Probability Semantics for Quantifier Logic

By supplying propositional calculus with a probability semantics we showed, in our 1996, that finite stochastic problems can be treated by logic-theoretic means equally as well as by the usual set-theoretic ones. In the present paper we continue the investigation to further the use of logical notions in probability theory. It is shown that quantifier logic, when supplied with a probability semantics, is capable of treating stochastic problems involving countably many trials.     

How to Be an Ethical Expressivist

Expressivism promises an illuminating account of the nature of normative judgment. But worries about the details of expressivist semantics have led many to doubt whether expressivism's putative advantages can be secured. Drawing on insights from linguistic semantics and decision theory, I develop a novel framework for implementing an expressivist semantics that I call ordering expressivism. I argue that by systematically interpreting the orderings that figure in analyses of normative terms in terms of the basic practical attitude of conditional weak preference, the expressivist can explain the semantic properties of normative sentences in terms of the logical properties of that attitude. Expressivism's problems with capturing the logical relations among normative sentences can be reduced to the familiar, more tractable problem of explaining certain coherence constraints on preferences. Particular attention is given to the interpretation of wide‐scope negation. The proposed solution is also extended to other types of embedded contexts—most notably, disjunctions.     

The Justification of the Logical Laws Revisited

The proof-theoretic analysis of logical semantics undermines the received view of proof theory as being concerned with symbols devoid of meaning, and of model theory as the sole branch of logical theory entitled to access the realm of semantics. The basic tenet of proof-theoretic semantics is that meaning is given by some rules of proofs, in terms of which all logical laws can be justified and the notion of logical consequence explained. In this paper an attempt will be made to unravel some aspects of the issue and to show that this justification as it stands is untenable, for it relies on a formalistic conception of meaning and fails to recognise the fundamental distinction between semantic definitions and rules of inference. It is also briefly suggested that the profound connection between meaning and proofs should be approached by first reconsidering our very notion of proof.     

Convex MV-Algebras: Many-Valued Logics Meet Decision Theory

This paper introduces a logical analysis of convex combinations within the framework of ?ukasiewicz real-valued logic. This provides a natural link between the fields of many-valued logics and decision theory under uncertainty, where the notion of convexity plays a central role. We set out to explore such a link by defining convex operators on MV-algebras, which are the equivalent algebraic semantics of ?ukasiewicz logic. This gives us a formal language to reason about the expected value of bounded random variables. As an illustration of the applicability of our framework we present a logical version of the Anscombe–Aumann representation result.     

Context-Sensitivity in Jain Philosophy: A Dialogical Study of Siddharṣigaṇi’s <Emphasis Type="Italic">Commentary on the Handbook of Logic</Emphasis>

In classical India, Jain philosophers developed a theory of viewpoints (naya-vāda) according to which any statement is always performed within and dependent upon a given epistemic perspective or viewpoint. The Jainas furnished this epistemology with an (epistemic) theory of disputation that takes into account the viewpoint in which the main thesis has been stated. The main aim of our paper is to delve into the Jain notion of viewpoint-contextualisation and to develop the elements of a suitable logical system that should offer a reconstruction of the Jainas’ epistemic theory of disputation. A crucial step of our project is to approach the Jain theory of disputation with the help of a theory of meaning for logical constants based on argumentative practices called dialogical logic. Since in the dialogical framework the meaning of the logical constants is given by the norms or rules for their use in a debate, it provides a meaning theory closer to the Jain context-sensitive disputation theory than the main-stream formal model-theoretic semantics.     

A logic of intentions and beliefs

Intentions are an important concept in Artificial Intelligence and Cognitive Science. We present a formal theory of intentions and beliefs based on Discourse Representation Theory that captures many of their important logical properties. Unlike possible worlds approaches, this theory does not assume that agents are perfect reasoners, and gives a realistic view of their internal architecture; unlike most representational approaches, it has anobjective semantics, and does not rely on anad hoc labeling of the internal states of agents. We describe a minimal logic for intentions and beliefs that is sound and complete relative to our semantics. We discuss several additional axioms, and the constraints on the models that validate them.     

Conditionals in causal decision theory

This paper explores the possibility that causal decision theory can be formulated in terms of probabilities of conditionals. It is argued that a generalized Stalnaker semantics in combination with an underlying branching time structure not only provides the basis for a plausible account of the semantics of indicative conditionals, but also that the resulting conditionals have properties that make them well-suited as a basis for formulating causal decision theory. Decision theory (at least if we omit the frills) is not an esoteric science, however unfamiliar it may seem to an outsider. Rather it is a systematic exposition of the consequences of certain well-chosen platitudes about belief, desire, preference and choice. It is the very core of our common-sense theory of persons, dissected out and elegantly systematized. (David Lewis, Synthese 23:331–344, 1974, p. 337). A small distortion in the analysis of the conditional may create spurious problems with the analysis of other concepts. So if the facts about usage favor one among a number of subtly different theories, it may be important to determine which one it is. (Robert Stalnaker, A Defense of Conditional Excluded Middle, pp. 87–104, 1980, p. 87)      

The original sin of proof-theoretic semantics

Proof-theoretic semantics is an alternative to model-theoretic semantics. It aims at explaining the meaning of the logical constants in terms of the inference rules that govern their behaviour in proofs. We argue that this must be construed as the task of explaining these meanings relative to a logic, i.e., to a consequence relation. Alas, there is no agreed set of properties that a relation must have in order to qualify as a consequence relation. Moreover, the association of a consequence relation to a logical calculus is not as straightforward as it may seem. We show that these facts are problematic for the proof-theoretic project but the problems can be solved. Our thesis is that the consequence relation relevant for proof-theoretic semantics is the one given by the sequent-to-sequent derivability relation in Gentzen systems.

     

A Lewisian Semantics for S2

This paper sets out a semantics for C.I. Lewis's logic S2 based on the ontology of his 1923 paper ‘Facts, Systems, and the Unity of the World’. In that article, worlds are taken to be maximal consistent systems. A system, moreover, is a collection of facts that is closed under logical entailment and conjunction. In this paper, instead of defining systems in terms of logical entailment, I use certain ideas in Lewis's epistemology and philosophy of logic to define a class of models in which systems are taken to be primitive elements but bear certain relations to one another. I prove soundness and completeness for S2 over this class of models and argue that this semantics makes sense of at least a substantial fragment of Lewis's logical theory.     

Ideal Worlds — Wishful Thinking in Deontic Logic

The ideal world semantics of standard deontic logic identifies our obligations with how we would act in an ideal world. However, to act as if one lived in an ideal world is bad moral advice, associated with wishful thinking rather than well-considered moral deliberation. Ideal world semantics gives rise to implausible logical principles, and the metaphysical arguments that have been put forward in its favour turn out to be based on a too limited view of truth-functional representation. It is argued that ideal world semantics should be given up in favour of other, more plausible uses of possible worlds for modelling normative subject-matter.     

A future for the thin red line

The thin red line (TRL) is a theory about the semantics of future-contingents. The central idea is that there is such a thing as the ??actual future??, even in the presence of indeterminism. It is inspired by a famous solution to the problem of divine foreknowledge associated with William of Ockham, in which the freedom of agents is argued to be compatible with God??s omniscience. In the modern branching time setting, the theory of the TRL is widely regarded to suffer from several fundamental problems. In this paper we propose several new TRL semantics, each with differing degrees of success. This leads up to our final semantics, which is a cross between the TRL and supervaluationism. We discuss the notions of truth, validity and semantic consequence which result from our final semantics, and demonstrate some of its pleasing results. This account, we believe, answers the main objection in the literature, and thus places the TRL on the same level as any other competing semantics for future contingents.     

Logical inference in English: A preliminary analysis

The perfect fit of syntactic derivability and logical consequence in first-order logic is one of the most celebrated facts of modern logic. In the present flurry of attention given to the semantics of natural language, surprisingly little effort has been focused on the problem of logical inference in natural language and the possibility of its completeness. Even the traditional theory of the syllogism does not give a thorough analysis of the restricted syntax it uses.My objective is to show how a theory of inference may be formulated for a fragment of English that includes a good deal more than the classical syllogism. The syntax and semantics are made as formal and as explicit as is customary for artificial formal languages. The fragment chosen is not maximal but is restricted severely in order to provide a clear overview of the method without the cluttering details that seem to be an inevitable part of any grammar covering a substantial fragment of a natural language. (Some readers may feel the details given here are too onerous.)I am especially concerned with quantifier words in both object and subject position, with negation, and with possession. I do not consider propositional attitudes or the modalities of possibility and necessity, although the model-theoretic semantics I use has a standard version to deal with such intensional contexts.An important point of methodology stressed in earlier publications (Suppes, 1976; Suppes & Macken, 1978; Suppes, 1979) is that the semantic representation of the English sentences in the fragment uses neither quantifiers nor variables, but only constants denoting given sets and relations, and operations on sets and relations.In the first section, I rapidly sketch the formal framework of generative syntax and model-theoretic semantics, with special attention to extended relation algebras. The second section states the grammar and semantics of the fragment of English considered. The next section is concerned with developing some of the rules of inference. The results given are quite incomplete. The final section raises problems of extension. Classical logic is a poor guide for dealing with inferences involving high-frequency function words such as of, to, a, in, for, with, as, on, at, and by. Indeed, the line between logical and nonlogical inference in English seems to be nonexistent or, if made, highly arbitrary in character-much more so than has been claimed by those critical of the traditional analyticsynthetic tradition.No theorems on soundness or completeness are considered because of the highly tentative and incomplete character of the rules of inference proposed. However, because of the variable-free semantics used, soundness is easy to establish for the rules given.The research reported here has been supported in part by National Science Foundation Grant No. SED77-09698.     

A postulate-based analysis of comparative preference statements

There has been a growing interest in the study of preferences for their utility in solving problems related to decision making. Most of the preference representation languages developed in the literature are based on comparative preference statements since they offer a simple and intuitive way for expressing preferences. They can be further interpreted following different semantics, imparting a greater flexibility on how outcomes can be compared. So far the main objective has been to rank-order the set of outcomes given a set of comparative preference statements and one or several semantics. Tackling this problem from a different angle, we look into the behavioural aspects of the preference semantics and statements by attempting to formalise the intuition behind them using postulates studied in preference logics and non-monotonic reasoning. We select the postulates w.r.t. three criteria: coherence, syntax independence and inference. Thus, our analysis provides a means to determine those properties that are satisfied for a given preference semantics.     

Why Propositions Cannot be Sets of Truth-supporting Circumstances

No semantic theory satisfying certain natural constraints can identify the semantic contents of sentences (the propositions they express), with sets of circumstances in which the sentences are true–no matter how fine-grained the circumstances are taken to be. An objection to the proof is shown to fail by virtue of conflating model-theoretic consequence between sentences with truth-conditional consequence between the semantic contents of sentences. The error underlines the impotence of distinguishing semantics, in the sense of a truth-based theory of logical consequence, and semantics, in the sense of a theory of meaning.     

What Tipper is Ready for: A Semantics for Incomplete Predicates

This paper presents a precise semantics for incomplete predicates such as “ready”. Incomplete predicates have distinctive logical properties that a semantic theory needs to accommodate. For instance, “Tipper is ready” logically implies “Tipper is ready for something”, but “Tipper is ready for something” does not imply “Tipper is ready”. It is shown that several approaches to the semantics of incomplete predicates fail to accommodate these logical properties. The account offered here defines contexts as structures containing an element called a proposition set, which contains atomic propositions and negations of atomic propositions. The condition under which “Tipper is ready” is true in a context is defined in terms of the contents of the proposition set for the context. On this account, the content of the context pertinent to a conversation must be determined not by what speakers have in mind but by relations of objective relevance.     

Verification, falsification, and the logic of enquiry

Our starting point is Michael Luntley's falsificationist semantics for the logical connectives and quantifiers: the details of his account are criticised but we provide an alternative falsificationist semantics that yields intuitionist logic, as Luntley surmises such a semantics ought. Next an account of the logical connectives and quantifiers that combines verificationist and falsificationist perspectives is proposed and evaluated. While the logic is again intuitionist there is, somewhat surprisingly, an unavoidable asymmetry between the verification and falsification conditions for negation, the conditional, and the universal quantifier. Lastly we are lead to a novel characterization of realism.     

A Theory of Truthmaker Content I: Conjunction,Disjunction and Negation

I develop a basic theory of content within the framework of truthmaker semantics and, in the second part, consider some of the applications to subject matter, common content, logical subtraction and ground.     

概念语义与弗雷格迷题消解

弗雷格因同一替换律讨论而提出了涵义与指称的理论,这个理论后来引出了弗雷格迷题。弗雷格迷题的形成有多方原因,直接指称论对弗雷格理论批评是主要原因之一,以至于可以说,这是产生于直接指称论哲学立场的迷题。尽管如此,弗雷格理论确有不足。最重要的是,弗雷格理论只有关于涵义与指称的理论,即只有语言层面的理论,而缺少认知层面的理论。这个不足使得在弗雷格理论基础上解决同一替换律问题难有令人满意的结果,让"迷题"多添了几分"迷"的色彩。这里将给出一个新的方案:在弗雷格理论的基础上,增加有关概念的理论,以概念和内涵、涵义等这些概念的形式刻画为中心,建立可以消解弗雷格迷题的形式语义学,即概念语义。通过概念语义可以在不同层次上对弗雷格迷题的消解给出统一回答。