On Dualities and Equivalences between Physical Theories

The main aim of this paper is to make a remark about the relation between (i) dualities between theories, as ‘duality’ is understood in physics and (ii) equivalence of theories, as ‘equivalence’ is understood in logic and philosophy. The remark is that in physics, two theories can be dual, and accordingly get called ‘the same theory’, though we interpret them as disagreeing—so that they are certainly not equivalent, as ‘equivalent’ is normally understood. So the remark is simple, but, I shall argue, worth stressing, since often neglected. My argument for this is based on the account of duality developed by De Haro. I also spell out how this remark implies a limitation of proposals (both traditional and recent) to understand theoretical equivalence as either logical equivalence or a weakening of it.

Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic.

COMPOUND PROPOSITIONAL LAW FOR LOGICAL EQUIVALENCE, TAUTOLOGY AND CONTRADICTION

This paper presents a Compound Propositional Law for Logical Equivalence, Tautology and Contradiction. The proposed Law is developed with the help of negation, disjunction, conjunction, exclusive or, conditional statement and bi-conditional statement. The idea of research is taken from de-Morgan law. This proposed law is important and useful for Logical Equivalence, Tautology and Contradiction for the research purpose because these are the rare cases in the field of research. This article aims to help readers understand the compound proposition and proposition equivalence in conducting research. This article discusses propositions that are relevant for proposition equivalence. Six main compound propositions are distinguished and an overview is given in the article. Hence, it is observed from the result and discussion that the compound proposition law is a good achievement in discrete structure for the logical Equivalence, Tautology and Contradiction purpose.

On the strongest three-valued paraconsistent logic contained in classical logic and its dual

$\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ is a three-valued paraconsistent propositional logic that is essentially the same as J3. It has the most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic. However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned. In this paper, properties concerning the logical equivalence relation of a logic are used to distinguish $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ from the others. As one of the bonuses of focusing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent and commutative laws for conjunction and disjunction. For most properties of $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ that have been proposed as desirable properties of a reasonable paraconsistent propositional logic, its paracomplete analogue has a comparable property. In this paper, properties concerning the logical equivalence relation of a logic are also used to distinguish the paracomplete analogue of $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ from the other three-valued paracomplete propositional logics with those comparable properties.

First among equals: co-hyperintensionality for structured propositions

Theories of structured meanings are designed to generate fine-grained meanings, but they are also liable to overgenerate structures, thus drawing structural distinctions without a semantic difference. I recommend the proliferation of very fine-grained structures, so that we are able to draw any semantic distinctions we think we might need. But, in order to contain overgeneration, I argue we should insert some degree of individuation between logical equivalence and structural identity based on structural isomorphism. The idea amounts to forming an equivalence class of different structures according to one or more formal criteria and designating a privileged element as a representative of all the elements, i.e., a first among equals. The proposed method helps us to a cluster of notions of co-hyperintensionality. As a test case, I consider a recent objection levelled against the act theory of structured propositions. I also respond to an objection against my methodology.

Is ‘Would If’ Hyperintensional?

This chapter responds to Fine’s arguments that counterfactual conditionals are hyperintensional because the result of substituting a truth-functionally equivalent antecedent can change the truth-value of a counterfactual. Fine’s descriptions of his alleged examples of hyperintensionality are inconsistent but seem consistent as a result of contextual shifts between different components, for which there is independent evidence, and which can be explained by the application of the suppositional heuristic, which makes some worlds verifying the antecedent relevant. Different senses of ‘hyperintensional’ are distinguished, one based on sameness of intension, another on logical equivalence, but the objection to Fine’s arguments works for both.

WITTGENSTEIN’S ELIMINATION OF IDENTITY FOR QUANTIFIER-FREE LOGIC

One of the central logical ideas in Wittgenstein’s Tractatus logico-philosophicus is the elimination of the identity sign in favor of the so-called “exclusive interpretation” of names and quantifiers requiring different names to refer to different objects and (roughly) different variables to take different values. In this paper, we examine a recent development of these ideas in papers by Kai Wehmeier. We diagnose two main problems of Wehmeier’s account, the first concerning the treatment of individual constants, the second concerning so-called “pseudo-propositions” (Scheinsätze) of classical logic such as $a=a$ or $a=b \wedge b=c \rightarrow a=c$ . We argue that overcoming these problems requires two fairly drastic departures from Wehmeier’s account: (1) Not every formula of classical first-order logic will be translatable into a single formula of Wittgenstein’s exclusive notation. Instead, there will often be a multiplicity of possible translations, revealing the original “inclusive” formulas to be ambiguous. (2) Certain formulas of first-order logic such as $a=a$ will not be translatable into Wittgenstein’s notation at all, being thereby revealed as nonsensical pseudo-propositions which should be excluded from a “correct” conceptual notation. We provide translation procedures from inclusive quantifier-free logic into the exclusive notation that take these modifications into account and define a notion of logical equivalence suitable for assessing these translations.

Logics of Synonymy

We investigate synonymy in the strong sense of content identity (and not just meaning similarity). This notion is central in the philosophy of language and in applications of logic. We motivate, uniformly axiomatize, and characterize several “benchmark” notions of synonymy in the messy class of all possible notions of synonymy. This class is divided by two intuitive principles that are governed by a no-go result. We use the notion of a scenario to get a logic of synonymy (SF) which is the canonical representative of one division. In the other division, the so-called conceptivist logics, we find, e.g., the well-known system of analytic containment (AC). We axiomatize four logics of synonymy extending AC, relate them semantically and proof-theoretically to SF, and characterize them in terms of weak/strong subject matter preservation and weak/strong logical equivalence. This yields ways out of the no-go result and novel arguments—independent of a particular semantic framework—for each notion of synonymy discussed (using, e.g., Hurford disjunctions or homotopy theory). This points to pluralism about meaning and a certain non-compositionality of truth in logic programs and neural networks. And it unveils an impossibility for synonymy: if it is to preserve subject matter, then either conjunction and disjunction lose an essential property or a very weak absorption law is violated.