Negation, material equivalence, and conditioned nonconjunction: completeness and duality (original) (raw)
Related papers
On Constructive Connectives and Systems
Logical Methods in Computer Science, 2010
Canonical inference rules and canonical systems are defined in the framework of non-strict single-conclusion sequent systems, in which the succeedents of sequents can be empty. Important properties of this framework are investigated, and a general nondeterministic Kripke-style semantics is provided. This general semantics is then used to provide a constructive (and very natural), sufficient and necessary coherence criterion for the validity of the strong cut-elimination theorem in such a system. These results suggest new syntactic and semantic characterizations of basic constructive connectives. CC Creative Commons 2 A. AVRON AND O. LAHAV 4 A. AVRON AND O. LAHAV Definition 2.4. An etcr ⊢ for L is structural if for every L-substitution σ and every T and E, if T ⊢ E then σ(T ) ⊢ σ(E). ⊢ is finitary iff the following condition holds for every T and
The logic with unsharp implication and negation
arXiv (Cornell University), 2023
It is well-known that intuitionistic logics can be formalized by means of Brouwerian semilattices, i.e. relatively pseudocomplemented semilattices. Then the logical connective implication is considered to be the relative pseudocomplement and conjunction is the semilattice operation meet. If the Brouwerian semilattice has a bottom element 0 then the relative pseudocomplement with respect to 0 is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the Ascending Chain Condition, which is trivially satisfied in finite semilattices, and introduce the connective negation x 0 as the set of all maximal elements z satisfying x ∧ z = 0 and the connective implication x → y as the set of all maximal elements z satisfying x ∧ z ≤ y. Such a negation and implication is "unsharp" since it assigns to one entry x or to two entries x and y belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication, respectively, still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. Several examples are presented. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.
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.
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.