The Logical Deduction of Doctrine

The idea that Roman Catholic doctrines for which there is no early testimony can be explained as logical deductions from undoubtedly early teachings is usually dismissed as obviously false. By invoking the logical properties of doctrines expressed as explicit generalizations, however, and by distinguishing deductions in which all the assumptions represent Apostolic doctrine from those in which all the doctrinal assumptions are Apostolic, a way is found to deduce the disputed doctrines while leaving the immutability of doctrine intact. Although a theory of theological development is thus not needed to justify doctrinal additions, developments in theology nevertheless often motivate the authoritative pronouncements cited by doctrinal deductions. Finally, it is argued that a correct understanding of such deductions improves the prospects for reunion between those whose doctrinal axioms coincide even if differing historical information has rendered them incapable of following the same chain of deductions.     

Sequent-systems and groupoid models. I

The purpose of this paper is to connect the proof theory and the model theory of a family of propositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related toBCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hubert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.The first part of the paper, which follows, contains the first two sections, which deal with sequent-systems and Hubert-formulations. The second part, due to appear in the next issue of this journal, will contain the third section, which deals with groupoid models.     

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.     

An Admissible Semantics for Propositionally Quantified Relevant Logics

The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p-instantiations of A. It is also shown that without the admissibility qualification many of the systems considered are semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t. The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras.     

Logics for Epistemic Programs

We construct logical languages which allow one to represent a variety of possible types of changes affecting the information states of agents in a multi-agent setting. We formalize these changes by defining a notion of epistemic program. The languages are two-sorted sets that contain not only sentences but also actions or programs. This is as in dynamic logic, and indeed our languages are not significantly more complicated than dynamic logics. But the semantics is more complicated. In general, the semantics of an epistemic program is what we call aprogram model. This is a Kripke model of ‘actions’,representing the agents' uncertainty about the current action in a similar way that Kripke models of ‘states’ are commonly used in epistemic logic to represent the agents' uncertainty about the current state of the system. Program models induce changes affecting agents' information, which we represent as changes of the state model, called epistemic updates. Formally, an update consists of two operations: the first is called the update map, and it takes every state model to another state model, called the updated model; the second gives, for each input state model, a transition relation between the states of that model and the states of the updated model. Each variety of epistemic actions, such as public announcements or completely private announcements to groups, gives what we call an action signature, and then each family of action signatures gives a logical language. The construction of these languages is the main topic of this paper. We also mention the systems that capture the valid sentences of our logics. But we defer to a separate paper the completeness proof. The basic operation used in the semantics is called the update product. A version of this was introduced in Baltag et al. (1998), and the presentation here improves on the earlier one. The update product is used to obtain from any program model the corresponding epistemic update, thus allowing us to compute changes of information or belief. This point is of interest independently of our logical languages. We illustrate the update product and our logical languages with many examples throughout the paper.     

From IF to BI

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O’Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural rôle, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural rôle.     

The Logic of Location

I consider the idea of a propositional logic of location based on the following semantic framework, derived from ideas of Prior. We have a collection L of locations and a collection S of statements such that a statement may be evaluated for truth at each location. Typically one and the same statement may be true at one location and false at another. Given this semantic framework we may proceed in two ways: introducing names for locations, predicates for the relations among them and an “at” preposition to express the value of statements at locations; or introduce statement operators which do not name locations but whose truth-conditional effect depends on the truth or falsity of embedded statements at various locations. The latter is akin to Prior’s approach to tense logic. In any logic of location there will be some basic operators which we can define. By ringing the changes on the topology of locations, different logical systems may be generated, and the challenge for the logician is then in each case to find operators, axioms and rules yielding a proof theory adequate to the semantics. The generality of the approach is illustrated with familiar and not so familiar examples from modal, tense and place logic, mathematics, and even the logic of games.

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一类命题逻辑的一般弱框架择类语义

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The Unity of Language and Logic in Wittgenstein's Tractatus

The purpose of this paper is to offer an interpretation of the Tractatus’ proof of the unity of logic and language. The kernel of the proof is the thesis that the sole logical constant is the general propositional form. I argue that the Grundgedanke, the existence of the sole fundamental operation N and the analyticity thesis, together with the fact that the operation NN can always be seen as having no specific formal difference between its result and its base, imply that NN is intrinsic to every elementary proposition. I also argue that the picture theory of proposition is an account of the generation of propositions via naming, and that its crucial idea is that naming is the instantiation of the form of a name, which consists in arbitrarily picking out an object as the meaning of the name from those objects sorted out by the form of the name. It follows that the existential quantifier, that is, NN, is intrinsic to naming (and therefore to every elementary proposition). It is then proven that the sole logical constant is the general propositional form. This, together with the truth‐functionality of logical necessity, implies that logic and language are unified via a general rule – logical syntax.     

Supervaluations and the Propositional Attitude Constraint

For the sentences of languages that contain operators that express the concepts of definiteness and indefiniteness, there is an unavoidable tension between a truth-theoretic semantics that delivers truth conditions for those sentences that capture their propositional contents and any model-theoretic semantics that has a story to tell about how indetifiniteness in a constituent affects the semantic value of sentences which imbed it. But semantic theories of both kinds play essential roles, so the tension needs to be resolved. I argue that it is the truth theory which correctly characterises the notion of truth, per se. When we take into account the considerations required to bring model theory into harmony with truth theory, those considerations undermine the arguments standardly used to motivate supervaluational model theories designed to validate classical logic. But those considerations also show that celebration would be premature for advocates of the most frequently encountered rival approach – many-valued model theory.     

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.     

Logic with numbers

Many people regard utility theory as the only rigorous foundation for subjective probability, and even de Finetti thought the betting approach supplemented by Dutch Book arguments only good as an approximation to a utility-theoretic account. I think that there are good reasons to doubt this judgment, and I propose an alternative, in which the probability axioms are consistency constraints on distributions of fair betting quotients. The idea itself is hardly new: it is in de Finetti and also Ramsey. What is new is that it is shown that probabilistic consistency and consequence can be defined in a way formally analogous to the way these notions are defined in deductive (propositional) logic. The result is a free-standing logic which does not pretend to be a theory of rationality and is therefore immune to, among other charges, that of “logical omniscience”.     

Inexact Knowledge with Introspection

Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidean. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson’s luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. We formulate a generalized version of the semantics, called token semantics, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resource-sensitive logic with implications for the representation of common knowledge in situations of bounded rationality.     

An Axiomatisation of a Pure Calculus of Names

A calculus of names is a logical theory describing relations between names. By a pure calculus of names we mean a quantifier-free formulation of such a theory, based on classical propositional calculus. An axiomatisation of a pure calculus of names is presented and its completeness is discussed. It is shown that the axiomatisation is complete in three different ways: with respect to a set theoretical model, with respect to Le?niewski??s Ontology and in a sense defined with the use of axiomatic rejection. The independence of axioms is proved. A decision procedure based on syntactic transformations and models defined in the domain of only two members is defined.     

Logics of public communications

Multi-modal versions of propositional logics S5 or S4—commonly accepted as logics of knowledge—are capable of describing static states of knowledge but they do not reflect how the knowledge changes after communications among agents. In the present paper (part of broader research on logics of knowledge and communications) we define extensions of the logic S5 which can deal with public communications. The logics have natural semantics. We prove some completeness, decidability and interpretability results and formulate a general method that solves certain kind of problems involving public communications—among them well known puzzles of Muddy Children and Mr. Sum & Mr. Product. As the paper gives a formal logical treatment of the operation of restriction of the universe of a Kripke model, it contributes also to investigations of semantics for modal logics. This paper was originally published as Plaza, J. A. (1989). Logics of public communications. In M. L. Emrich, M. S. Pfeifer, M. Hadzikadic, & Z.W. Ras (Eds.), Proceedings of the fourth international symposium on methodologies for intelligent systems: Poster session program (pp. 201–216). Publisher: Oak Ridge National Laboratory, ORNL/DSRD-24. Research partly supported by NSF Grant CCR-8702307 and PSC-CUNY Grant 668283.     

How to be a structuralist all the way down

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.     

On A Neglected Path to Intuitionism

According to Quine, in any disagreement over basic logical laws the contesting parties must mean different things by the connectives or quantifiers implicated in those laws; when a deviant logician ‘tries to deny the doctrine he only changes the subject’. The standard (Heyting) semantics for intuitionism offers some confirmation for this thesis, for it represents an intuitionist as attaching quite different senses to the connectives than does a classical logician. All the same, I think Quine was wrong, even about the dispute between classicists and intuitionists. I argue for this by presenting an account of consequence, and a cognate semantic theory for the language of the propositional calculus, which (a) respects the meanings of the connectives as embodied in the familiar classical truth-tables, (b) does not presuppose Bivalence, and with respect to which (c) the rules of the intuitionist propositional calculus are sound and complete. Thus the disagreement between classicists and intuitionists, at least, need not stem from their attaching different senses to the connectives; one may deny the doctrine without changing the subject. The basic notion of my semantic theory is truth at a possibility, where a possibility is a way that (some) things might be, but which differs from a possible world in that the way in question need not be fully specific or determinate. I compare my approach with a previous theory of truth at a possibility due to Lloyd Humberstone, and with a previous attempt to refute Quine’s thesis due to John McDowell.     

Wittgenstein's ab-Notation: An Iconic Proof Procedure

This paper systematically outlines Wittgenstein's ab-notation. The purpose of this notation is to provide a proof procedure in which ordinary logical formulas are converted into ideal symbols that identify the logical properties of the initial formulas. The general ideas underlying this procedure are in opposition to a traditional conception of axiomatic proof and are related to Peirce's iconic logic. Based on Wittgenstein's scanty remarks concerning his ab-notation, which almost all apply to propositional logic, this paper explains how to extend his method to a subset of first-order formulas, namely, formulas that do not contain dyadic sentential connectives within the scope of any quantifier.     

A Temporal Logic for Sortals

With the past and future tense propositional operators in its syntax, a formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. A completeness proof for the system is constructed and its absolute consistency proved. The completeness proof is given relative to a notion of logical validity provided by an intensional semantic system, which assumes an approach to sortals from a modern form of conceptualism.     

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.