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Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Predicative Foundations of Arithmetic
 Journal of Philosophical Logic
, 1995
"... Predicative mathematics in the sense originating with Poincaré andWeylbegins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or settheoretic standpoint, this appears problematic, for, as the story is usu ..."
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Predicative mathematics in the sense originating with Poincaré andWeylbegins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or settheoretic standpoint, this appears problematic, for, as the story is usually told, impredicative
A Formal Theory for KnowledgeBased Product Model Representation
, 1996
"... The field of design science attempts to place engineering design on a more formal, rigorous footing. This paper introduces recent work by the author in this area. ArtifactCentered Modeling (ACM) is a general framework intended to partition the design endeavor in manageable sections. A fundamental p ..."
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The field of design science attempts to place engineering design on a more formal, rigorous footing. This paper introduces recent work by the author in this area. ArtifactCentered Modeling (ACM) is a general framework intended to partition the design endeavor in manageable sections. A fundamental part of ACM is the representation of information about products being designed. The Axiomatic Information Model for Design (AIMD) is a formal theory about product information based on axiomatic set theory. AIMD provides formal bases for quantities, features, parts and assemblies, systems, and subassemblies; these are all notions essential to design. It is not a product modeling system per se, but rather a logic of product structure whose axioms define criteria for determining the logical validity of product models. A previous version of the theory has been found to contain logical inconsistencies; the version presented herein addresses those problems. A complete axiomatization of the new th...
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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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
Empirical Adequacy and Ramsification
 British Journal for Philosophy of Science
, 2004
"... Structural realism has been proposed as an epistemological position interpolating between realism and sceptical antirealism about scientific theories. The structural realist who accepts a scientific theory Θ thinks that Θ is empirically correct, and furthermore is a realist about the ‘structural co ..."
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Structural realism has been proposed as an epistemological position interpolating between realism and sceptical antirealism about scientific theories. The structural realist who accepts a scientific theory Θ thinks that Θ is empirically correct, and furthermore is a realist about the ‘structural content ’ of Θ. But what exactly is ‘structural content’? One proposal is that the ‘structural content ’ of a scientific theory may be associated with its Ramsey sentence ℜ(Θ). However, Demopoulos and Friedman argued, using ideas drawn from Newman’s earlier criticism of Russell’s structuralism, that this move fails to achieve an interesting intermediate position between realism and antirealism. Rather, ℜ(Θ) adds little content beyond the instrumentalistically acceptable claim that the theory Θ is empirically adequate. Here, we formulate carefully the crucial claim of Demopoulos and Friedman, and show that the Ramsey sentence ℜ(Θ) is true just in case Θ possesses a full model which is empirically correct and satisfies a certain cardinality condition on its theoretical domain. This suggests that structural realism is not a position significantly different
Toward a Theory of SecondOrder Consequence
 JOURNAL OF FORMAL LOGIC
, 1999
"... We develop an account of logical consequence for the secondorder language of set theory in the spirit of Boolos’s plural interpretation of monadic secondorder logic. ..."
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We develop an account of logical consequence for the secondorder language of set theory in the spirit of Boolos’s plural interpretation of monadic secondorder logic.
What does it mean to say that logic is formal?
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this