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Ideas for 'Truth and the Past', 'Philosophies of Mathematics' and 'Causality and Explanation'

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9 ideas

5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
8195 Undecidable statements result from quantifying over infinites, subjunctive conditionals, and the past tense [Dummett]
     Full Idea: I once wrote that there are three linguistic devices that make it possible for us to frame undecidable statements: quantification over infinity totalities, as expressed by word such as 'never'; the subjunctive conditional form; and the past tense.
     From: Michael Dummett (Truth and the Past [2001], 4)
     A reaction: Dummett now repudiates the third one. Statements containing vague concepts also appear to be undecidable. Personally I have no problems with deciding (to a fair extent) about 'never x', and 'if x were true', and 'it was x'.
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
10111 Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
     Full Idea: A hallmark of our realist stance towards the natural world is that we are prepared to assert the Law of Excluded Middle for all statements about it. For all statements S, either S is true, or not-S is true.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4)
     A reaction: Personally I firmly subscribe to realism, so I suppose I must subscribe to Excluded Middle. ...Provided the statement is properly formulated. Or does liking excluded middle lead me to realism?
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10129 A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman]
     Full Idea: A 'model' of a theory is an assignment of meanings to the symbols of its language which makes all of its axioms come out true.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7)
     A reaction: If the axioms are all true, and the theory is sound, then all of the theorems will also come out true.
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
10105 Differences between isomorphic structures seem unimportant [George/Velleman]
     Full Idea: Mathematicians tend to regard the differences between isomorphic mathematical structures as unimportant.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
     A reaction: This seems to be a pointer towards Structuralism as the underlying story in mathematics. The intrinsic character of so-called 'objects' seems unimportant. How theories map onto one another (and onto the world?) is all that matters?
5. Theory of Logic / K. Features of Logics / 2. Consistency
10119 Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman]
     Full Idea: Consistency is a purely syntactic property, unlike the semantic property of soundness.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6)
10126 A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman]
     Full Idea: If there is a sentence such that both the sentence and its negation are theorems of a theory, then the theory is 'inconsistent'. Otherwise it is 'consistent'.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7)
5. Theory of Logic / K. Features of Logics / 3. Soundness
10120 Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman]
     Full Idea: Soundness is a semantic property, unlike the purely syntactic property of consistency.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6)
5. Theory of Logic / K. Features of Logics / 4. Completeness
10127 A 'complete' theory contains either any sentence or its negation [George/Velleman]
     Full Idea: If there is a sentence such that neither the sentence nor its negation are theorems of a theory, then the theory is 'incomplete'. Otherwise it is 'complete'.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7)
     A reaction: Interesting questions are raised about undecidable sentences, irrelevant sentences, unknown sentences....
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / b. The Heap paradox ('Sorites')
8194 Surely there is no exact single grain that brings a heap into existence [Dummett]
     Full Idea: There is surely no number n such that "n grains of sand do not make a heap, although n+1 grains of sand do" is true.
     From: Michael Dummett (Truth and the Past [2001], 4)
     A reaction: It might be argued that there is such a number, but no human being is capable of determing it. Might God know the value of n? On the whole Dummett's view seems the most plausible.