7,401 research outputs found

    Some Strong Conditionals for Sentential Logics

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    In this article I define a strong conditional for classical sentential logic, and then extend it to three non-classical sentential logics. It is stronger than the material conditional and is not subject to the standard paradoxes of material implication, nor is it subject to some of the standard paradoxes of C. I. Lewis’s strict implication. My conditional has some counterintuitive consequences of its own, but I think its pros outweigh its cons. In any case, one can always augment one’s language with more than one conditional, and it may be that no single conditional will satisfy all of our intuitions about how a conditional should behave. Finally, I suspect the strong conditional will be of more use for logic rather than the philosophy of language, and I will make no claim that the strong conditional is a good model for any particular use of the indicative conditional in English or other natural languages. Still, it would certainly be a nice bonus if some modified version of the strong conditional could serve as one. I begin by exploring some of the disadvantages of the material conditional, the strict conditional, and some relevant conditionals. I proceed to define a strong conditional for classical sentential logic. I go on to adapt this account to Graham Priest’s Logic of Paradox, to S. C. Kleene’s logic K3, and then to J. Łukasiewicz’s logic Ł, a standard version of fuzzy logic

    A 4-valued logic of strong conditional

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    How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions (Section 1), we argue that a stronger account of conditional can be obtained in two steps: firstly, by reminding its historical roots inside modal logic and set-theory (Section 2); secondly, by revising the meaning of logical values, thereby getting rid of the paradoxes of material implication whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections (Section 3). Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed through a broader definition of logical consequence that encompasses both a normal relation of truth propagation and a weaker relation of falsity non-propagation from premises to conclusion (Section 3)

    Intuitionism and the Modal Logic of Vagueness

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    Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness

    Some Concerns Regarding Ternary-relation Semantics and Truth-theoretic Semantics in General

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    This paper deals with a collection of concerns that, over a period of time, led the author away from the Routley–Meyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment

    Non-Fregean Logics of Analytic Equivalence (II)

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    This paper presents the main assumptions of Andrzej Grzegorczyk’s last research project concerning the logic of synonymity. It shows that the basis of logic of analytic equivalence, presented in the first part of the work, fully corresponds with these assumptions.The project is supported by the Polish National Science Centre grant Logic and language experience 2011/03/B/HS1/04580

    Questions and Answers about Oppositions

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    A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth

    The Football of Logic

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    An analogy is made between two rather different domains, namely: logic, and football. Starting from a comparative table between the two activities, an alternative explanation of logic is given in terms of players, ball, goal, and the like. Our main thesis is that, just as the task of logic is preserving truth from premises to the conclusion, footballers strive to keep the ball as far as possible until the opposite goal. Assuming this analogy may help think about logic in the same way as in dialogical logic, but it should also present truth-values in an alternative sense of speech-acts occurring in a dialogue. The relativity of truth-values is focused by this way, thereby leading to an additional way of logical pluralism

    Suszko's Problem: Mixed Consequence and Compositionality

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    Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for non-permeable logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.Comment: Keywords: Suszko's thesis; truth value; logical consequence; mixed consequence; compositionality; truth-functionality; many-valued logic; algebraic logic; substructural logics; regular connective
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