Are Logical Languages Compositional?

In this paper I argue that in contrast to natural languages, logical languages typically are not compositional. This does not mean that the meaning of expressions cannot be determined at all using some well-defined set of rules. It only means that the meaning of an expression cannot be determined without looking at its form. If one is serious about the compositionality of a logic, the only possibility I see is to define it via abstraction from a variable free language.     

An Application of Information Theory to the Problem of the Scientific Experiment

There are two basic approaches to the problem of induction:the empirical one, which deems that the possibility of induction depends on how theworld was made (and how it works) and the logical one, which considers the formation(and function) of language. The first is closer to being useful for induction, whilethe second is more rigorous and clearer. The purpose of this paper is to create an empiricalapproach to induction that contains the same formal exactitude as the logical approach.This requires: (a) that the empirical conditions for the induction are enunciatedand (b) that the most important results already obtained from inductive logic are againdemonstrated to be valid. Here we will be dealing only with induction by elimination,namely the analysis of the experimental confutation of a theory. The result will bea rule of refutation that takes into consideration all of the empirical aspect of theexperiment and has each of the asymptotic properties which inductive logic has shown tobe characteristic of induction.     

Algebraic foundations for the semantic treatment of inquisitive content

In classical logic, the proposition expressed by a sentence is construed as a set of possible worlds, capturing the informative content of the sentence. However, sentences in natural language are not only used to provide information, but also to request information. Thus, natural language semantics requires a logical framework whose notion of meaning does not only embody informative content, but also inquisitive content. This paper develops the algebraic foundations for such a framework. We argue that propositions, in order to embody both informative and inquisitive content in a satisfactory way, should be defined as non-empty, downward closed sets of possibilities, where each possibility in turn is a set of possible worlds. We define a natural entailment order over such propositions, capturing when one proposition is at least as informative and inquisitive as another, and we show that this entailment order gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators. Just as in classical logic, these semantic operators are then associated with the logical constants in a first-order language. We explore the logical properties of the resulting system and discuss its significance for natural language semantics. We show that the system essentially coincides with the simplest and most well-understood existing implementation of inquisitive semantics, and that its treatment of disjunction and existentials also concurs with recent work in alternative semantics. Thus, our algebraic considerations do not lead to a wholly new treatment of the logical constants, but rather provide more solid foundations for some of the existing proposals.     

Criteria for logical formalization

The article addresses two closely related questions: What are the criteria of adequacy of logical formalization of natural language arguments, and what gives logic the authority to decide which arguments are good and which are bad? Our point of departure is the criticism of the conception of logical formalization put forth, in a recent paper, by M. Baumgartner and T. Lampert. We argue that their account of formalization as a kind of semantic analysis brings about more problems than it solves. We also argue that the criteria of adequate formalization need not be based on truth conditions associated with logical formulas; in our view, they are better based on structural (inferential) grounds. We then put forward our own version of the criteria. The upshot of the discussion that follows is that the quest for an adequate formalization in a suitable logical language is best conceived of as the search for a Goodmanian reflective equilibrium.     

Synonymy and Intra-Theoretical Pluralism

The starting point of this paper is a version of intra-theoretical (logical) pluralism that was recently proposed by Hjortland [2013]. In a first move, I use synonymy-relations to formulate an intuitively compelling objection against Hjortland's claim that, if one uses a single calculus to characterise the consequence relations of the paraconsistent logic LP and the paracomplete logic K3, one immediately obtains multiple consequence relations for a single language and hence a reply to the Quinean charge of meaning variance. In a second move, I explain how a natural generalisation of the notion of synonymy (adapted to the 3-sided sequent-calculus used by Hjortland) can be used to counter this objection, but I also show how the solution can be turned into an equally devastating ‘one logic after all’ type of objection. Finally, I propose the general diagnosis that these problems could only arise in the presence of conceptual distinctions that are too coarse to accommodate coherent pluralist theses. The latter leads to the general methodological recommendation that the conceptual resources used to think and talk about logic should be kept in line with the formal resources that are used to define and describe a logical theory.     

Gebrauchssprache und logik. eine philosophiehistorische notiz zu frege und lotze

Die Zusammenhänge die zwischen G. Freges und R. H. Lotzes logischen Lehren bestehen, sind, wie die gemeinsame Beurteilung der Gebrauchssprache zeigt, noch tiefer als allgemein angenommen. Insbesondere die von Frege konzipierte logische Sprachkritik ist in drei Punkten von Lotze beeinflußt. Lotze fordert nämlich die strenge Trennung von Logik und Gebrauchssprache. Daneben spielt der Begriff des Logischeinfachen eine zentrale Rolle in seiner Logik. Schließlich unterscheidet er den objektiven Gedanken von seiner Färbung.

The connexions that exist between the logical doctrines of G. Frege and R. H. Lotze are, as shows their common treatment of natural language, deeper than is generally admitted. In particular, the logical criticism of language conceived by Frege is influenced in three points by Lotze. Firstly, Lotze postulates the strict separation of logic and natural language. Furthermore, the idea of logical simplicity plays an important role in his logic. Finally, he distinguishes objective thought from its tone.     

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.     

关于实质蕴涵“如果4那么B”的一些思考

这篇文章尝试辩解布尔（亚里士多德式）逻辑中关于实质蕴涵的反直观定义。主要问题是：自然语言和逻辑定义的“蕴涵”的隐喻之间的联系是什么？为了更好地理解该定义和避免一些潜在的困惑,本文呈现出不同论点,并论述了“A→B”的经典定义的合理性。     

Classical harmony: Rules of inference and the meaning of the logical constants

The thesis that, in a system of natural deduction, the meaning of a logical constant is given by some or all of its introduction and elimination rules has been developed recently in the work of Dummett, Prawitz, Tennant, and others, by the addition of harmony constraints. Introduction and elimination rules for a logical constant must be in harmony. By deploying harmony constraints, these authors have arrived at logics no stronger than intuitionist propositional logic. Classical logic, they maintain, cannot be justified from this proof-theoretic perspective. This paper argues that, while classical logic can be formulated so as to satisfy a number of harmony constraints, the meanings of the standard logical constants cannot all be given by their introduction and/or elimination rules; negation, in particular, comes under close scrutiny.     

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.     

面向汉语认知与功能特征的语法逻辑

本文研究范畴语法的两种扩充,一是从认知特征角度的扩充,将范畴语法扩充为认知特征范畴语法,通过具有完全性的逻辑证明解决了一些不合语言事实的句子判别问题;二是从功能特征角度的扩充,提出逻辑推理的形式和进一步将二者统一的可能性问题。     

Why Combine Logics?

Combining logics has become a rapidly expanding entreprise that is inspired mainly by concerns about modularity and the wish to join together tailor made logical tools into more powerful but still manageable ones. A natural question is whether it offers anything new over and above existing standard languages.By analysing a number of applications where combined logics arise, we argue that combined logics are a potentially valuable tool in applied logic, and that endorsements of standard languages often miss the point. Using the history of quantified modal logic as our main example, we also show that the use of combined structures and logics is a recurring theme in the analysis of existing logical systems.     

From reasonable preferences,via argumentation,to logic

This article demonstrates that typical restrictions which are imposed in dialogical logic in order to recover first-order logical consequence from a fragment of natural language argumentation are also forthcoming from preference profiles of boundedly rational players, provided that these players instantiate a specific player type and compute partial strategies. We present two structural rules, which are formulated similarly to closure rules for tableaux proofs that restrict players' strategies to a mapping between games in extensive forms (i.e., game trees) and proof trees. Both rules are motivated from players' preferences and limitations; they can therefore be viewed as being player-self-imposable. First-order logical consequence is thus shown to result from playing a specific type of argumentation game. The alignment of such games with the normative model of the Pragma-dialectical theory of argumentation is positively evaluated. But explicit rules to guarantee that the argumentation game instantiates first-order logical consequence have now become gratuitous, since their normative content arises directly from players' preferences and limitations. A similar naturalization for non-classical logics is discussed.     

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.     

Wie Natürlich ist das System der Natürlichen Deduktion?

How natural is natural deduction?– Gentzen's system of natural deduction intends to fit logical rules to the effective mathematical reasoning in order to overcome the artificiality of deductions in axiomatic systems (¶ 2). In spite of this reform some of Gentzen's rules for natural deduction are criticised by psychologists and natural language philosophers for remaining unnatural. The criticism focuses on the principle of extensionality and on formalism of logic (¶ 3). After sketching the criticism relatively to the main rules, I argue that the criteria of economy, simplicity, pertinence etc., on which the objections are based, transcend the strict domain of logic and apply to arguments in general (¶ 4). (¶ 5) deals with Frege's critique of the concept of naturalness as regards logic. It is shown that this concept means a regression into psychologism and is exposed to the same difficulties as are: relativity, lack of precision, the error of arguing from `is' to `ought' (the naturalistic fallacy). Despite of these, the concept of naturalness plays the role of a diffuse ideal which favours the construction of alternative deductive systems in contrast to the platonic conception of logic (¶ 6).     

Bilattices with Implications

In a previous work we studied, from the perspective of Abstract Algebraic Logic, the implicationless fragment of a logic introduced by O. Arieli and A. Avron using a class of bilattice-based logical matrices called logical bilattices. Here we complete this study by considering the Arieli-Avron logic in the full language, obtained by adding two implication connectives to the standard bilattice language. We prove that this logic is algebraizable and investigate its algebraic models, which turn out to be distributive bilattices with additional implication operations. We axiomatize and state several results on these new classes of algebras, in particular representation theorems analogue to the well-known one for interlaced bilattices.     

Traditional Logic,Modern Logic and Natural Language

In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’.