Counting distinctions: on the conceptual foundations of Shannon’s information theory

Categorical logic has shown that modern logic is essentially the logic of subsets (or “subobjects”). In “subset logic,” predicates are modeled as subsets of a universe and a predicate applies to an individual if the individual is in the subset. Partitions are dual to subsets so there is a dual logic of partitions where a “distinction” [an ordered pair of distinct elements (u, u′) from the universe U] is dual to an “element”. A predicate modeled by a partition π on U would apply to a distinction if the pair of elements was distinguished by the partition π, i.e., if u and u′ were in different blocks of π. Subset logic leads to finite probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the logic of partitions is to assign to a partition the number of distinctions made by a partition normalized by the total number of ordered |U|² pairs from the finite universe. That yields a notion of “logical entropy” for partitions and a “logical information theory.” The logical theory directly counts the (normalized) number of distinctions in a partition while Shannon’s theory gives the average number of binary partitions needed to make those same distinctions. Thus the logical theory is seen as providing a conceptual underpinning for Shannon’s theory based on the logical notion of “distinctions.” This paper is dedicated to the memory of Gian-Carlo Rota—mathematician, philosopher, mentor, and friend.     

Modus Tollens,Modus Shmollens: Contrapositive reasoning and the pragmatics of negation

The utterance of a negative statement invites the pragmatic inference that some reason exists for the proposition it negates to be true; this pragmatic inference paves the way for the logically unexpected Modus Shmollens inference: “If p then q; not-q; therefore, p.” Experiment 1 shows that a majority of reasoners endorse Modus Shmollens from an explicit major conditional premise and a negative utterance as a minor premise: e.g., reasoners conclude that “the soup tastes like garlic” from the premises “If a soup tastes like garlic, then there is garlic in the soup; Carole tells Didier that there is no garlic in the soup they are eating.” Experiment 2 shows that this effect is mediated by the derivation of a pragmatic inference from negation. We discuss how theories of conditional reasoning can integrate such a pragmatic effect.     

Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa’s school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the “core” of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency.     

A Logical Transmission Principle for Conclusive Reasons

Dretske's conclusive reasons account of knowledge is designed to explain how epistemic closure can fail when the evidence for a belief does not transmit to some of that belief's logical consequences. Critics of Dretske dispute the argument against closure while joining Dretske in writing off transmission. This paper shows that, in the most widely accepted system for counterfactual logic (David Lewis's system VC), conclusive reasons are governed by an informative, non-trivial, logical transmission principle. If r is a conclusive reason for believing p in Dretske's sense, and if p logically implies q, and if p and q satisfy one additional condition, it follows that r is a conclusive reason for believing q. After introducing this additional condition, I explain its intuitive import and use the condition to shed new light on Dretske's response to scepticism, as well as on his distinction between the so-called ‘lightweight’ and ‘heavyweight’ implications of a piece of perceptual knowledge.     

Supervaluationism and Logical Consequence: A Third Way

It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness. This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion.     

The Tractatus on Logical Consequence

Abstract: I discuss the account of logical consequence advanced in Wittgenstein's Tractatus. I argue that the role that elementary propositions are meant to play in this account can be used to explain two remarkable features that Wittgenstein ascribes to them: that they are logically independent from one another and that their components refer to simple objects. I end with a proposal as to how to understand Wittgenstein's claim that all propositions can be analysed as truth functions of elementary propositions.     

Three-Valued Temporal Logic <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">t</Emphasis></Subscript> and Future Contingents

Prior’s three-valued modal logic Q was developed as a philosophically interesting modal logic. Thus, we should be able to modify Q as a temporal logic. Although a temporal version of Q was suggested by Prior, the subject has not been fully explored in the literature. In this paper, we develop a three-valued temporal logic Q _t and give its axiomatization and semantics. We also argue that Q _t provides a smooth solution to the problem of future contingents. Presented by Daniele Mundici     

形式化量子力学不必使用量子逻辑

一般认为,标准量子力学需要使用一套它自己的逻辑系统,即量子逻辑。量子逻辑采用与一般逻辑系统不同的语义规则,因此和古典逻辑无法兼容。此篇文章将呈现一套量子力学的严格形式基础,它是对古典二值逻辑之保守扩充;保守扩充意指比原先之逻辑系统强,但较强的原因为它有较多之词汇。此套逻辑为三值逻辑。古典逻辑中为真的句子仍然为真。古典逻辑中为假的句子将被区分为强性假与中性。第三个真值一中性一考虑了非本征态情况中之观察句。本文详列了物理的公理并显示它们具有一个模型。此提案的可行性说明了量子逻辑是不必要的,并且存在一个共同的逻辑架构可提供给数学、非量子物理及量子力学使用。     

What is <Emphasis Type="Italic">the</Emphasis> Logic of Inference?

The topic of this paper is the question whether there is a logic which could be justly called the logic of inference. It may seem that at least since Prawitz, Dummett and others demonstrated the proof-theoretical prominency of intuitionistic logic, the forthcoming answer is that it is this logic that is the obvious choice for the accolade. Though there is little doubt that this choice is correct (provided that inference is construed as inherently single-conclusion and complying with the Gentzenian structural rules), I do not think that the usual justification of it is satisfactory. Therefore, I will first try to clarify what exactly is meant by the question, and then sketch a conceptual framework in which it can be reasonably handled. I will introduce the concept of ‘inferentially native’ logical operators (those which explicate inferential properties) and I will show that the axiomatization of these operators leads to the axiomatic system of intuitionistic logic. Finally, I will discuss what modifications of this answer enter the picture when more general notions of inference are considered. Presented by Hannes Leitgeb     

Taming the runabout imagination ticket

The ‘puzzle of imaginative use’ (Kind and Kung in Knowledge through imagination, Oxford University Press, Oxford, 2016) asks: given that imagination is arbitrary escape from reality, how can it have any epistemic value? In particular, imagination seems to be logically anarchic, like a runabout inference ticket: one who imagines A may also imagine whatever B pops to one’s mind by free mental association. This paper argues that at least a certain kind of imaginative exercise—reality-oriented mental simulation—is not logically anarchic. Showing this is part of the task of solving the puzzle. Six plausible features of imagination, so understood, are listed. Then a formal semantics is provided, whose patterns of logical validity and invalidity model the six features.

     

Archetypal Forms of Inference

A form (or pattern) of inference, let us say, explicitlysubsumes just such particular inferences as are instances of the form, and implicitly subsumes thoseinferences with a premiss and conclusion logically equivalent to the premiss and conclusion of an instanceof the form in question. (For simplicity we restrict attention to one-premiss inferences.) A form ofinference is archetypal if it implicitly subsumes every correct inference. A precise definition (Section 1)of these concepts relativizes them to logics, since different logics classify different inferences ascorrect, as well as ruling differently on the matter of logical equivalence which entered into the definitionof implicit subsumption. When relativized to classical propositional logic, we find (Section 2) thatall but a handful of `degenerate' inference forms turn out to be archetypal, whereas matters are verydifferent in this respect for the case of intuitionistic propositional logic (Sections 3 and 4), and an interestingstructure emerges in this case (the poset of equivalence classes of inference forms, with respect tothe equivalence relation of implicitly subsuming the same inferences). Thus a more accurate, if excessivelylong-winded title would be 'Archetypal and Non-Archetypal Forms of Inference in Classical andIntuitionistic Propositional Logic'. Some left-overs are postponed for a final discussion (Section 5).The overall intention is to introduce a new subject matter rather than to have the last word on thequestions it raises; indeed several significant questions are left as open problems.     

The cost of closure: logical realism,anti-exceptionalism,and theoretical equivalence

Philosophers of science often assume that logically equivalent theories are theoretically equivalent. I argue that two theses, anti-exceptionalism about logic (which says, roughly, that logic is not a priori, that it is revisable, and that it is not special or set apart from other human inquiry) and logical realism (which says, roughly, that differences in logic reflect genuine metaphysical differences in the world) make trouble for this commitment, as well as a closely related commitment to theories being closed under logical consequence. I provide three arguments. The first two show that anti-exceptionalism about logic provides an epistemic challenge to both the closure and the equivalence claims; the third shows that logical realism provides a metaphysical challenge to both the closure and the equivalence claims. Along the way, I show that there are important methodological upshots for metaphysicians and philosophers of logic, in particular, lessons about certain conceptions of naturalism as constraining the possibilities for metaphysics and the philosophy of logic.

     

The Pentus Theorem for Lambek Calculus with Simple Nonlogical Axioms

The Lambek calculus introduced in Lambek [6] is a strengthening of the type reduction calculus of Ajdukiewicz [1]. We study Associative Lambek Calculus L in Gentzen style axiomatization enriched with a finite set Γ of nonlogical axioms, denoted by L(Γ).It is known that finite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages (see Buszkowski [2]). Then we confine nonlogical axioms to sequents of the form p → q, where p and q are atomic types. For calculus L(Γ) we prove interpolation lemma (modifying the Roorda proof for L [10]) and the binary reduction lemma (using the Pentus method [9] with modification from [3]). In consequence we obtain the weak equivalence of the Context-Free Grammars and grammars based on L(Γ).     

Deviance and Vice: Strength as a Theoretical Virtue in the Epistemology of Logic

This paper is about the putative theoretical virtue of strength, as it might be used in abductive arguments to the correct logic in the epistemology of logic. It argues for three theses. The first is that the well‐defined property of logical strength is neither a virtue nor a vice, so that logically weaker theories are not—all other things being equal—worse or better theories than logically stronger ones. The second thesis is that logical strength does not entail the looser characteristic of scientific strength, and the third is that many modern logics are on a par—or can be made to be on a par—with respect to scientific strength.     

意义理论

Research into logical syntax provides us the knowledge of the structure of sentences, while logical semantics provides a window into uncovering the truth of sentences. Therefore, it is natural to make sentences and truth the central concern when one deals with the theory of meaning logically. Although their theories of meaning differ greatly, both Michael Dummett’s theory and Donald Davidson’s theory are concerned with sentences and truth and developed in terms of truth. Logical theories and methods first introduced by G. Frege underwent great developments during the past century and have played an important role in expanding these two scholars’ theories of meaning. Translated by Ma Minghui from Zhexue Yanjiu 哲学研究 (Philosophical Research), 2006, (7): 53–61     

Expressivism and Moral Dilemmas: A Response to Marino

Simon Blackburn’s expressivist logic of attitudes aims to explain how we can use non-assertoric moral judgements in logically valid arguments. Patricia Marino has recently argued that Blackburn’s logic faces a dilemma: either it cannot account for the place of moral dilemmas in moral reasoning or, if it can, it makes an illicit distinction between two different kinds of moral dilemma. Her target is the logic’s definition of validity as satisfiability, according to which validity requires an avoidance of attitudinal inconsistency. Against Marino’s arguments, I contend that expressivists following Blackburn are able to show how we appreciate the validity of arguments found in dilemma-contexts, and that Marino’s argument concerning the distinction between contingent moral dilemmas and logical moral dilemmas rests on a mistake concerning the logical representation of a contingent dilemma.     

Making Sense of Sense Containment

Proposition 5.122 of Wittgenstein’s Tractatus (‘If p follows from q, the sense of [p] is contained in the sense of [q]’) has been the source of much puzzlement among interpreters, so much so that no fully satisfactory account is yet available. This is unfortunate, if only because the containment account of logical consequence has a venerable tradition behind it. Pasquale Frascolla’s interpretation of proposition 5.122 is based on a valid argument and one true premise. However, the argument explains sense containment only in an indirect way, leaving some crucial questions unanswered. Besides, Frascolla does not address the issue of how to make sense containment notationally perspicuous, an essential theme in Wittgenstein’s reflections. In this paper, we elaborate on Frascolla’s account by looking at the issue through the Tractarian notion of logical space. Our analysis shows that, for containment to be fully appreciated, one should adopt a negative perspective on the notion of sense (which is taken into consideration also by Wittgenstein), in line with the exclusionary theory of conceptual content, as labelled by Ian Rumfitt. Besides this, we introduce and discuss two methods—one envisaged by Wittgenstein himself—for making sense containment notationally perspicuous.     

Is there a Bayesian justification of hypothetico-deductive inference?

Many philosophers have claimed that Bayesianism can provide a simple justification for hypothetico-deductive (H-D) inference, long regarded as a cornerstone of the scientific method. Following up a remark of van Fraassen (1985), we analyze a problem for the putative Bayesian justification of H-D inference in the case where what we learn from observation is logically stronger than what our theory implies. Firstly, we demonstrate that in such cases the simple Bayesian justification does not necessarily apply. Secondly, we identify a set of sufficient conditions for the mismatch in logical strength to be justifiably ignored as a “harmless idealization”. Thirdly, we argue, based upon scientific examples, that the pattern of H-D inference of which there is a ready Bayesian justification is only rarely the pattern that one actually finds at work in science. Whatever the other virtues of Bayesianism, the idea that it yields a simple justification of a pervasive pattern of scientific inference appears to have been oversold.     

A new theory of content I: Basic content

Philosophers of science as divergent as the inductivist Carnap and the deductivist Popper share the notion that the (logical) content of a proposition is given by its consequence class. I claim that this notion of content is (a) unintuitive and (b) inappropriate for many of the formal needs of philosophers of science. The basic problem is that given this notion of content, for any arbitraryp andq, (p Vq) will count as part of the content of bothp andq. In other words, any arbitraryp andq share some common content. This notion of content has disastrous effects on, for instance, Carnap's attempts to explicate the notion of confirmation in terms of probabilistic favorable relevance, and Popper's attempts to define verisimilitude. After briefly reviewing some of the problems of the traditional notion of content I present an alternative notion of (basic) content which (a) better fits our intuitions about content and (b) better serves the formal needs of philosophers of science.     

Truth, Warrant and Superassertibility

In a recent paper on Truth, Knowability and Neutrality Timothy Kenyon sets out to defend the coherence of a putative anti-realist truth-predicate, superassertibility, due to Wright (1992, 1999), against a number of Wright’s critics. By his own admission, the success of Kenyon’s defensive strategies turns out to hinge upon a realist conception of absolute warrant which conflicts with the anti-realist character of the original proposal, based, as it was, on a notion of defeasible warrant. Kenyon’s potential success in resisting Wright’s critics brings a pyrrhic victory: either way, realism wins. Here I argue that the link between superassertibility and defeasible warrant can be restored in a way which clarifies the consistency of the pair, provided that the notion of inference to superassertibility is properly understood in logical terms. As one might expect, the requisite notion is not classical in character. As one might not expect, the notion cannot be properly construed in intuitionist terms either. Hence, I propose an alternative logical framework which, I believe, is at least formally adequate to the representation of superassertibility on the basis of defeasible warrant. The price to be paid for rejecting the ‘third way’ proposed here in favour of either of the two more traditional logical options is precisely that indicated by Kenyon’s discussion.