames 8. Map of the Relationships Between Modal Logics 9. The General Axiom Two Dimensional Semantics Provability Logics Advanced Modal Logic Bisimulation Modal Logic and Games A list describing the best known of these logics follows. Conditional and Relevance Logics The founder of modal logic, C. Possible Worlds Semantics The purpose of logic is to characterize the difference between valid and invalid arguments. Diagram of Modal Logics. Bibliography Texts on modal logic with philosophers in mind include Hughes and Cresswell , , , Chellas , Fitting and Mendelsohn , Garson , Girle , and Humberstone An excellent bibliography of historical sources can be found in Hughes and Cresswell Anderson, A.

Barcan Marcus , R. Bartrett and R. Gibson eds. Belnap, N. Perloff, and M. Gabbay and F. Guenthner eds. Reidel, — Benthem, J. Blackburn, P. Bonevac, D. Boolos, G. Bressan, A. Bull, R. Reidel, 1— Carnap, R. Chicago Press. Carnielli, W. Chagrov, A. Chalmers, D. Chalmers ed.

Garcia-Carpintero and J. Chellas, B. Cresswell, M. Goble ed. Cocchiarella, N. Corsi, G. Crossley, J and L. Fitting, M. Gabbay, D. Guenthner, F. Reidel, Garson, J. Girle, R. Grim, P. Denis, P. Goldblatt, R. Gabbay and J. Woods eds. Harel, D. Hayaki, R. Hintikka, J. Hilpinen, R. Hughes, G. Humberstone, L. Although there are many different systems of presentation for logic, there is comparatively little disagreement about which concepts these systems should respect. For present purposes, the key components of a logical system are its logical constants and its consequence relation.

Various different accounts have been proposed for the empirical sciences e. Fine Secondly, we should note that glory need not be transitive: a sequence of glorious revolutions may amount to an inglorious revolution. This could hap- pen if the relative significance of the key components changes sufficiently for some components to cease to be key, or if preservation is itself non-transitive. However, this is less likely in the logical than the empirical case, since the range of possible key components is more narrowly constrained. Of course, inglorious revolutions can cancel each other out, so that characterization is straightforwardly non-transitive.

More- over, the difference between successors and competitors is imprecise; indeed if the terms are understood with sufficient latitude, any successor may be seen as a competitor, since its advocacy is in competition to die-hard de- fence of the old theory, and vice versa, since a successful competitor succeeds the old theory. The adoption of a logic which is a conservative expansion of the antecedent system an extended logic can only represent a revolution if the new material is of key significance. Hence, if the new constants of an ex- tended logic formalize hitherto extra-logical and thereby non-key material, its adoption will be non-revolutionary; but if they formalize material hitherto formalized by the existing constants, the new system will be paragloriously revolutionary.

Note that the question of what a constant formalizes, and thereby the precise delimitation of paraglorious from static extensions, is set- tled by the parsing theory, not by the formal system alone. The consequence relation is always at least apparently preserved because all logical systems have a conception of consequence.

Yet the characterization of consequence could undergo inglorious revolution. It might seem that, in contrast to the constants, any change of consequence relation must be glorious, since the new relation will still be a consequence relation. Most commentators have argued that inglorious revolutions are impossible in mathematics. The ground for denying that inglorious revolutions occur in mathematics is that the discipline is cumulative in a way that empirical science is not: both dis- ciplines discard old material, but mathematicians never really throw it away.

Quaternions or conic sections may be of no greater interest to the modern mathematician than phlogiston or caloric are to the modern physicist, but their legitimacy is not disputed. With rough logic this is much clearer: our concern is with a specific range of research programmes concerned with the formalization of natural argumentation, which are situated within a vast hinterland of smooth logic results.

In this fashion inglorious revolutions are possible within a cumulative discipline. Gray , Hence the ordered pairs are really as indicated by the first letter, but appear to be as indicated by the second. Reality and appearance coincide on the diagonal, hence these sit- uations are how the original four situations were initially understood. Much of the problem here is that where there is genuine confusion or disagreement about the status of a revolution, we will tend to use the same term before and after the revolution: either to describe something which endures through the revolution, or to mis describe two distinct but similar things.

Hence the dispute becomes one of how and whether the meaning of that term has changed. Lakatos inherited from Popper an account of objectivity in terms of the process of discovery, rather than the objects discovered; something of considerable utility in the formal and social sciences, in which the former is much more readily accessible than the latter. It is said to be empirically progressive if some of the excess content is corroborated—if some of the predictions come true ibid.

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Research programmes are progressive if both theoretically and empirically progressive, and degenerating otherwise ibid. If anything, it is easier to describe a logical analogue for empirical content than a mathematical one, since, unlike mathematics, rough logic always has an application. Hence the empirical content of a logical theory is its formalization of inference patterns in natural argumentation where the intuitive validity of these is sufficiently well-entrenched to resist being over- turned in favour of a simpler calculus. When a new theory offers a plausible formalization of patterns of inference hitherto ignored, or judged ill-formed, or unconvincingly paraphrased, it exhibits excess empirical content.

A research programme endures through the sequence of theories of which it is composed as a continuous programmatic component. This consists of two sets of methodological rules: the negative heuristic which counsels against cer- tain lines of enquiry and the positive heuristic which advocates others Lakatos , 48 ff. The chief task of the negative heuristic is to defend the hard core of the programme, that is those propositions fundamental to its character ibid.

The hard core contains the key features of a theory which must be retained in any revision if the successor theory is to belong to the same pro- gramme. Hence a revolutionary change of theory will be glorious iff the hard core is unchanged, paraglorious iff the hard core is monotonically and conser- vatively increased, and inglorious iff the hard core is contracted or revised.

The negative heuristic protects the hard core by ensuring that inferences from contrary evidence are directed not at the hard core but at a protective belt of auxiliary hypotheses: initial conditions, observational assumptions and the like ibid. The research programme is deemed successful if these moves can be achieved progressively; unsuccessful, if they involve degeneration. This assessment of success works to rationalize the conventionalist strategy of pre- serving some propositions from criticism.

We are justified in doing so if the programme thereby exhibits progress, but if we can only do so at the expense of degeneration we may be obliged to revise or abandon our hard core. The other characteristic feature of a research programme is its positive heuristic. A research programme without a positive heuristic would warrant the methodological anarchy recommended by the later Feyer- abend One particular strength of the positive heuristic is that it permits practitioners to postpone consid- eration of apparent refutations of a progressive programme.

Thus anomalies only command atten- tion when the programme is in infancy or degeneration. A good illustration of this is provided by the considerable success of the classical logic research programme in the first half of the twentieth century, which was not signifi- cantly impeded by known anomalies such as the paradoxes of self-reference and of material implication Priest a, f. An issue that is especially pertinent to the rational reconstruction of the development of logic is what one might call the nesting of one research pro- gramme within another. For logic not only develops within its own research programmes, it is also assumed in the development of many other programmes in other disciplines.

We require a more detailed account of scientific devel- opment, distinguishing between the different scopes, or depths of focus, that a research programme may have. However, they would also subscribe—albeit more loosely—to some general research programme of the whole discipline of organic chemistry. If there are theoretical organic chemists within that programme who entertain the prospect of more wholesale revision, the hard core of the programme will be much smaller. Two features of this picture are immediately striking.

First, the hard core of the general programme will be a proper subset of the hard core of the specialized programme. This should tend to limit the size of the hard core and permit wide-ranging speculation as to the direction of future research. For practical purposes, so that a programme may be kept within manageable bounds, it is convenient to augment the hard core of a research programme by additional, conven- tional assumptions. This strategy is permissible within the more specialized programmes of sub-disciplines and specific projects, but methodologically vi- cious if adopted with respect to the discipline as a whole, since it would rule out potentially progressive revision.

Within specialized programmes individ- ual researchers may harmlessly differ over which aspects of the hard core are conventional. The latter would represent an irrevisable finished science. As an official view, this would have attained a state presumably unattainable by mere mortals;22 as a conventional view, it would represent the cessation of scientific curiosity. On a realist account of science this end point as an official view must be unique. A research programme with an empty hard core would represent the conceptual starting point for science suggested by Carte- sian scepticism.

More practical research programmes are situated between these extremes. Programmes with very small hard cores containing only the most general principles would resemble Foucauldian epistemes Foucault? The content of the hard core of an episteme would be contained within the hard core of all contemporaneous research programmes, making it hard to charac- terize, and especially hard to revise. Close to the other extreme are research programmes concerned with fine-tuning a theory or developing a specific ap- plication. Here most of the content of the theory would be contained in the hard core, although much of this would be assumed by convention.

The array imposes a partial ordering, rather than a total ordering, on research programmes, thereby accommodating incompatible programmes at the same stage of development. For any given programme in the array we can identify a cone of programmes with hard cores which properly include the hard core of the initial programme.

Where the initial programme has the right degree of generality we shall call this cone a research tradition. The overall development of logic is too broad to be assimilated into a single coherent tradition. For example, any starting point from which we could develop both Brouwerian intuitionism, in which certain principles of mathematical intuition are conceptually prior to logic, and classical logicism, in which classical logic is conceptually prior to all of mathematics, would have a hard core little larger than that of the prevailing episteme.

Although it is important to acknowledge the assumptions that the two programmes share, there is insufficient community of content for the cone of programmes containing them both to be a research tradition. Within a given logical research tradition we shall be concerned with re- search programmes at several different depths, which may be outlined as fol- lows. First, there is the initial programme, which characterizes the whole tra- dition, since its hard core is contained within that of all programmes within the tradition. At this stage, the content of the protective belt may still be fairly confused.

If the programme is progressive, successive revisions will yield a more com- pletely articulated logical theory. Much of this theory may then be placed in the hard core by convention, to facilitate fine-tuning the theory. When this has been attained, the whole logical theory will have earned at least a conventional place within the hard core of successor programmes applying the logic to more specific disciplines.

Where a system can be characterized as an extension of a more primitive system, this development will be more piecemeal. Hence, within the classical research programme, the propositional and first order systems are regarded as having attained an optimal fit with natural argumentation, and are placed in the hard core while work continues on issues that are still contentious, such as higher order quantifiers or modal extensions. We can now diagnose the thesis that logic is irrevisable as a confusion between research programmes of different depths within the same tradition.

From the perspective of a more developed programme, a specific system may be taken as irrevisable, but that programme exists within a tradition in which logic may be revised, hence it will always be conceptually possible to revise the system by adopting an ingloriously revolutionary programme within the tradition. We can now see that the research programmes of a logical conser- vative and a logical reformer differ not so much in the content of their logical theories, as in the partition of this content into hard core and protective belt.

The conservative insists on placing the whole formal system within the hard core, and redirecting any apparently conflicting evidence at aspects of the parsing and background theories within the protective belt. As a conven- tional expedient this could be advantageous, but the conservative regards this as an official view. Thus the supposed irrevisability of logic is relativized to the research programme of the logical conservative.

Within that programme, logic is immune to revision, but the programme is not unique, and not guaran- teed to succeed. In this sense, both Kant and Frege were justified in regarding logic as non-revisable, despite having different logics. Conversely, a move from classical to non-bivalent logic would be outlawed by the negative heuristic of the conservative programme, but advocated by that of some re- form programmes.

Since both programmes are progressing, we are not yet motivated to abandon either. The move to an extended logic need not induce a change of research pro- gramme: since extended logics do not conflict with the rules of the logic from which they are derived, the syntactic component of the hard core of the re- search programme of that logic may be preserved.

Hence an extension may be an admissible change of theory within a research programme. Of course, this is not to say that such a move will always be welcome: the positive heuris- tic may point elsewhere or the extension may lead to a conflict with hard core aspects of other areas, such as proof theory or semantics, or inferential goals or background theories. If the hard cores of logical research programmes contained all the rules of inference of their formal systems, the adoption of a non-conservatively revisionary system would always require a change of programme.

However, at the stage of a research tradition at which logical reform is entertained, we have argued that the hard core should contain only a partial characterization of the system. Hence glorious deviations should not always initiate a new programme. An important requirement for this model of scientific change is an ac- count of when research programmes and traditions should be abandoned. In essence, the story is the same as that for change of theory within a research programme: a programme should only be replaced by a rival with greater heuristic or explanatory power, that is, if the rival can explain everything that the original programme does, as well as some novelties.

However, nov- elties may be obvious as such only in retrospect, particularly when they turn on the reinterpretation of elements of the original programme or tradition. Moreover, a later theory within a defeated programme or tradition may be able to make a comeback; only if no such reply is forthcoming should a pro- gramme or tradition be abandoned. An eventually superior rival may be slow to draw level with and overtake a well-established programme or tradition. The Philosophy of Alternative Logics 15 The positive heuristic of a programme need not have been exhausted for the programme to be superseded by a more successful rival, although the explana- tory potential of a moribund theory should not be overlooked.

In practice, this is unlikely to be a problem as the development of a progressive programme or tradition is likely to hasten the degeneration of its rivals, since its novel facts will represent anomalies for the rivals. Furthermore, it can be productive to work simultaneously on rival programmes within a tradition, or even on rival traditions Lakatos , n. This account of theory change is slow, but sure.

Indeed it is crucial that this should happen, lest we fall into a sceptical relativism. Thus we are now in a position to answer a concern raised by a con- ventionalist account: that in logic a research programme or tradition may be able to defend itself against refutation indefinitely by repeated employment of a strong negative heuristic. However good its negative heuristic, a programme or tradition cannot survive indefinitely in the face of a more explanatory rival.

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Yet where the negative heuristic is especially strong, as in logic, the transition may be very slow. This tardiness motivates a methodological commitment to scientific pluralism; science cannot advance without competition between pro- grammes. It is particularly important that no theory is permitted to achieve a position of hegemony which permits it to dispatch potential rivals before they have developed sufficiently to pose a threat.

Some commentators, for example, Priest a, ff. Loosely speaking, one system of logic recaptures another if it is possible to specify a subsystem of the former system which exhibits the same patterns of inference as the latter system. This has been invoked by several proponents of non-classical logics to argue that their system retains K as a limit case, and is therefore a methodologically progressive successor to K.

In this section we shall advance and defend a new and more precise account of recapture and the character of its reception by the proponents of the recapturing system. We will then indicate some of the applications of classical recapture which this account makes possible. Our account of recapture builds on an account of the equivalence of conse- quence systems developed in Aberdein Equivalence consists in a one-to-one correspondence between equiv- alence classes of the wffs of the systems which preserves the partitions of the classes of inferences into valid and invalid subclasses: Definition 1 L1 is a proper reduct of L2 iff L1 and L2 are inequivalent, W1 is defined on a proper subset of the class of constants of L2 and V1 contains precisely those elements of V2 which contain only elements of W1.

Hence, reduction is the inverse of conservative extension. Formally, we may say L1 extends L2 iff L1 and L2 are inequivalent and L1 is equivalent to a logic which has a proper reduct which is equivalent to L2. However, reducts are not the only sort of contractions that may be defined upon formal systems; the definition may be generalized as follows: Definition 2 L1 is a proper subsystem of L2 iff L1 and L2 are inequivalent, W1 is a proper subset of W2 and V1 contains precisely those elements of V2 which contain only elements of W1.

The metaphors of strength, size and inclusion which so often illustrate the mereology of logical systems suffer from an ambiguity: there is a tension be- tween a deductive characterization, a measure of how much may be deduced from how little, and an expressive characterization, a measure of the subtlety of the distinctions which can be preserved.

In short, reducts are exclusively generated by reducing the set of constants upon which the class of wffs is based, but subsystems may also be generated by reducing the class of wffs in some other way. For example, K is a subsystem of intuitionistic logic, J. But this subsystem has either an extra axiom or an extra rule of inference. This apparatus provides the means for a formal account of recapture. If L2 is K, then L1 is a classical recapture logic.

The Philosophy of Alternative Logics 17 That is to say that if one system recaptures another we may express within it some finite constraint by which a subsystem equivalent to the recaptured system may be generated. For example, we can see that J is a classical re- capture logic, with the constraint of decidability.

The relevance system R has also been claimed to recapture K, with the constraints of negation con- sistency and primality see Mortensen Quantum logic also recaptures K, with the constraint of compatibility. Indeed, many non-classical logics are classical recapture logics: exactly which will turn on which constraints are deemed expressible. It has even been suggested that the recapture of K is a necessary criterion of logicality, in which case all logics would be classical recapture logics.

Some attempt to reject it outright or deny its significance, others em- brace it, while others see recapture results as motivating the reduction of the recapturing system to a conservative extension. Thus, before recapture can contribute to the understanding of how logical systems change, we must dis- tinguish amongst the variety of responses that advocates of a system may make to the prospect of recapturing a prior system typically K.

We shall order these responses by analogy with a spectrum of political attitudes: rad- ical left, centre left, centre right and reactionary right. This is a formal not a sociological analogy: we do not intend to imply that views on logic may be correlated to political allegiance pace some sociologists of scientific knowl- edge. The spectrum of attitudes to the recapture of the prior system L may be summarized by the table in Fig. I The most extreme attitude is the radical left: formal repudiation of recap- ture status.

Individuals of this tendency deny that their system recaptures the prior system, claiming that no suitable recapture constraint is express- ible in the new system. If classical recapture were a criterion of logicality, then a radical-left response could only be embraced by quitting the discipline of logic. Yet such a criterion must be open to doubt, since some familiar programmes include proponents from the radical left.

Proponents of this stance argue that the formal equivalence between a subsystem of their system and another system is irrel- evant, since the other system cannot be understood as formalizing anything intelligible in terms of their theory. Hence some advocates of J regard the double-negation translation of K into their system as no more than a curios- ity, since they reject the cogency of classical concepts. To defend a position on the centre left one must demonstrate that conceding more than a technical significance to recapture will induce an intolerable tension between successful problem-solving within the programme and the retention of its key non-formal components, such as the central aspects of its parsing theory.

Thus, although a recapture con- straint can be articulated, it does not correspond to any plausible feature of natural argumentation. On the centre right recapture is embraced as evidence of the status of the new system as a methodologically progressive successor. The meaning invari- ance of all key terms is welcomed in this context, and recapture is understood as establishing the old system as a limit case of its successor. By contrast, left-wing recapture involves a far more comprehensive rejection of the old system, by which its intelligibility is denied, and it is ultimately to be dismissed as an incoherent wrong turning.

This is much more plausible behaviour in a competitor than a successor theory, and suggests left-wing recapture as a criterion for this tricky distinction. This is corroborated by the enthusiasm shown for classical recapture amongst systems typically promoted as succeeding K, and the op- position shown by its self-proclaimed competitors. Most non-classical logics have been defended as successors to K by at least some of their advocates.

Conversely, the most credible left-wing stance is from proponents of J, and it is this system which has the greatest claim to be a true competitor to K, rather than a would-be successor. Least radical of all are the reactionary right, who argue that the subsystem of the new system equivalent to the old system is actually a proper reduct of the new system, that is, that the new system should be understood as extending the old system.

Hence the status quo is maintained: the old system is still generally sound, but can be extended to cover special cases. In this case there is no rivalry between the systems cf. Modal logic may be understood as having successfully completed a move from the centre right to the reactionary right: although it is now understood as extending K, its early protagonists conceived it as a prospective successor system.

Thus, if the reactionary stance is technically feasible, it is the only plausible response to recapture. This represents a dualism with the radical stance, which is also mandated by properties of the chosen formal system. L2 does not rival L1.

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No far left centre left The theory of L2 is a competitor to that of L1. The reactionary agrees with the left-wingers that the constants of the new system have different meanings from those of the old. The difference is that the left wing think that the new meanings must replace the old, whereas reactionar- ies believe that they can be assimilated into an augmented system through employment alongside the old meanings.

The greater the difference between the new and the old constants, the more difficult it is to maintain a centrist position. The full range of options may be seen more clearly as a flow chart, shown in Fig. This chart has been devised to display the consequences of a change of theory in which a specific formal system L2 replaces another L1.

However, it should be stressed that, in the practical development of logical research programmes, a dialectic exists between the choice of formal system and the attitude taken to the recapture of the prior system. Hence, providing that enough of the formal system remains within the revisable part of a logical research programme, there are always two alternatives: embrace the consequences of the formal system, or change the system to resist them.

With this picture in place, we can begin to outline some of the uses to which it may be put. In the first place, we now have the resources to draw some fundamental distinctions between different sorts of theory change. Hence, while certain outcomes are necessitated by formal features, other outcomes are underdetermined by such data alone. Solely on formal data we can observe that rivalry must oc- cur unless one system conservatively extends the other, and that competition must occur unless one system recaptures the other. In its most primitive form this amounts to seeking to acknowledge the anomalies without altering the theory ibid.

Monster-adjustment redefines the purported counterexample into terms which no longer conflict with the theory. Finally, monster-exploiting is the employment of anomalies as motivation for theo- retical innovation and development. Primitive exception-barring, monster- barring and monster-adjustment are strategies from the negative heuristic: they represent increasingly sophisticated methods for resisting the pressure for change exerted by an anomaly.

He imaginatively reconstructs the dialectic implicit in the development of this area of mathematics as a classroom dialogue. The methods discussed above are introduced in turn as increasingly sophisticated responses to the puzzle cases. For instance, if polyhedra are defined to be surfaces rather than solids, then hollow solids no longer count as polyhedra.

Alternatively, puzzle cases may be reconciled with the conjecture by monster-adjustment. In this way the small stellated dodecahedron may be seen to satisfy the Euler conjecture if its faces are counted as sixty triangles, but not if they are counted as twelve pentagrams ibid. Monster- Monster- Grid Low Exploiting. Low High Group Group Fig. III An illustration of the spirit behind this sequence of methods is provided by David Bloor , ff. Group measures the strength of the boundary separating the society from the rest of the world.

High grid, low group societies are preoccupied with internal divisions and indifferent to the actions of strangers. Low grid, high group societies have strong social cohesion, but little internal order, and are inclined to be hostile to strangers. Such open hostility will not work in high grid, high group societies since an excluded stranger might be exploited by an- other sub-group.

Hence individuals within these societies will seek either to justify overall exclusion of the stranger, or to assimilate him into their own sub-group. This structure may be represented diagrammatically Douglas , 82 ff. A diagram of this kind Fig. Thus primitive exception-barring corresponds to indifference, monster-barring to fear and aggression, monster-adjustment to assimilation, exception-barring to well-motivated exclusion and monster-exploiting to op- portunistic exploitation. This picture assembles the different responses into an implicit hierarchy, from decadent primitive exception-barring, through isola- tionist monster-barring, aristocratic exception-barring and whiggish monster- adjusting to free-market monster-exploiting.

So far we have followed Bloor and diverged from Lakatos, for whom socio- logical factors are irrelevant to rational reconstruction in the central assump- tion of the strong programme in the sociology of scientific knowledge: that theories resemble the societies which produce them, thereby associating each strategy with a society in which it is expected to be typical. However, we can retain this picture as an account of the heuristic practices characteristic of dif- ferent stages in the development of research programmes, while abstaining on this sociological assumption.

III as an heuristic context. It is difficult for Bloor to explain how the same societies, the same institu- tions, and even the same individuals can simultaneously contribute to multiple disciplines occupying different heuristic contexts. By decoupling sociological context from heuristic context, it becomes easier to see why each strategy will be hard to defend away from its home quadrant. The underlying point is more easily accepted: a methodological move that does little more than isolate anomalies will not be of much use in an heuristic context in which diversity and experimentation are encouraged.

In this section we shall explain and illustrate the levels of this hierarchy. At the first level is brute indifference to the problem: primitive exception-barring. We can find plenty of examples in logic of refusal to acknowledge the exis- tence of a problem, particularly in the early stages of the development of a programme.

Responses to the paradoxes of implication in the early devel- opment of the classical programme furnish several examples. The next step up are responses which are not revisionary of the formal system. The first of these, delimitation of the subject matter, consists in rul- ing the puzzle cases to be inappropriate for logical formalization. This could be either monster-barring, or, if sufficiently systematic, exception-barring.

The monster-barring variant is typical of contexts where the overwhelming concern is maintenance of the boundary of logicality. Where the emphasis is on describing the limitations of formalization, rather than merely maintaining them, more systematic, and thereby exception- barring, responses result. This assessment, and that of the Begriffsschrift as a contri- bution to a high grid, high group enterprise, is reinforced by the swift recogni- tion by other researchers in the same programme of the incompatibility of this proposal with the heuristic context then occupied by their programme.

This strategy sets out to reinterpret the anomaly in order to reconcile it with the formal system central to the research programme and thus employs monster-adjustment. At its most subtle, this species of monster-adjustment can take the form of an admonition to understand formalized propositions in a particular way, rather than explicit paraphrase. That these three methods can be so closely related is further corroboration for the taxonomy, since they share a heuristic context.

Also employing monster-adjustment are the various proposals to preserve classical logic by a more complicated semantics. More extensive revision of the classical logical programme may still be required. All the above exam- ples of either of the two monster-adjusting steps available to logicians occur in sophisticated and highly structured programmes, generally in response to more radical competitor proposals: high grid, high group heuristic contexts. These typically take the form of a switch to an extended logic in which a satisfactory treatment of the anomalies may be developed.

Numer- ous examples can be furnished by most logical research traditions, involving extension by various sorts of quantifiers, identity functions, set-membership operators and alethic, deontic, temporal, doxastic and other modal operators. This strategy is monster-exploiting—in a modest way—and potentially pro- gressive, although not all anomalies will yield to this treatment. Most of the extensions listed above have been accompanied by rearguard claims that the resulting system is no longer purely logical, or even intelligible.

Examples of both moves may be found in Quine: his claim , 68 that higher-order quantification is mathematics, not logic is of the former kind, whereas his opposition to quantified modal logic Quine is an example of the latter kind. These moves correspond to monster-adjusting and exception-barring moves respectively that they are so controversial suggests that extending a logic is a tactic from a different heuristic context.

Indeed, it is a low grid, low group move—monster-exploiting—in the modest sense that it requires acknowledgement that the formal system is not set in stone. This assessment of conservative extension is clearest where it is the most radical of all proposed responses to the anomaly. A change of inferential goals not motivated by the adoption of incompatible background theories would yield a novel research programme which was not really a competitor to the original, and therefore treated at this level of the hierarchy.

First, Kuhn distinguishes only two heuristic contexts: normal science and crisis. Secondly, normal science is taken by Kuhn to be constitutive of, and dominant within, a whole discipline, not just of a research programme or tradition within a discipline. In the fourth level of the hierarchy we find the responses employing a non- conservative revision of logic. The first of these is restriction of the logic: avoidance of the anomaly by moving to a logic which lacks previously valid inferences and theorems.

This exclusion of the puzzle cases from treatment is systematic, and thereby exception-barring, provided that the calculus re- sulting from the restriction has a finite, well-behaved presentation without which the restriction would be blatantly degenerating. As the revision in- volved cuts deep, solely exception-barring uses of restriction are out of tune with the heuristic context necessary for their deployment, and are seldom encountered as serious reform proposals.

The heuristic context sufficient for restriction characteristically results in a more substantial revision. This is the second sort of non-conservative revi- sionary response: wholesale revision, in which elements of the logical theory beyond the formal calculus are exposed to criticism, and reformulated in re- sponse. These elements, which include metalogical concepts, such as that of consequence, background theories and the inferential goal, are predominantly situated within the hard core of mature programmes.

How does wholesale revision work? Judicious restriction can permit clar- ification, precisification and disambiguation of previously confused concepts. Hence, in Lakatosian terms, the search for motivation for exception-barring steps can lead to a revision through proof analysis of the primitive conjecture here the claim that a given logic is adequate for the for- malization of natural argumentation , and thus constitute monster-exploiting.

What it undoubtedly shows is the adoption of an heuristic context in which more radical methods than had previously been deemed legitimate could be entertained. Finally, we come to a strategy more radical than any yet addressed: change of subject matter cf. We saw above that a change of inferential goal in which the background theories are preserved can occur at the conservatively revisionary level of the hierarchy. But changes of goal can also be precipitated by a non-conservative revision of the background theories. Typically this will alter the motivation of the whole logical enterprise, move the problem into a different area, and change the subject matter of logic.

In so far as goals and the background theo- ries which justify them are deep within the hard core of a programme, their non-conservative revision must initiate a change of research programme, and probably of research tradition. It is superseded by the question of which background theories obtain, and thereby of which goal is being pursued. The proper place for settling disputes of this sort is at the level at which the background theories conflict, not at the level of the different calculi. Any divergence at the latter level is understandable but derivative: they have been designed to meet different specifications.

Therefore the dispute is no longer in the discipline of logic, but rather in whatever discipline threw up the conflicting background theories. However, it is not impossible for goals and background theories to be revised without a change of programme or tradition , if the positive heuristic is specified in sufficiently general terms.

Hence there is a crucial difference between responding to a problem with a novel positive heuristic whereby the goal and background theories are radi- cally changed, and gradually adjusting the goal and background theories, in co-evolution with other aspects of a logical research tradition, while preserv- ing the positive heuristic. The latter move may be understood as wholesale revision, the previous level of the hierarchy, but the former is more profound, and can only be represented as a change in the subject matter of logic, the final level of the hierarchy.

It is important to observe that the non-conservative revision of background theories involved in a change of subject matter need not entail an inglorious revolution in the formal system. Many different non-classical systems have been promoted, particularly in recent years. One might men- tion: modal and multi-modal systems, including alethic, temporal, deontic, epistemic and doxastic modalities; paracomplete62 and many-valued logics; free logic; fuzzy logic; second-order logic; non-monotonic and dynamic logics; resource-sensitive and linear logics; and many other systems.

To stay within a manageable length, and to retain some unity of focus, we have restricted our case studies to a much smaller range. We have concentrated on systems which have been seriously proposed as rival organons to propositional K. The focus on the propositional case is because it is where the classical programme is at its strongest, and because the choice of quantifiers is seldom as fundamental as that of propositional constants. In the second case study we turn to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics.

The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform programme. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals. We shall begin by setting out the distinguishing features of the formal system, and of the two most important programmes: mathematical constructivism and semantic anti-realism. More detailed exegesis exploring the differences and important similarities of these programmes will follow. The origins of intuitionistic logic lie in constructivist philosophy of mathe- matics.

Like much contemporary philosophy of mathematics, constructivism originated as a response to the crisis in the foundations of mathematics caused by the discovery of set-theoretic paradoxes induced by the unrestricted ap- plication of infinitistic methods. Several different schools of constructivism may be identified, but they all achieve this nar- rowing of focus by arguing that the statements of mathematics should be understood in terms of proof rather than classical truth. This makes assert- ing the existence of mathematical objects illegitimate unless there are proofs of the existence of specific examples of each such object, that is to say a means of constructing the object in finitely many steps.

There is a sharp divide be- tween most constructivists and mainstream philosophy of mathematics since constructivism is generally revisionary of mathematics, claiming that certain hitherto acceptable areas of mathematics should be discarded. Hence it would be begging the ques- tion against the intuitionist to regard the existence of classically grounded constructivist programmes as an argument against intuitionism: so to argue would be to presume the priority of classical logic, which the intuitionist specifically disputes Haack , Logic is then no more than a formalization of the language used to describe this activity: if permitted to run unchecked it risks outstripping the intuitions constitutive of mathemat- ics.

For, if constructions are the only warrant for mathematical assertions, the occurrence, in any non- finite domain, of mathematical propositions for which we can construct neither a proof nor a refutation, conflicts with the unrestricted assertion of lem. And the establishment of the lack of a construction establishing the lack of a construction of the proof of a proposition cannot be transformed into a proof for that proposition, contradicting dne.

This amounts to saying that A cannot be shown to be unprovable, which is clearly too weak to establish that A is provable, hence the failure of dne. The first complete axiomatization of a logic meeting the con- straints of the bhk interpretation was developed by Heyting , f. If we temporarily disregard the variant interpretations given to the constants and atomic propositions of the two sys- tems, we can observe that J is a proper subcalculus of K: all theorems and valid inferences of the former hold in the latter, but not vice versa.

One consequence is that the connectives and quantifiers may not be interde- fined in J as they are in K. Although the consensus is to regard this as justified by the bhk interpretation, some con- structivists have demurred. Hence Johansson omits this rule from his system, yielding minimal logic, a proper subcalculus of J, which also satisfies the constructivist constraints.

Some super-intuitionistic subcalculi of K have also been promoted as formalizing constructive reasoning, but none of these systems has attracted the same degree of support as J see van Dalen , ff. This alternative focus on knowability, rather than the narrower notion of provability, makes the programme more readily applica- ble to non-mathematical discourse. It has all the properties of explicit knowledge, save only that it is not explicit.

For the classicist, all propositions have truth values, including proposi- tions whose truth values we are not in a position to ascertain. These so- called verification-transcendent propositions must be either true or false, even though there are no means of determining which.

Logical Options: An Introduction to Classical and Alternative Logics

The crux is a demonstra- tion of the untenability of this position: the manifestation argument. But the truth conditions of verification-transcendent propositions cannot be fully stated. Hence, if meaning is truth-conditional, the meaning of these propo- sitions cannot be fully manifested, thus the propositions cannot be properly understood. Yet such propositions are not unintelligible, so meaning cannot be expressed in terms of classical truth. Instead, Dummett promotes an alternative theory which reduces the mean- ing of terms to the conditions for their warranted assertibility.

This permits an anti-realist account of verification-transcendent propositions which does not forfeit their meaningfulness.

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In particular, it motivates the adoption of J, since that calculus preserves warranted assertibility—by reasoning parallel to that of the bhk interpretation—and is the most natural result of linking the meanings of the logical constants to their assertibility conditions. Alter- natively, but to the same effect, the semantic anti-realist programme can be conceived of as retaining a truth-conditional account of meaning, but with a radically revised account of truth. Hence the anti-realist argues that all truths are in principle knowable, whether by replacing the notion of truth with that of warranted assertibility or by subjecting it to epistemic constraint.

This is not the place for a thorough critique of semantic anti-realism, but we will note certain immediate lines of response. How- ever, the manifestation argument is a challenge not to this principle, but to the thesis that truth may transcend knowability. Dummett himself b, 83 f. The propositions of this theory would respect a logic which was neither K nor J, but rather dual to J: dne would be admissible, but double- negation introduction would not be, and so forth. However, this proposal accepts the revisionary force of the manifestation argument; it merely chan- nels it in an unexpected direction.

Some step of this kind may well be required anyway, to accommodate empirical discourse, which offers inde- pendent motivation for this meaning theory. The two historically substantive programmes outlined above do not ex- haust the possible applications of J as a rough logic. Since the deducibility relation of J is a proper sub- relation of that of K, in such a programme J would be sound with respect to classical semantics, although perhaps tolerably incomplete.

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Something of this kind has been suggested as a response to the sorites paradox Put- nam , f. Although there has been some subsequent discussion, no fully articulated programme has yet emerged. Sketchy as this programme is—and it may well remain so—it still serves to demonstrate that the formalism of J does not in itself necessitate the sweep- ing revisions generally promoted on its behalf.

Although this shows that J could in principle be promoted within a logical theory which was otherwise substantially classical, in practice its adoption has been advocated as resulting from dramatic revisions of classical background theories. The closest relationship that can obtain between two logics is equivalence, but J is inequivalent to K. The two systems may be formulated with the same atomic propositions, the same constants at least typographically , and therefore equiform classes of wffs and of sequents.

However, the two classes of sequents would be partitioned into valid and invalid subclasses in a different fashion, hence J would appear to be non-conservatively revisionary of K. The only difficulty with this assessment is that there are several well known ways of embedding K into J. Each of these approaches is a variation on the double-negation translation, which maps classical wffs to intuitionistic wffs in such a way that the validity of sequents in which the wffs occur is preserved, in a sense to be made precise below.

If this were so, it would be either because K was equivalent to a proper reduct of J, by the double- negation translation, or because J was equivalent to an established extension of K, by one of the translations into S4. The gmt translation can be shown to preserve deducibility as well Epstein , , contra Haack , However, it is a translation into S4: there is no corresponding map from S4 to J.

Hence J has not been shown to be equivalent to S4. The more serious proposal is that a double-negation translation might es- tablish that K is equivalent to a proper reduct of J. All four mappings are required for equivalence. It seems unlikely, although conceivable, that any mapping sufficiently ingenious to preserve both validity and invalidity could be found. Moreover, it can be shown that the double-negation translations do not preserve any of the presently available semantics for J, so any such pro- posal would also require perhaps unattainable semantic innovation Epstein , Establishing the equivalence of K to this subsystem would not show that J extended K, but unremarkably that J extended a system non-conservatively revisionary of K.

The next major question about how J is related to K is whether J recap- tures K. In formal terms this is easy to answer. The radical-left strategy of en- suring that recapture does not work is unavailable without revising J, since J recaptures K, so both programmes must be on the centre left. However, this assessment is somewhat overhasty: it is possible to make more productive use of the recapture result.

Yet, although this hostility may be maintained by some intuitionists, in general the situation is more eirenic. In both programmes it is generally conceded that there is a domain of propositions for which K is applicable for example: Brouwer , ; Dummett b, ; b, This would suggest that remarks in- imical to recapture should be taken as hyperbole, leaving open the possibility of a centre-right attitude. There are several reasons why the intuitionist should welcome recapture, but are they enough for a centre-right attitude? For a long time intuitionists were obliged to appeal to K to prove results in the metalogic of J, such as the completeness of the first order system.

Until intuitionistically acceptable proofs were produced Veldman ; de Swart , this provoked the clas- sical criticism that the intuitionist was indulging in a practice that he wished to deny to others Tennant , f. Even now that an intu- itionistic metalogic is practicable, a case may be made that the intuitionist should retain a classical metalanguage, at least as an alternative to the intu- itionistic version.

For, as Dummett points out, insistence on the employment of the logic of a reform proposal throughout the metalanguage serves to in- sulate the proposal from criticism, and at the cost of handicapping its ability to persuade the practitioners of other systems of its merits Dummett , The Gentiles should not really need a translation in this instance, since the deducibility relation of J is a sub-relation of that of K, which ensures that all intuitionistically valid proofs are classically valid too.

Dummett , 55 wishes to maintain the stronger claim, and argues that the metalogic should be as neutral as possible. The crucial difference between the two claims, which Tennant , accuses Dummett of having missed, is that the former cannot ground the latter without opening K, as much as any other system, to the accusation that it is seeking to resist criticism through question-begging self-justification.

The above argument is reprised in the analysis of the constants employed in the bhk interpretation: unless they are understood classically, the inter- pretation cannot explain intuitionistic usage to the classicist Makinson , Fortunately, the domain in which the interpretation is carried out is ef- fectively decidable, and thereby recaptured in J. In addition, the Brouwerian account of logic as subordinate to mathematics should be seen as favourable towards classical recapture. If logic is merely the a posteriori codification of valid modes of mathematical reasoning, there can be no objection to some aspects of this reasoning fitting more than one codification cf.

Heyting , This, with the points above, motivates the retention of K as a limit case of J, that is, centre-right recapture. However, against this suggestion it should be recalled that centre-right recapture would require the intelligibility of the inferential goal of K—epistemically unconstrained truth—within the theory of J. At least some proponents of J would regard this as unsustainable, relegating J to centre-left classical recapture. Conversely, it might be possible to move even further to the right, at least within the constructivist programme.

Most constructivists have followed Brouwer in holding that classical mathematical results remain unjustified until a constructive proof is forthcoming. However, there is an alternative tradi- tion in which these results are regarded as having their own, weaker, sort of legitimacy. The logic of real mathematics would then be the stricter system resulting from an extension by independent, constructive, notions of disjunc- tion and existential quantification: J. However, it remains strictly hypothetical: not only does it rely on an equivalence relation that we have no reason to believe obtains, it would also require an argument that disjunction and existential quantification are not intersystemically invariant that is, cannot be identified between K and J.