By a consequence relation on a set L of formulas we understand a relation I ââ¬â c p(L) x L satisfying the conditions called 'Overlap', 'Dilution', and 'Cut for Sets' at p.15 of [25]; we do not repeat the conditions here since we are simply fixing notation and the concept of a consequence relation is well known in any case. (The characterization in [25] amounts to that familiar from Tarski's work, except that there is no 'finitariness' restriction to the effect (...) that when I I ââ¬â A, for I c L, A c L, we must have I o I ââ¬â A for some finite I o c I . The presence or absence of this condition makes no difference to anything that follows.) Each language L to be considered will be a sentential language whose formulas are built in the usual way by application of it-ary (primitive) connectives to it simpler formulas, starting with the simplest formulas ââ¬â the propositional variables (or 'sentence letters') ââ¬â not constructed with the aid of connectives. We assume, as usual, that there are countably many such variables, and they will be denoted by p, q, r, ... possibly with numerical subscripts. A consequence relation I- on such an L has the Unrestricted Interpolation Property when for any A, C c L with A I ââ¬â C, there exists B c L with A I ââ¬â B and B I ââ¬â C, such that C is constructed only out of such propositional variables as occur both in A and in C. (Such a B is called an interpolant for A and C.) Note that we take the usual notational liberties here, writing "A I ââ¬â C" (and the like) for "iAi I ââ¬â C", "I, A I ââ¬â C" to mean "I u iAi I ââ¬â C", and "I ââ¬â C" to mean "8 I ââ¬â C". Further, we sometimes abbreviate the claim that A I ââ¬â B and B I ââ¬â C to "A I ââ¬â B I ââ¬â C", and when C is A itself, we always write this simply as "A ââ¬â IIââ¬â B".. (shrink)
Am(B m B). Specifically I was wondering whether for every BCI-provable formula A there is a B for which the inset formula was provable. If you want to read about this issue, which I..
In a 1990 paper, George Hughes axiomatized the logic determined by the class of all frames in which each point has a reflexive successor, and raised various questions along the way, one of which is answered incorrectly here by means of an interestingly fallacious argument.
Graham Priest has asked whether the consequence relation associated with the Anderson–Belnap system of Tautological Entailment,1 in the language with connectives ¬, ∧, ∨, and countably many propositional variables as tomic formulas, maximal amongst the substitution-invariant relevant consequence relations on this language. Here a consequence relation is said to be relevant just in case whenever for a set of formulas Γ and formula B, we have Γ B only if some propositional variable occurring in B occurs in at least one (...) formula in Γ. (It follows that relevant consequence relations are atheorematic in the sense that whenever Γ B for some such consequence relation , Γ = ∅.) Here I write up in more detail the upshot of the conversation – returning an affirmative answer to Priest’s question – about this in the common room that Greg Restall and I were participating in last Friday [ = October 6, 2006], dotting some “i”s and crossing some “t”s (and adding the odd further reflection). (shrink)
Impossible worlds are regarded with understandable suspicion by most philosophers. Here we are concerned with a modal argument which might seem to show that acknowledging their existence, or more particularly, the existence of some hypothetical (we do not say “possible”) world in which everything was the case, would have drastic effects, forcing us to conclude that everything is indeed the case—and not just in the hypothesized world in question. The argument is inspired by a metaphysical (rather than modal-logical) argument of (...) Paul Kabay’s which would have us accept this unpalatable conclusion, though its details bear a closer resemblance to a line of thought developed by Jc Beall, in response to which Graham Priest has made some philosophical moves which are echoed in our diagnosis of what goes wrong with the present modal argument. Logical points of some interest independent of the main issue arise along the way. (shrink)
We study a range of issues connected with the idea of replacing one formula by another in a fixed (linguistic) context. The replacement core of a consequence relation ⊢ is the relation holding between a set of formulas { A 1 , ..., A m , ...} and a formula B when for every context C (·), we have C ( A 1 ), ..., C ( A m ), ... ⊢ C ( B ). Section 1 looks at some (...) differences between which inferences are lost on passing to the replacement cores of the classical and intuitionistic consequence relations. For example, we find that while the inference from A and B to , sanctioned by both these initial consequence relations, is retained on passage to the replacement core in the classical case, it is lost in the intuitionistic case. Further discussion of these two (and some other) logics occupies Sections 3 and 4. Section 2 looks at the m = 1 case, describing A as replaceable by B according to ⊢ when B is a consequence of A by the replacement core of ⊢, and inquiring as to which choices of ⊢ render this induced replaceability relation symmetric. Section 5 investigates further conceptual refinements— such as a contrast between horizontal and vertical replaceability—suggested by some work of R. B. Angell and R. Harrop (and a comment on the latter by T. J. Smiley) in the 1950s and 1960s. Appendix 1 examines a related aspect of term-for-term replacement in connection with identity in predicate logic. Appendix 2 is a repository for proofs which would otherwise clutter up Section 3. (shrink)
Recently, an improvement in respect of simplicity was found by Rohan French over extant translations faithfully embedding the smallest congruential modal logic (E) in the smallest normal modal logic (K). After some preliminaries, we explore the possibility of further simplifying the translation, with various negative findings (but no positive solution). This line of inquiry leads, via a consideration of one candidate simpler translation whose status was left open earlier, to isolating the concept of a minimally congruential context. This amounts, roughly (...) speaking, to a context exhibiting no logical properties beyond those following from its being congruential (i.e., from its yielding provably equivalent results when provably equivalent formulas are inserted into the context). On investigation, it turns out that a context inducing a translation embedding E faithfully in K need not be minimally congruential in K. Several related minimality conditions are noted in passing, some of them of considerable interest in their own right (in particular, minimal normality). The paper is exploratory, raising more questions than it settles; it ends with a list of open problems. (shrink)
It will be an essential resource for philosophers, mathematicians, computer scientists, linguists, or any scholar who finds connectives, and the conceptual issues surrounding them, to be a source of interest.This landmark work offers both ...
The purpose of the present note is to advertise an interesting conjecture concerning a well-known translation in modal logic, by confirming a (highly restricted) special case of the conjecture.
Jean-Yves Béziau (‘Classical Negation can be Expressed by One of its Halves’, Logic Journal of the IGPL 7 (1999), 145–151) has given an especially clear example of a phenomenon he considers a sufficiently puzzling to call the ‘paradox of translation’: the existence of pairs of logics, one logic being strictly weaker than another and yet such that the stronger logic can be embedded within it under a faithful translation. We elaborate on Béziau’s example, which concerns classical negation, as well as (...) giving some additional background (especially from intuitionistic logic) to the example. Our interest is more on the logical exploration of the phenomenon Béziau’s case exemplifies than on the question of whether that phenomenon is (even prima facie ) paradoxical, though in Section 5 we do approach the latter question – somewhat obliquely – by considering an analogous phenomenon which it is hard to find puzzling. (shrink)
We discuss aspects of the logic of negation bearing on an issue raised by Jean-Yves Béziau, recalled in §1. Contrary- and subcontrary-forming operators are introduced in §2, which examines some of their logical behaviour, leading on naturally to a consideration in §3 of dual intuitionistic negation (as well as implication), and some further operators related to intuitionistic negation. In §4, a historical explanation is suggested as to why some of these negation-related connectives have attracted more attention than others. The remaining (...) sections (§§5, 6) briefly address a question about a certain notion of global contrariety and the provision of Kripke semantics for the various operators in play in our discussion. (shrink)
The phrase ‘autoepistemic logic’ was introduced in Moore [1985] to refer to a study inspired in large part by criticisms in Stalnaker [1980] of a particular nonmonotonic logic proposed by McDermott and Doyle.1 Very informative discussions for those who have not encountered this area are provided by Moore [1988] and the wide-ranging survey article Konolige [1994], and the scant remarks in the present introductory section do not pretend to serve in place of those treatments as summaries of the field. A (...) good deal of the material omitted here pertains to the specifically nonmonotonic nature of autoepistemic logic as standardly developed, but as we shall urge, there is from one point of view nothing distinctively nonmonotonic about the basic motivating ideas of the subject. (shrink)
Our object is to study the interaction between mereology and David Lewis’ theory of subject-matters, elaborating his observation that not every subject matter is of the form: how things stand with such-and-such a part of the world. After an informal introduction to this point in Section 1, we turn to a formal treatment of the partial orderings arising in the two areas – the part-whole relation, on the one hand, and the relation of refinement amongst partitions of the set (...) of worlds, on the other. (We follow Lewis – approximately – in identifying subject-matters with such partitions.) We emphasize a certain duality, formulated in (2.6) and (2.7) in Section 2, between the corresponding lattice operations – mereological joins with partition-lattice meets, mereological meets with partition-lattice joins. Section 3 presents some issues that are raised by consideration of the informally familiar idea of logical subtraction. These include, in particular, a problem about the need for a notion of independence different from the usual logical notion(s) going by that name. The apparatus of Section 2 promises to throw some light on this problem, as we indicate in Section 4. Section 5 ties up some loose ends and suggests an area in which further work would be desirable. (shrink)
We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction (fission, ‘cotensor’, par) against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules (...) are what we call ipsilateral: applying such a rule does not make any (sub)formula change sides—from the left to the right of the sequent separator or vice versa. Some possibilities for a semantic characterization of the resulting logic are also explored. The paper concludes with three open questions. (shrink)
We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we (...) show that for the pure logic of one of these implicational connectives two-in general distinct-consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the 'propositional operations' associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives. (shrink)
The logic of ‘elsewhere,’ i.e., of a sentence operator interpretable as attaching to a formula to yield a formula true at a point in a Kripke model just in case the first formula is true at all other points in the model, has been applied in settings in which the points in question represent spatial positions (explaining the use of the word ‘elsewhere’), as well as in the case in which they represent moments of time. This logic is applied here (...) to the alethic modal case, in which the points are thought of as possible worlds, with the suggestion that its deployment clarifies aspects of a position explored by John Divers un-der the name ‘modal agnosticism.’ In particular, it makes available a logic whose Halldén incompleteness explicitly registers the agnostic element of the position – its neutrality as between modal realism and modal anti-realism. (shrink)
We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we (...) show that for the pure logic of one of these implicational connectives two-in general distinct-consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the 'propositional operations' associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives. (shrink)
Geach’s rich paper ‘A Program for Syntax’ introduced many ideas into the arena of categorial grammar, not all of which have been given the attention they warrant in the thirty years since its first publication. Rather surprisingly, one of our findings (Section 3 below) is that the paper not only does not contain a statement of what has widely come to be known as “Geach’s Rule”, but in fact presents considerations which are inimical to the adoption of the rule in (...) question. With regard to at least some amongst the numerous other points extracted here from Geach’s discussion, we shall not be able to reach so definitive a conclusion, and content ourselves with giving the issues an airing. (shrink)
Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free (...) formula. Thus it would seem that any detour going via formulas with conjunction can be avoided, which raises a puzzle: how is this consistent with the point from Section 1? Sections 3 and 4 raise and discuss this puzzle. In fact, the puzzle turns out on closer inspection not to be so puzzling after all, but it does serve as a convenient centrepiece around which to organize a discussion of the phenomenon illustrated by the Bull?Prior example. Section 5 notes that Prior's observation can be extended to the case of the result of adding disjunction to Bull's logic, while Section 6 includes some further remarks aimed at diagnosing one source of possible residual puzzlement. A subtext of our discussion?spanning several of the notes?is that this work by Bull and Prior has been overlooked, their results having to be rediscovered, by many algebraists and logicians in more recent years. (shrink)
The partitions of a given set stand in a well known one-to-onecorrespondence with the equivalence relations on that set. We askwhether anything analogous to partitions can be found which correspondin a like manner to the similarity relations (reflexive, symmetricrelations) on a set, and show that (what we call) decompositions – of acertain kind – play this role. A key ingredient in the discussion is akind of closure relation (analogous to the consequence relationsconsidered in formal logic) having nothing especially to do (...) with thesimilarity issue, and we devote a final section to highlighting some ofits properties. (shrink)
A form (or pattern) of inference, let us say, explicitlysubsumes just such particular inferences as are instances of the form, and implicitly subsumes thoseinferences with a premiss and conclusion logically equivalent to the premiss and conclusion of an instanceof the form in question. (For simplicity we restrict attention to one-premiss inferences.) A form ofinference is archetypal if it implicitly subsumes every correct inference. A precise definition (Section 1)of these concepts relativizes them to logics, since different logics classify different inferences ascorrect, (...) as well as ruling differently on the matter of logical equivalence which entered into the definitionof implicit subsumption. When relativized to classical propositional logic, we find (Section 2) thatall but a handful of `degenerate' inference forms turn out to be archetypal, whereas matters are verydifferent in this respect for the case of intuitionistic propositional logic (Sections 3 and 4), and an interestingstructure emerges in this case (the poset of equivalence classes of inference forms, with respect tothe equivalence relation of implicitly subsuming the same inferences). Thus a more accurate, if excessivelylong-winded title would be 'Archetypal and Non-Archetypal Forms of Inference in Classical andIntuitionistic Propositional Logic'. Some left-overs are postponed for a final discussion (Section 5).The overall intention is to introduce a new subject matter rather than to have the last word on thequestions it raises; indeed several significant questions are left as open problems. (shrink)
This paper recalls some applications of two-dimensional modal logic from the 1980s, including work on the logic of Actually and on a somewhat idealized version of the indicative/subjunctive distinction, as well as on absolute and relative necessity. There is some discussion of reactions this material has aroused in commentators since. We also survey related work by Leslie Tharp from roughly the same period.
We explore in an experimental spirit the prospects for extending classical propositional logic with a new operator P intended to be interpreted when prefixed to a formula as saying that formula in question is at least partly true. The paradigm case of something which is, in the sense envisaged, false though still partly true is a conjunction one of whose conjuncts is false while the other is true. Ideally, we should like such a logic to extend classical logic – or (...) any fragment thereof under consideration – conservatively, to be closed under uniform substitution (of arbitrary formulas for sentence letters or propositional variables), and to allow the substitutivity of provably equivalent formulas salva provabilitate. To varying degrees, we experience some difficulties only with this last (congruentiality) desideratum in the two four-valued logics we end up giving our most extended consideration to. (shrink)
We assemble material from the literature on matrix methodology for sentential logic?without claiming to present any new logical results?in order to show that Gödel once made (or at least, is quoted as having made) an uncharacteristically ill-considered remark in this area.
We explore a relation we call anticipation between formulas, where A anticipates B (according to some logic) just in case B is a consequence (according to that logic, presumed to support some distinguished implicational connective ) of the formula AB. We are especially interested in the case in which the logic is intuitionistic (propositional) logic and are much concerned with an extension of that logic with a new connective, written as a, governed by rules which guarantee that for any formula (...) B, aB is the (logically) strongest formula anticipating B. The investigation of this new logic, which we call ILa, will confront us on several occasions with some of the finer points in the theory of rules and with issues in the philosophy of logic arising from the proposed explication of the existence of a connective (with prescribed logical behaviour) in terms of the conservative extension of a favoured logic by the addition of such a connective. Other points of interest include the provision of a Kripke semantics with respect to which ILa is demonstrably sound, deployed to establish certain unprovability results as well as to forge connections with C. Rauszer's logic of dual intuitionistic negation and dual intuitionistic implication, and the isolation of two relations (between formulas), head-implication and head-linkage, which, though trivial in the setting of classical logic, are of considerable significance in the intuitionistic context. (shrink)
Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in §1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between the two (...) implications in the more familiar language of modal logic. In §§2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and – after a digression (in §4) on the intuitionistic analogue of the material in §3 – consider them together in §5, closing with a discussion in §6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent §§1–6 from becoming overly distended. (shrink)
Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of (...) what (in Section 2) we call a contra-classical modal logic, an unusual logic boasting a connective (" demi-negation" ) whose double application is equivalent to a single application of the negation connective. Pondering the example points the way to a general characterization of contra-classicality (Theorems 3.3 and 4.6). In an Appendix (Section 5), we look at one alternative to classical logic as the target for such translational assimilation, intuitionistic logic, calling logics which resist the assimilation, in this case, contra- intuitionistic. We will show that one such logic is classical logic itself, thereby strengthening a result of Wojcicki's to the effect that the consequence relation of classical logic cannot be faithfully embedded by any connective-by-connective translation into that of intuitionistic logic. (What the "faithfully" means here is that not only is the translation of anything provable in the 'source' logic.. (shrink)
Whether assent (acceptance) and dissent (rejection) are thought of as speech acts or as propositional attitudes, the leading idea of rejectivism is that a grasp of the distinction between them is prior to our understanding of negation as a sentence operator, this operator then being explicable as applying to A to yield something assent to which is tantamount to dissent from A. Widely thought to have been refuted by an argument of Frege"s, rejectivism has undergone something of a revival in (...) recent years, especially in writings by Huw Price and Timothy Smiley. While agreeing that Frege"s argument does not refute the position, we shall air some philosophical qualms about it in Section 5, after a thorough examination of the formal issues in Sections 1–4. This discussion draws on – and seeks to draw attention to – some pertinent work of Kent Bendall in the 1970s. (shrink)
A sentence mentioning an object can be regarded as saying any one of several things about that object, without thereby being ambiguous. Some of the (logical) repercussions of this commonplace observation are recorded, and some critical discussion is provided of views which would appear to go against it.
Given a 1-ary sentence operator , we describe L - another 1-ary operator - as as a left inverse of in a given logic if in that logic every formula is provably equivalent to L. Similarly R is a right inverse of if is always provably equivalent to R. We investigate the behaviour of left and right inverses for taken as the operator of various normal modal logics, paying particular attention to the conditions under which these logics are conservatively extended (...) by the addition of such inverses, as well as to the question of when, in such extensions, the inverses behave as normal modal operators in their own right. (shrink)
If a certain semantic relation (which we call local consequence) is allowed to guide expectations about which rules are derivable from other rules, these expectations will not always be fulfilled, as we illustrate. An alternative semantic criterion (based on a relation we call global consequence), suggested by work of J.W. Garson, turns out to provide a much better — indeed a perfectly accurate — guide to derivability.
