Negation, material equivalence, and conditioned nonconjunction: completeness and duality (original) (raw)
Two restrictions on possible connectives
UCLA Working Papers in Linguistics, Theories of Everything, 17(18), p.154-162, 2012
If languages could lexicalize arbitrary truth tables as sentential connectives, we should be able to find a great variety of connectives in the world's languages. However, very few connectives are typologically attested, as has long been known. For example, no known language lexicalizes if-and-only-if, 's schmor, a connective that returns true exactly when two or more of its arguments are true. In fact, makes the point that the only bona fide non-unary connectives are ∧ and ∨. This typological puzzle calls for an explanation, and indeed several proposals have been suggested in the literature. 1 We examine two approaches to restricting the possible connectives. Both follow Mc-Cawley (1972), and more specifically , in assuming that connectives can only see the set of truth values of their syntactic arguments. As Gazdar notes, this eliminates sensitivity to ordering and repetitions. The first approach, growing out of and , takes the notion of choice as its starting point: if O is a connective and A its argument, then O(A) ∈ A. For example, if all the arguments are true, A = {1}, and O(A) is 1. We will refer to this as the choice-based approach. 2 The second approach, growing out of Faltz (1978, 1985), takes ordering as its starting point: the domain of truth values is assumed to be ordered, with 0 < 1, and a connective can only choose the maximum or minimum element within its argument. We will refer to this as the ordering-based approach. 3 In the classical domain, choice and ordering seem to predict the same sentential connectives. The perspectives they offer are different, though, offering the hope of divergent predictions if we go beyond the classical domain. A tempting place to look is non-classical semantics, used to implement the Frege-Strawson program for presupposition. 4 The challenge here is that there are many trivalent extensions of the classical operators, but only one c 2012 This is an open-access article distributed under the terms of a Creative Commons Non-Commercial License
Logica Universalis, 2011
Combined connectives arise in combined logics. In fibrings, such combined connectives are known as shared connectives and inherit the logical properties of each component. A new way of combining connectives (and other language constructors of propositional nature) is proposed by inheriting only the common logical properties of the components. A sound and complete calculus is provided for reasoning about the latter. The calculus is shown to be a conservative extension of the original calculus. Examples are provided contributing to a better understanding of what are the common properties of any two constructors, say disjunction and conjunction.
A 4-valued logic of strong conditional
South American Journal of Logic, 2017
How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions, we argue that a proper account of conditional can be obtained by extending the logical notation of Frege's ideography. From a dialogical redefinition of truth-values as moves in a game, it becomes possible to characterize the logical meaning of " If " , and only " If ". That is: by getting rid of the paradoxes of material implication, whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections. Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed in this algebraic and game-theoretical light.
A Natural Negation Completion of Urquhart's Many-Valued Logic C
Journal of Philosophical Logic, 1998
A → (A → B)) → (A → B), (A → ¬A) → ¬A, respectively]. And the system CI is, we think, interesting from two different points of view: (a) As suggested by Urquhart, multivalent logics can be understood "as the logics of inference from multisets" ([7], p. 106; see [3] and references there). According to this suggestion, C (as remarked by Urquhart himself) and, so, CI seem more adequate than Lw to this "multiset interpretation". (b) In the "concluding remarks" of their reference work on contractionless logic [4], Ono and Komori recommend the study of superintuitionistic logics without the contraction axiom. Now, CI is one of the most interesting items in this class. In what follows we provide Routley-Meyer type relational semantics (see ) for CI with negation defined either as a primitive connective or by means of a falsity constant. In this sense, we note that the reader can find in the development of these semantics for CI some technical "detours" not required in the case of the standard semantics: unlike the
Against Conjunctive Properties
Acta Analytica, 2020
I put in question in this article the existence of conjunctive properties. In the second section, after having provided a characterization of conjunctive properties, I develop an argument based on the principle of ontological parsimony: if we accept that there are conjunctive properties in the universe then, ceteris paribus, our ontology turns out to be less ontologically parsimonious than if we reject them. Afterwards, in the third section, I distinguish between maximalist and non-maximalist and reductionist and non-reductionist theories of conjunctive properties., as well as between reductionist and non-reductionist theories that reject conjunctive properties. Such distinctions help to clarify the options at hand when accepting or rejecting conjunctive properties. In light of these distinctions, I then tackle two objections against the argument from ontological parsimony. Finally, in the remaining sections, I deal with two arguments defended by D. M. Armstrong for the existence of conjunctive properties: the argument from infinite complexity and the argument from causal powers. I show that there are several ways to resist these arguments and their conclusion.
Monofunctional Sequent and Negation-Tree Systems for the Standard Propositional Calculus
We define as mono-functional a formal system which provides grammatical accommodation and regulatory decision-procedural mechanisms for only one function symbol. In the history of Modern Logic, many pioneers have been keen on constructing such idioms for the material implication and material equivalence functions (respectively constituting an Implicative and an Equivalential calculus language) for the standard sentential logic. Such systems were axiomatic, in the Hilbert style, in earlier periods in the history of modern logic. Those are properly considered Fragments of the Sentential Logicor, syntactically approached, calculi for such fragments -since their proven theses are subsets of the set of theses of the standard two-valued Sentential Logic. The emergence of proof-theoretic approaches, originated by Gerhard Gentzen, resulted in the development of sequent-and natural-deduction formal systems. The systems I present here have a prooftheoretic visage to them but they can be thought of as stimulated by the standard modeltheoretic view -so that, when desired, soundness and completeness results can be motivated with reference to the familiar truth table. The proof-theoretic approach opts for a radically different view of logical procedures -placing foundational emphasis on derivability as a process and capturing the meanings of logical constants by means of procedural rule-based operations for