Dialogical logic

2018 ◽  
Author(s):  
Erik C.W. Krabbe
Keyword(s):  
Winning Strategy ◽  
Logical Consequence ◽  
Logical Form ◽  
Logical Constant ◽  
Critical Dialogue ◽  
Logical Constants ◽  
Dialogue Game ◽  
Dialogical Logic ◽  
Formal Dialogue

Dialogical logic characterizes logical constants (such as ‘and’, ‘or’, ‘for all’) by their use in a critical dialogue between two parties: a proponent who has asserted a thesis and an opponent who challenges it. For each logical constant, a rule specifies how to challenge a statement that displays the corresponding logical form, and how to respond to such a challenge. These rules are incorporated into systems of regimented dialogue that are games in the game-theoretical sense. Dialogical concepts of logical consequence can then be based upon the concept of a winning strategy in a (formal) dialogue game: B is a logical consequence of A if and only if there is a winning strategy for the proponent of B against any opponent who is willing to concede A. But it should be stressed that there are several plausible (and non-equivalent) ways to draw up the rules.

Logical constants

2018 ◽  
Author(s):  
Timothy McCarthy
Keyword(s):  
Fundamental Problem ◽  
Logical Consequence ◽  
Logical Form ◽  
Logical Truth ◽  
Semantic Property ◽  
Truth Conditions ◽  
Property A ◽  
Logical Constants ◽  
Language L

A fundamental problem in the philosophy of logic is to characterize the concepts of ‘logical consequence’ and ‘logical truth’ in such a way as to explain what is semantically, metaphysically or epistemologically distinctive about them. One traditionally says that a sentence p is a logical consequence of a set S of sentences in a language L if and only if (1) the truth of the sentences of S in L guarantees the truth of p and (2) this guarantee is due to the ‘logical form’ of the sentences of S and the sentence p. A sentence is said to be logically true if its truth is guaranteed by its logical form (for example, ‘2 is even or 2 is not even’). There are three problems presented by this picture: to explicate the notion of logical form or structure; to explain how the logical forms of sentences give rise to the fact that the truth of certain sentences guarantees the truth of others; and to explain what such a guarantee consists in. The logical form of a sentence may be exhibited by replacing nonlogical expressions with a schematic letter. Two sentences have the same logical form when they can be mapped onto the same schema using this procedure (‘2 is even or 2 is not even’ and ‘3 is prime or 3 is not prime’ have the same logical form: ‘p or not-p’). If a sentence is logically true then each sentence sharing its logical form is true. Any characterization of logical consequence, then, presupposes a conception of logical form, which in turn assumes a prior demarcation of the logical constants. Such a demarcation yields an answer to the first problem above; the goal is to generate the demarcation in such a way as to enable a solution of the remaining two. Approaches to the characterization of logical constants and logical consequence are affected by developments in mathematical logic. One way of viewing logical constanthood is as a semantic property; a property that an expression possesses by virtue of the sort of contribution it makes to determining the truth conditions of sentences containing it. Another way is proof-theoretical: appealing to aspects of cognitive or operational role as the defining characteristics of logical expressions. Broadly, proof-theoretic accounts go naturally with the conception of logic as a theory of formal deductive inference; model-theoretic accounts complement a conception of logic as an instrument for the characterization of structure.

scholarly journals Tonk Strikes Back∗

2005 ◽  
Vol 3 ◽  
Author(s):  
Denis Bonnay ◽  
Benjamin Simmenauer
Keyword(s):  
Logical Constant ◽  
Double Line ◽  
Rule Based ◽  
Logical Constants ◽  
Inferential Role ◽  
The Right

What is a logical constant? In which terms should we characterize the meaning of logical words like “and”, “or”, “implies”? An attractive answer is: in terms of their inferential roles, i.e. in terms of the role they play in building inferences. More precisely, we favor an approach, going back to Dosen and Sambin, in which the inferential role of a logical constant is captured by a double line rule which introduces it as reflecting structural links (for example, multiplicative conjunction reflects comma on the right of the turnstyle). Rule-based characterizations of logical constants are subject to the well known objection of Prior’s fake connective, tonk. We show that some double line rules also give rise to such pseudo logical constants. But then, we are able to find a property of a double line rules which guarantee that it defines a genuine logical constant. Thus we provide an alternative answer to Belnap’s requirement of conservatity in terms of a local requirement on double line rules.

Truthmakers and Consequence

2017 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Winning Strategy ◽  
Logical Consequence ◽  
Game Plan ◽  
Soundness And Completeness ◽  
Truthmaker Semantics ◽  
Definition Of ◽  
Point Of Departure

We compare Tarski’s notion of logical consequence (preservation of truth) with that of Prawitz (transformability of warrants for assertion). The latter is our point of departure for a definition of consequence in terms of the transformability of truthmakers (verifications) relative to all models. A sentence’s Tarskian truth-in-M coincides with its having an M-relative truthmaker. An M-relative truthmaker serves as a winning strategy or game plan for player T in the ‘material game’ played on that sentence against the background of the model M. We enter conjectures about soundness and completeness of Classical Core Logic with respect to the notion of consequence that results when the domain is required to be decidable. We consider whether the truthmaker semantics threatens a slide to realism. We work with examples of core proofs whose premises are given M-relative truthmakers; and show how these can be systematically transformed into a truthmaker for the proof’s conclusion.

Finite Axiomatizability using additional predicates

1958 ◽  
Vol 23 (3) ◽  
pp. 289-308 ◽  
Author(s):  
W. Craig ◽  
R. L. Vaught
Keyword(s):  
Order Logic ◽  
The Other ◽  
First Order Logic ◽  
Logical Constant ◽  
Finite Axiomatizability ◽  
Logical Constants ◽  
First Order

By a theory we shall always mean one of first order, having finitely many non-logical constants. Then for theories with identity (as a logical constant, the theory being closed under deduction in first-order logic with identity), and also likewise for theories without identity, one may distinguish the following three notions of axiomatizability. First, a theory may be recursively axiomatizable, or, as we shall say, simply, axiomatizable. Second, a theory may be finitely axiomatizable using additional predicates (f. a.+), in the syntactical sense introduced by Kleene [9]. Finally, the italicized phrase may also be interpreted semantically. The resulting notion will be called s. f. a.+. It is closely related to the modeltheoretic notion PC introduced by Tarski [16], or rather, more strictly speaking, to PC∩ACδ.For arbitrary theories with or without identity, it is easily seen that s. f. a.+ implies f. a.+ and it is known that f. a.+ implies axiomatizability. Thus it is natural to ask under what conditions the converse implications hold, since then the notions concerned coincide and one can pass from one to the other.Kleene [9] has shown: (1) For arbitrary theories without identity, axiomatizability implies f. a.+. It also follows from his work that : (2) For theories with identity which have only infinite models, axiomatizability implies f. a.+.

scholarly journals Logic in Natural Language: Commitments and Constraints

Disputatio ◽  
2020 ◽  
Vol 12 (58) ◽  
pp. 277-308
Author(s):  
Gil Sagi
Keyword(s):  
Natural Language ◽  
Logical Consequence ◽  
Logical Form ◽  
Semantic Role ◽  
Semantic Constraints

Abstract In his new book, Logical Form, Andrea Iacona distinguishes between two different roles that have been ascribed to the notion of logical form: the logical role and the semantic role. These two roles entail a bifurcation of the notion of logical form. Both notions of logical form, according to Iacona, are descriptive, having to do with different features of natural language sentences. I agree that the notion of logical form bifurcates, but not that the logical role is merely descriptive. In this paper, I focus on formalization, a process by which logical form, on its logical role, is attributed to natural language sentences. According to some, formalization is a form of explication, and it involves normative, pragmatic, as well as creative aspects. I present a view by which formalization involves explicit commitments on behalf of a reasoner or an interpreter, which serve the normative grounds for the evaluation of a given text. In previous work, I proposed the framework of semantic constraints for the explication of logical consequence. Here, I extend the framework to include formalization constraints. The various constraints then serve the role of commitments. I discuss specific issues raised by Iacona concerning univocality, co-reference and equivocation, and I show how our views on these matters diverge as a result of our different starting assumptions.

The Metaphysics of Logical Consequence

2018 ◽  
Author(s):  
Alexandra Zinke
Keyword(s):  
Logical Consequence ◽  
Current Debate ◽  
Philosophy Of Logic ◽  
Logical Constants ◽  
Classical Definition ◽  
Central Notion ◽  
New Vision ◽  
Definition Of ◽  
Special Case

The book discusses the central notion of logic: the concept of logical consequence. It shows that the classical definition of consequence as truth preservation in all models must be restricted to all admissible models. The challenge for the philosophy of logic is therefore to supplement the definition with a criterion for admissible models. The problem of logical constants, so prominent in the current debate, constitutes but a special case of this much more general demarcation problem. The book explores the various dimensions of the problem of admissible models and argues that standard responses are unwarranted. As a result, it develops a new vision of logic, suggesting in particular that logic is deeply imbued with metaphysics.

Boghossian and Casalegno on Understanding and Inference1

2020 ◽  
pp. 108-116
Author(s):  
Paul Boghossian ◽  
Timothy Williamson
Keyword(s):  
Logical Constant ◽  
Logical Constants

In response to Boghossian’s objections in Chapter 6, this chapter defends counterexamples offered by Paolo Casalegno and the author to an inferentialist account of what it is to understand a logical constant, on which Boghossian relied in his explanation of our entitlement to reason according to basic logical principles. The importance for understanding is stressed of non-inferential aspects of the use of logical constants, for example in the description of a perceived scene. Boghossian’s criteria for individuating concepts are also queried, as is the viability of hybrid accounts which mix inferential accounts of the use of some terms with non-inferential accounts of other terms.

On recursively enumerable and arithmetic models of set theory

1958 ◽  
Vol 23 (4) ◽  
pp. 408-416 ◽  
Author(s):  
Michael O. Rabin
Keyword(s):  
Binary Relation ◽  
Set Theory ◽  
Simple Proof ◽  
Side Product ◽  
Logical Constant ◽  
Logical Constants ◽  
Recursive Models ◽  
Recursively Enumerable ◽  
Models Of Set Theory ◽  
Countable Models

In this note we shall prove a certain relative recursiveness lemma concerning countable models of set theory (Lemma 5). From this lemma will follow two results about special types of such models.Kreisel [5] and Mostowski [6] have shown that certain (finitely axiomatized) systems of set theory, formulated by means of the ϵ relation and certain additional non-logical constants, do not possess recursive models. Their purpose in doing this was to construct consistent sentences without recursive models. As a first corollary of Lemma 5, we obtain a very simple proof, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems. Namely, set theory with the single non-logical constant ϵ does not possess any recursively enumerable model. Thus we get, as a side product, an easy example of a consistent sentence containing a single binary relation which does not possess any recursively enumerable model; this sentence being the conjunction of the (finitely many) axioms of set theory.

scholarly journals Introducing Identity

2021 ◽  
Author(s):  
Owen Griffiths ◽  
Arif Ahmed
Keyword(s):  
Standard Approach ◽  
Logical Constant ◽  
Inference Rules ◽  
Elimination Rule ◽  
Leibniz’S Law ◽  
Logical Constants ◽  
Introduction Rule

AbstractThe best-known syntactic account of the logical constants is inferentialism . Following Wittgenstein’s thought that meaning is use, inferentialists argue that meanings of expressions are given by introduction and elimination rules. This is especially plausible for the logical constants, where standard presentations divide inference rules in just this way. But not just any rules will do, as we’ve learnt from Prior’s famous example of tonk, and the usual extra constraint is harmony. Where does this leave identity? It’s usually taken as a logical constant but it doesn’t seem harmonious: standardly, the introduction rule (reflexivity) only concerns a subset of the formulas canvassed by the elimination rule (Leibniz’s law). In response, Read [5, 8] and Klev [3] amend the standard approach. We argue that both attempts fail, in part because of a misconception regarding inferentialism and identity that we aim to identify and clear up.

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