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A revengeimmune solution to the semantic paradoxes
 Journal of Philosophical Logic
"... The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(〈A〉) ↔ A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ord ..."
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Cited by 24 (6 self)
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The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(〈A〉) ↔ A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(〈A〉) with A within the language. The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are “defective”. We can in fact define a hierarchy of “defectiveness ” predicates within the language; contrary to claims that any solution to the paradoxes just breeds further paradoxes (“revenge problems”) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various ”levels of defectiveness ” can all be made coherent together within a single object language. 1
There are no abstract objects
 In
, 2008
"... Suppose you start out inclined towards the hardheaded view that the world of material objects is the whole of reality. You elaborate: ‘Everything there is is a material object: the sort of thing you could bump into; the sort of thing for which it would be sensible to ask how much it weighs, what sh ..."
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Cited by 5 (1 self)
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Suppose you start out inclined towards the hardheaded view that the world of material objects is the whole of reality. You elaborate: ‘Everything there is is a material object: the sort of thing you could bump into; the sort of thing for which it would be sensible to ask how much it weighs, what shape it is, how fast it is moving, and how far it is from other material objects. There is nothing else. ’ You develop some practice defending your thesis from the expected objections, from believers in ghosts, God, immaterial souls, Absolute Space, and so on. None of this practice will do you much good the first time you are confronted with the following objection: What about numbers and properties? These are obviously not material objects. It would be crazy to think that you might bump into the number two, or the property of having many legs. One would have to be confused to wonder how much these items weigh, or how far away they are. But obviously there are numbers and properties. Surely even you don’t deny that there are four prime numbers between one and ten, or that spiders and insects share many important anatomical properties. 1 But these wellknown truths evidently imply that there are numbers, and that there are properties. So
1 PYTHAGOREAN POWERS or A CHALLENGE TO PLATONISM
"... I have tried to apprehend the Pythagorean power by which number holds sway above the flux. Bertrand Russell, Autobiography, vol. 1, Prologue. The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot ..."
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I have tried to apprehend the Pythagorean power by which number holds sway above the flux. Bertrand Russell, Autobiography, vol. 1, Prologue. The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects
Philosophy of Mathematics: Making a Fresh Start
"... ABSTRACT: The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment ..."
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ABSTRACT: The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the effectiveness of mathematics in natural science.