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28
Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an appro ..."
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Cited by 57 (2 self)
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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...
Logical Pluralism
 To appear, Special Logic issue of the Australasian Journal of Philosophy
, 2000
"... Abstract: A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism th ..."
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Cited by 23 (5 self)
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Abstract: A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism there is not one true logic; there are many. There is not always a single answer to the question “is this argument valid?” 1 Logic, Logics and Consequence Anyone acquainted with contemporary Logic knows that there are many socalled logics. 1 But are these logics rightly socalled? Are any of the menagerie of nonclassical logics, such as relevant logics, intuitionistic logic, paraconsistent logics or quantum logics, as deserving of the title ‘logic ’ as classical logic? On the other hand, is classical logic really as deserving of the title ‘logic ’ as relevant logic (or any of the other nonclassical logics)? If so, why so? If not, why not? Logic has a chief subject matter: Logical Consequence. The chief aim of
Secondorder logic and foundations of mathematics
 The Bulletin of Symbolic Logic
, 2001
"... We discuss the differences between firstorder set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if secondorder logic is understood in its full semantics capable of characterizing categorical ..."
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Cited by 18 (3 self)
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We discuss the differences between firstorder set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if secondorder logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. Firstorder set theory and secondorder logic are not radically different: the latter is a major fragment of the former. 1
Generalized Kripke models for epistemic logic
 Theoretical Aspects of Reasoning about Knowledge: Proc. Fourth Conference
, 1992
"... In this paper a generalization of Kripke models is proposed for systemizing the study of the many different epistemic notions that appear in the literature. The generalized Kripke models explicitly represent an agent's epistemic states to which the epistemic notions refer. Two central epistemic ..."
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Cited by 10 (1 self)
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In this paper a generalization of Kripke models is proposed for systemizing the study of the many different epistemic notions that appear in the literature. The generalized Kripke models explicitly represent an agent's epistemic states to which the epistemic notions refer. Two central epistemic notions are identified: objective (S5) knowledge and rational introspective (KD45) belief. Their interaction is determined and a notion of justified true belief is explained in terms of them. The logic of this notion of justified true belief is shown to be S4.2, which is in accordance with a conjecture by Wolfgang Lenzen. The logic of justified belief is also determined. 1
Word and objects
 Noûs
, 2002
"... The aim of this essay is to show that the subjectmatter of ontology is richer than one might have thought. Our route will be indirect. We will argue that there are circumstances under which standard firstorder regimentation is unacceptable, and that more appropriate varieties of regimentation lead ..."
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Cited by 7 (6 self)
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The aim of this essay is to show that the subjectmatter of ontology is richer than one might have thought. Our route will be indirect. We will argue that there are circumstances under which standard firstorder regimentation is unacceptable, and that more appropriate varieties of regimentation lead to unexpected kinds of ontological commitment. Quine has taught us that ontological inquiry—inquiry as to what there is—can be separated into two distinct tasks. 1 On the one hand, there is the problem of determining the ontological commitments of a given theory; on the other, the problem of deciding what theories to accept. The objects whose existence we have reason to believe in are then the ontological commitments of the theories we have reason to accept. Regarding the former of these two tasks, Quine maintains that a firstorder theory is committed to the existence of an object satisfying a certain predicate if and only if some object satisfying that predicate must be admitted among the values of the theory’s variables in order for the theory to be true. Quine’s criterion is extremely attractive, but it applies only to theories that are couched in firstorder languages. Offhand this is not a serious constraint, because most of our theories have straightforward firstorder regimentations. But here we shall see that there is a special kind of tension between regimenting our discourse in a firstorder language and allowing our quantifiers to range over absolutely everything. 2 We will proceed on the assumption that absolutely unrestricted quantification is possible, and show that an important class of English sentences resists firstorder regimentation. This will lead us to develop alternate languages of regimentation, languages containing plural
Absolute Identity and Absolute Generality
"... The aim of this chapter is to tighten our grip on some issues about quantification by analogy with corresponding issues about identity on which our grip is tighter. We start with the issues about identity. I In conversations between native speakers, words such as ‘same ’ and ‘identical ’ do not usua ..."
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Cited by 2 (0 self)
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The aim of this chapter is to tighten our grip on some issues about quantification by analogy with corresponding issues about identity on which our grip is tighter. We start with the issues about identity. I In conversations between native speakers, words such as ‘same ’ and ‘identical ’ do not usually cause much difficulty. We take it for granted that others use them with the same sense as we do. If it is unclear whether numerical or qualitative identity is intended, a brief gloss such as ‘one thing not two ’ for the former or ‘exactly alike ’ for the latter removes the unclarity. In this paper, numerical identity is intended. A particularly conscientious and logically aware speaker might explain what ‘identical ’ means in her 1 mouth by saying: ‘Everything is identical with itself. If something is identical with something, then whatever applies to the former also applies to the latter. ’ It seems perverse to continue doubting whether ‘identical ’ in her mouth means identical (in our sense). Yet other interpretations are conceivable. For instance, she might have been speaking an odd idiolect in which ‘identical ’ means in love, under the misapprehension
Arguments for the Continuity Principle
, 2000
"... Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences ..."
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Cited by 2 (0 self)
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Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences . . . . . . . . . 15 3.2 Kripke's Schema and full PEM . . . . . . . . . . . . . . . . . 15 3.3 The KLST theorem . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion 19 1 The continuity principle There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the rst time in print in [Brouwer 1918]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fa
Nonclassical metatheory for nonclassical logics
, 2012
"... A number of authors have objected to the application of nonclassical logic to problems in philosophy on the basis that these nonclassical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible f ..."
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A number of authors have objected to the application of nonclassical logic to problems in philosophy on the basis that these nonclassical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for nonclassicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is to show that a large class of nonclassical logics are strong enough to formulate their own model theory in a corresponding nonclassical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid
Squeezing arguments
, 2010
"... Many of our concepts are introduced to us via, and seem only to be constrained by, roughandready explanations and some sample paradigm positive and negative applications. This happens even in informal logic and mathematics. Yet in some cases, the concepts in question – although only informally and ..."
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Many of our concepts are introduced to us via, and seem only to be constrained by, roughandready explanations and some sample paradigm positive and negative applications. This happens even in informal logic and mathematics. Yet in some cases, the concepts in question – although only informally and vaguely characterized – in fact have, or appear to have, entirely determinate extensions. Here’s one familiar example. When we start learning computability theory, we are introduced to the idea of an algorithmically computable function (from numbers to numbers) – i.e. one whose value for any given input can be determined by a stepbystep calculating procedure, where each step is fully determined by some antecedently given finite set of calculating rules. We are told that we are to abstract from practical considerations of how many steps will be needed and how much ink will be spilt in the process, so long as everything remains finite. We are also told that each step is to be ‘small’ and the rules governing it must be ‘simple’, available to a cognitively limited calculating agent: for we want an algorithmic procedure, stepbyminimalstep, to be idiotproof. For a classic elucidation of this kind, see e.g. Rogers (1967, pp. 1–5). Church’s Thesis, in