A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI, LV, or LF, and every antitone logic can be extended to LN, LV, or LF, where LI, LN, LV, and LF are logics axiomatized, respectively, by the schemas □α↔α, □α↔¬α, □α↔⊤, and □α↔⊥. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV, LF, and the minimum amphitone logic AM axiomatized by the schema □α→□β. These logics, (...) along with LI, LN, and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the nonmodal connectives. We close by discussing these divergences and conditions under which our results do carry over. (shrink)

Quine taught us the difference between a theory’s ontology and its ideology. Ontology is the things a theory’s quantifiers must range over if it is true, Ideology is the primitive concepts that must be used to state the theory. This allows us to split the theoretical virtue of parsimony into two kinds: ontological parsimony and ideological parsimony. My goal is help illuminate the virtue of ideological parsimony by giving a criterion for ideological innocence—a rule for when additional ideology does not (...) count against parsimony. I propose the expressive power innocence criterion: if the ideology of theory one is expressively equivalent to that of theory two, then neither is ideologically simpler than the other. In its favor I offer the argument from accuracy, showing that any account of a theoretical virtue that is supposed to make theories that have it more likely to be true than theories that do not must respect it. Next I consider its ramifications, eliminating rival views and passing judgment on some arguments from parsimony that can be found in the literature. Finally, I consider two objections. First: I address an objection arising from the possibility of languages with a ‘primitive’ operator that allows us to list a theory’s primitives in the object-language. Second: I address an objection raised by Nelson Goodman against attempts to reckon simplicity by expressive power. Both objections fail. (shrink)

A brief review of Carnap's formal program in philosophy of science.

There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power, translation, and notational variance. In doing so, I show how these investigations (...) help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language. (shrink)

The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following Williamson and Russell, we distinguish between logical and scientific strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientific strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories.

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...) accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic. (shrink)

What does it take for two logics to be mere notational variants? The present paper proposes a variety of different ways of cashing out notational variance, in particular isolating a constraint on any reasonable account of notational variance which makes plausible that the only kinds of translations which can witness notational variance are what are sometimes called definitional translations.

Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.

In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ‘no loss of generality’ problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53–54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. (...) The purpose of this article is to clarify Russell's claim and to solve the ‘no loss of generality’ problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (‘ramif-types’). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other hand, makes it possible to give us a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the ‘no loss of generality’ problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed. (shrink)

Mathematicians, physicists, and philosophers of physics often look to the symmetries of an object for insight into the structure and constitution of the object. My aim in this paper is to explain why this practice is successful. In order to do so, I present a collection of results that are closely related to (and in a sense, generalizations of) Beth’s and Svenonius’ theorems.

It has become commonplace to distinguish enumerative theories of welfare, which tell us which things are good for us, from explanatory theories, which tell us why the things that are good for us have that status. It has also been claimed that while hedonism and objective list theories are enumerative but not explanatory, desire satisfactionism is explanatory but not enumerative. In this paper, I argue that this is mistaken. When properly understood, every major theory of welfare is both enumerative and (...) explanatory. (shrink)

The purpose of the present note is to advertise an interesting conjecture concerning a well-known translation in modal logic, by confirming a (highly restricted) special case of the conjecture.

I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.

In a recent paper, Wigglesworth claims that syntactic criteria of theoretical equivalence are not appropriate for settling questions of equivalence between logical theories, since such criteria judge classical and intuitionistic logic to be equivalent; he concludes that logicians should use semantic criteria instead. However, this is an artefact of the particular syntactic criterion chosen, which is an implausible criterion of theoretical equivalence. Correspondingly, there is nothing to suggest that a more plausible syntactic criterion should not be used to settle questions (...) of equivalence between different logical theories; such a criterion is exhibited and shown to judge classical and intuitionistic logic to be inequivalent. (shrink)

Enumeration and explanation in theories of welfare, Analysis, doi.org/10.1093/analys/anx035, published 13 April 2017.

Logicians and philosophers of science have proposed various formal criteria for theoretical equivalence. In this paper, we examine two such proposals: definitional equivalence and categorical equivalence. In order to show precisely how these two well-known criteria are related to one another, we investigate an intermediate criterion called Morita equivalence.

The question of whether knowledge is definable in terms of belief, which has played an important role in epistemology for the last 50 years, is studied here in the framework of epistemic and doxastic logics. Three notions of definability are considered: explicit definability, implicit definability, and reducibility, where explicit definability is equivalent to the combination of implicit definability and reducibility. It is shown that if knowledge satisfies any set of axioms contained in S5, then it cannot be explicitly defined in (...) terms of belief. S5 knowledge can be implicitly defined by belief, but not reduced to it. On the other hand, S4.4 knowledge and weaker notions of knowledge cannot be implicitly defined by belief, but can be reduced to it by defining knowledge as true belief. It is also shown that S5 knowledge cannot be reduced to belief and justification, provided that there are no axioms that involve both belief and justification. (shrink)

Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it is true, at least according to one reasonable notion of theoretical (...) equivalence. Our clarification of Quine’s conjecture, however, exposes the shortcomings of his argument against many-sorted logic. (shrink)

Three notions of definability in multimodal logic are considered. Two are analogous to the notions of explicit definability and implicit definability introduced by Beth in the context of first-order logic. However, while by Beth’s theorem the two types of definability are equivalent for first-order logic, such an equivalence does not hold for multimodal logics. A third notion of definability, reducibility, is introduced; it is shown that in multimodal logics, explicit definability is equivalent to the combination of implicit definability and reducibility. (...) The three notions of definability are characterized semantically using (modal) algebras. The use of algebras, rather than frames, is shown to be necessary for these characterizations. (shrink)

In this paper we propose and defend the Synonymy account, a novel account of metaphysical equivalence which draws on the idea (Rayo in The Construction of Logical Space, Oxford University Press, Oxford, 2013) that part of what it is to formulate a theory is to lay down a theoretical hypothesis concerning logical space. Roughly, two theories are synonymous—and so, in our view, equivalent—just in case (i) they take the same propositions to stand in the same entailment relations, and (ii) they (...) are committed to the truth of the same propositions. Furthermore, we put our proposal to work by showing that it affords a better and more nuanced understanding of the debate between Quineans and noneists. Finally we show how the Synonymy account fares better than some of its competitors, specifically, McSweeney’s (Philosophical Perspectives 30(1):270–293, 2016) epistemic account and Miller’s (Philosophical Quarterly 67(269):772–793, 2017) hyperintensional account. (shrink)

The variety \ of semi-Heyting algebras was introduced by H. P. Sankappanavar [13] as an abstraction of the variety of Heyting algebras. Semi-Heyting algebras are the algebraic models for a logic HsH, known as semi-intuitionistic logic, which is equivalent to the one defined by a Hilbert style calculus in Cornejo :9–25, 2011) [6]. In this article we introduce a Gentzen style sequent calculus GsH for the semi-intuitionistic logic whose associated logic GsH is the same as HsH. The advantage of this (...) presentation of the logic is that we can prove a cut-elimination theorem for GsH that allows us to prove the decidability of the logic. As a direct consequence, we also obtain the decidability of the equational theory of semi-Heyting algebras. (shrink)

Anti-exceptionalism about logic takes logical theories to be continuous with scientific theories. Scientific theories are subject to criteria of theoretical equivalence. This article compares two types of theoretical equivalence – one syntactic and one semantic – in the context of logical anti-exceptionalism, and argues that the syntactic approach leads to undesirable consequences. The anti-exceptionalist should therefore take a semantic approach when evaluating whether logical theories, understood as scientific theories, are equivalent. This article argues for a particular semantic approach, in terms (...) of categorical equivalence, to determine whether logical theories are equivalent. (shrink)

In this paper we introduce a logic that we name semi Heyting–Brouwer logic, \, in such a way that the variety of double semi-Heyting algebras is its algebraic counterpart. We prove that, up to equivalences by translations, the Heyting–Brouwer logic \ is an axiomatic extension of \ and that the propositional calculi of intuitionistic logic \ and semi-intuitionistic logic \ turn out to be fragments of \.

Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which were defined by H. P. Sankappanavar as a generalization of Heyting algebras. We present a new, more streamlined set of axioms for semi-intuitionistic logic, which we prove translationally equivalent to the original one. We then study some formulas that define a semi-Heyting implication, and specialize this study to the case in which the formulas use only the lattice operators and the intuitionistic implication. We prove then that all the logics thus (...) obtained are equivalent to intuitionistic logic, and give their Kripke semantics. (shrink)

Genuine Paraconsistent logics \ and \ were defined in 2016 by Béziau et al, including only three logical connectives, namely, negation disjunction and conjunction. Afterwards in 2017 Hernández-Tello et al, provide implications for both logics and define the logics \ and \. In this work we continue the study of these logics, providing sound and complete Hilbert-type axiomatic systems for each logic. We prove among other properties that \ and \ satisfy a restricted version of the Substitution Theorem, and that (...) both of them are maximal with respect to Classical Propositional Logic. To conclude we make some comparisons between \ and \ and among other logics, for instance \ and some \s. (shrink)

The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

Interpretation is commonly used in mathematical logic to compare different theories and identify cases where two theories are for almost all intents and purposes the same. Similar techniques are used in the comparison between alternative logics although the links between these approaches are not transparent. This paper generalizes theoretical comparison techniques to the case of logical comparison using an extremely general approach to semantics that provides a very generous playing field upon which to make our comparisons. In particular, we aim (...) to develop the useful idea that interpretations should determine inner models. (shrink)

In this article, I explore the relation between theoretical and practical reason in the work of J.S. Mill. I argue that Mill holds that theoretical reason is subordinate to practical reason. Ultimately, this amounts to the claim that the norms of theoretical reason – those rules governing how we ought to believe – are grounded in considerations of utility.

The paper aims at developing the idea that the standard operator of noncontingency, usually symbolized by Δ, is a special case of a more general operator of dyadic noncontingency Δ(−, −). Such a notion may be modally defined in different ways. The one examined in the paper is __Δ__(B, A) = df ◊B ∧ (A ⥽ B ∨ A ⥽ ¬B), where ⥽ stands for strict implication. The operator of dyadic contingency __∇__(B, A) is defined as the negation of __Δ__(B, (...) A). Possibility (◊A) may be then defined as __Δ__(A, A), necessity (□A) as __∇__(¬A, ¬A) and standard monadic noncontingency (__Δ__A) as __Δ__( \({\textsf{T}}\), A). In the second section it is proved that the deontic system KD is translationally equivalent to an axiomatic system of dyadic noncontingency named KDΔ 2, and that the minimal system KΔ of monadic contingency is a fragment of KDΔ 2. The last section suggests lines for further inquiries. (shrink)

_ doi:10.1093/analys/anx072 _, published: 27 June 2017.

In an earlier paper entitled Synonymous Logics, the authors attempted to show that there are two modal logics so that each is exactly translatable into the other, but they are not translationally equivalent. Unfortunately, there is an error in the proof of this result. The present paper provides a new example of two such logics, and a proof of the result claimed in the earlier paper.