Temporal and atemporal truth in intuitionistic mathematics

In section 1 we argue that the adoption of a tenseless notion of truth entails a realistic view of propositions and provability. This view, in turn, opens the way to the intelligibility of theclassical meaning of the logical constants, and consequently is incompatible with the antirealism of orthodox intuitionism. In section 2 we show how what we call the potential intuitionistic meaning of the logical constants can be defined, on the one hand, by means of the notion of atemporal provability and, on the other, by means of the operator K of epistemic logic. Intuitionistic logic, as reconstructed within this perspective, turns out to be a part of epistemic logic, so that it loses its traditional foundational role, antithetic to that of classical logic. In section 3 we uphold the view that certain consequences of the adoption of atemporal notion of truth, despite their apparent oddity, are quite acceptable from an antirealist point of view.     

The Undecidability of Iterated Modal Relativization

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 ¹₁–complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 ⁰₁–complete. These results go via reduction to problems concerning domino systems.     

佛教逻辑学的论辩解释与认知解释——陈那、法称与因明

佛教逻辑有别于西方形式逻辑的一项重要特征,在于对论证前提为真的强调,并在此基础上提出了著名的"因三相"理论。论证前提的真,在陈那著作中又被理解为辩论主体将该前提确定为真。这种"确定"体现为文献中对"极成"(prasiddha)、"成"(siddha)、"决定"(ni?cita)、"见"(d???a)、"已知"(vidita)的强调。这些表达辩论者认知状态的词汇,皆可概括为佛教逻辑中的"认知算子"(epistemic operator)。本文通过研究东亚因明与法称两个传统对陈那《正理门论》中"决定同许"的理论设定的不同解释,说明因明传统在陈那奠定的方向上,进一步采取了"论辩解释"(dialectic interpretation),将"确定为真"解释为在辩论的情境中为辩论的双方承认为真(共许、同许)。法称《释量论自注》对陈那"决定同许"的解释,则表明由他开启的传统在陈那的基础上,进一步采取了"认知解释"(epistemic interpretation),将"确定为真"解释为在认识论的意义上得到确定(ni?cita/ni?caya,决定),即为有效认知的手段(量)所证成。这是因明传统与法称传统的一项根本差异。因明传统对辩论术的关注,极有可能反映了法称以前印度学界对陈那思想的诠释路径。     

Why Friedman's non-monotonic reasoning defies Hempel's covering law model

In this paper we will show that Hempel's covering law model can't deal very well with explanations that are based on incomplete knowledge. In particular the symmetry thesis, which is an important aspect of the covering law model, turns out to be problematic for these explanations. We will discuss an example of an electric circuit, which clearly indicates that the symmetry of explanation and prediction does not always hold. It will be argued that an alternative logic for causal explanation is needed. And we will investigate to what extent non-monotonic epistemic logic can provide such an alternative logical framework. Finally we will show that our non-monotonic logical analysis of explanation is not only suitable for simple cases such as the electric circuit, but that it also sheds new light on more controversial causal explanations such as Milton Friedman's explanation of the business cycle.We gratefully acknowledge the helpful suggestions and critical comments of Johan van Benthem, Bert Hamminga, Kevin Hoover, Theo Kuipers, and Rick Looyen.     

Knowledge of proofs

If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical constants. Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch's paradox:p and it will never be known thatp. It is argued that the intuitionist faces a dilemma: give up strongly entrenched common sense intuitions about such unknowable propositions, or give up verificationism. The third section considers one attempt to save intuitionism while partly giving up verificationism: keep the idea that a proposition is true iff there is a proof (verification) of it, and reject the idea that proofs must be recognizable in principle. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. This is the case with Dummett's meaning theoretical argument. At the same time the basic reason for regarding proofs as more than mere truth makers is lost.I am much indebted for comments to Lars Bergström, Per Martin-Löf, Wlodek Rabinowicz, Fredrik Stjernberg, Dag Westerståhl and Tim Williamson. I owe even more to the many seminars about truth and meaning, led by Dag Prawitz, at the philosophy department of Stockholm University. These were especially intense in the mideighties, when I was a graduate student.     

A Note on Bisimulation and Modal Equivalence in Provability Logic and Interpretability Logic

Provability logic is a modal logic for studying properties of provability predicates, and Interpretability logic for studying interpretability between logical theories. Their natural models are GL-models and Veltman models, for which the accessibility relation is well-founded. That’s why the usual counterexample showing the necessity of finite image property in Hennessy-Milner theorem (see [1]) doesn’t exist for them. However, we show that the analogous condition must still hold, by constructing two GL-models with worlds in them that are modally equivalent but not bisimilar, and showing how these GL-models can be converted to Veltman models with the same properties. In the process we develop some useful constructions: games on Veltman models, chains, and general method of transformation from GL-models/frames to Veltman ones.     

Epistemic justification and psychological realism

The main thesis of this paper is that it is not possible to determine the nature of epistemic justification apart from scientific psychological investigation. I call this view the strong thesis of methodological psychologism. Two sub-theses provide the primary support for this claim. The first sub-thesis is that no account of epistemic justification is correct which requires for the possession of at least one justified belief a psychological capacity which humans do not have. That is, the correct account of epistemic justification must be psychologically realistic. The second sub-thesis is that it is not possible to determine whether an account of epistemic justification is psychologically realistic apart from scientific psychological investigation. After defending these subtheses, I point out some interesting consequences of the overall thesis which present a challenge to traditional epistemology.     

Some Useful 16-Valued Logics: How a Computer Network Should Think

In Belnaps useful 4-valued logic, the set 2={T,F} of classical truth values is generalized to the set 4=(2)={,{T},{F},{T,F}}. In the present paper, we argue in favor of extending this process to the set 16=(4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR₂ with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN₃ with an information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN₃ coincide with the logic of FOUR₂, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arielis and Avrons notion of a logical bilattice and state a number of open problems for future research.Dedicated to Nuel D. Belnap on the occasion of his 75th Birthday     

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.

     

Formal systems of dialogue rules

Section 1 contains a survey of options in constructing a formal system of dialogue rules. The distinction between material and formal systems is discussed (section 1.1). It is stressed that the material systems are, in several senses, formal as well. In section 1.2 variants as to language form (choices of logical constants and logical rules) are pointed out. Section 1.3 is concerned with options as to initial positions and the permissibility of attacks on elementary statements. The problem of ending a dialogue, and of infinite dialogues, is treated in section 1.4. Other options, e.g., as to the number of attacks allowed with respect to each statement, are listed in section 1.5. Section 1.6 explains the concept of a chain of arguments.From section 2 onward four types of dialectic systems are picked out for closer study: D, E, Di and Ei. After a preliminary section on dialogue sequents and winning strategies, the equivalence of derivability in intuitionistic logic and the existence of a winning strategy (for the Proponent) on the strength of Ei is shown by simple inductive proofs.Section 3 contains a — relatively quick — proof of the equivalence of the four systems. It follows that each of them yields intuitionistic logic.     

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.     

An Internal Version of Epistemic Logic

Representing an epistemic situation involving several agents obviously depends on the modeling point of view one takes. We start by identifying the types of modeling points of view which are logically possible. We call the one traditionally followed by epistemic logic the perfect external approach, because there the modeler is assumed to be an omniscient and external observer of the epistemic situation. In the rest of the paper we focus on what we call the internal approach, where the modeler is one of the agents involved in the situation. For this approach we propose and axiomatize a logical formalism based on epistemic logic. This leads us to formalize some intuitions about the internal approach and about its connections with the external ones. Finally, we show that our internal logic is decidable and PSPACE-complete.     

Quantified Hintikka-style epistemic logic

This paper contains a formal treatment of the system of quantified epistemic logic sketched in Appendix II of Carlson (1983). Section 1 defines the syntax and recapitulates the model set rules and principles of the Appendix system. Section 2 defines a possible worlds semantics for this system, and shows that the Appendix system is complete with respect to this semantics. Section 3 extends the system by an explicit truth operatorT it is true that and considers quantification over nonexistent individuals. Section 4 formalizes the idea of variable identity criteria typical of Hintikkian epistemic logic.     

Interpolation in computing science: the semantics of modularization

The Interpolation Theorem, first formulated and proved by W. Craig fifty years ago for predicate logic, has been extended to many other logical frameworks and is being applied in several areas of computer science. We give a short overview, and focus on the theory of software systems and modules. An algebra of theories TA is presented, with a nonstandard interpretation of the existential quantifier . In TA, the interpolation property of the underlying logic corresponds with the quantifier combination property . It is shown how the Modularization Theorem, the Factorization Lemma and the Normal Form Theorem for module expressions can be proved in TA. Dedicated to the 50th anniversary of William Craig’s Interpolation Theorem.     

$${\in_K}$$: a Non-Fregean Logic of Explicit Knowledge

We present a new logic-based approach to the reasoning about knowledge which is independent of possible worlds semantics. \({\in_K}\) (Epsilon-K) is a non-Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom K _i φ → φ and some minimal conditions concerning common knowledge in a group. Knowledge is explicit and all forms of the logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the K-axiom, closure under logical connectives, etc. can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between \({\in_K}\) and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self-referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic “paradoxes” can be solved in \({\in_K}\). Every specific \({\in_K}\)-logic is defined as a certain extension of some underlying classical abstract logic.     

Formal systems of dialogue rules

Section 1 contains a survey of options in constructing a formal system of dialogue rules. The distinction between material and formal systems is discussed (section 1.1). It is stressed that the material systems are, in several senses, formal as well. In section 1.2 variants as to language form (choices of logical constants and logical rules) are pointed out. Section 1.3 is concerned with options as to initial positions and the permissibility of attacks on elementary statements. The problem of ending a dialogue, and of infinite dialogues, is treated in section 1.4. Other options, e.g., as to the number of attacks allowed with respect to each statement, are listed in section 1.5. Section 1.6 explains the concept of a chain of arguments.From section 2 onward four types of dialectic systems are picked out for closer study: D, E, Di and Ei. After a preliminary section on dialogue sequents and winning strategies, the equivalence of derivability in intuitionistic logic and the existence of a winning strategy (for the Proponent) on the strength of Ei is shown by simple inductive proofs.Section 3 contains a — relatively quick — proof of the equivalence of the four systems. It follows that each of them yields intuitionistic logic.     

Reasoning About Games

A mixture of propositional dynamic logic and epistemic logic that we call PDL + E is used to give a formalization of Artemov??s knowledge based reasoning approach to game theory, (KBR), [4, 5]. Epistemic states of players are represented explicitly and reasoned about formally. We give a detailed analysis of the Centipede game using both proof theoretic and semantic machinery. This helps make the case that PDL + E can be a useful basis for the logical investigation of game theory.     

Knowledge,certainty and probability

In this essay, I discuss some of the important logical principles governing the concepts of knowledge, certainty and probability. In the first section, I suggest a series of definitions of epistemic terms, employing as primitive the locution ‘p is epistemi‐cally possible to S’ In the second section, I develop an epistemic concept of probability and compare it to the concepts of certainty and knowledge. In the third section, I relate the epistemic concepts of certainty and probability to the quantifiers of traditional logic and to a non‐episteznic concept of probability. I conclude by noting similarities and differences between the two concepts of probability.