Combining Texts

All the ideas for 'Science and Method', 'Mathematical Truth' and 'The Concept of Logical Consequence'

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8 ideas

5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Split out the logical vocabulary, make an assignment to the rest. It's logical if premises and conclusion match [Tarski, by Rumfitt]
     Full Idea: Tarski made a division of logical and non-logical vocabulary. He then defined a model as a non-logical assignment satisfying the corresponding sentential function. Then a conclusion follows logically if every model of the premises models the conclusion.
     From: report of Alfred Tarski (The Concept of Logical Consequence [1936]) by Ian Rumfitt - The Boundary Stones of Thought 3.2
     A reaction: [compressed] This is Tarski's account of logical consequence, which follows on from his account of truth. 'Logical validity' is then 'true in every model'. Rumfitt doubts whether Tarski has given the meaning of 'logical consequence'.
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
X follows from sentences K iff every model of K also models X [Tarski]
     Full Idea: The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X.
     From: Alfred Tarski (The Concept of Logical Consequence [1936], p.417)
     A reaction: [see Idea 13343 for his account of a 'model'] He is offering to define logical consequence in general, but this definition fits what we now call 'semantic consequence', written |=. This it is standard practice to read |= as 'models'.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' is a sequence of objects which satisfies a complete set of sentential functions [Tarski]
     Full Idea: An arbitrary sequence of objects which satisfies every sentential function of the sentences L' will be called a 'model' or realization of the class L of sentences. There can also be a model of a single sentence is this way.
     From: Alfred Tarski (The Concept of Logical Consequence [1936], p.417)
     A reaction: [L' is L with the constants replaced by variables] Tarski is the originator of model theory, which is central to modern logic. The word 'realization' is a helpful indicator of what he has in mind. A model begins to look like a possible world.
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf]
     Full Idea: Most accounts of the concept of mathematical truth can be identified with serving one or another of either semantic theory (matching it to ordinary language), or with epistemology (meshing with a reasonable view) - always at the expense of the other.
     From: Paul Benacerraf (Mathematical Truth [1973], Intro)
     A reaction: The gist is that language pulls you towards platonism, and epistemology pulls you towards empiricism. He argues that the semantics must give ground. He's right.
6. Mathematics / A. Nature of Mathematics / 2. Geometry
One geometry cannot be more true than another [Poincaré]
     Full Idea: One geometry cannot be more true than another; it can only be more convenient.
     From: Henri Poincaré (Science and Method [1908], p.65), quoted by Stewart Shapiro - Philosophy of Mathematics
     A reaction: This is the culminating view after new geometries were developed by tinkering with Euclid's parallels postulate.
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan]
     Full Idea: Benacerraf argues that realists about mathematical objects have a nice normal semantic but no epistemology, and anti-realists have a good epistemology but an unorthodox semantics.
     From: report of Paul Benacerraf (Mathematical Truth [1973]) by Mark Colyvan - Introduction to the Philosophy of Mathematics 1.2
The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf]
     Full Idea: The principle defect of the standard (platonist) account of mathematical truth is that it appears to violate the requirement that our account be susceptible to integration into our over-all account of knowledge.
     From: Paul Benacerraf (Mathematical Truth [1973], III)
     A reaction: Unfortunately he goes on to defend a causal theory of justification (fashionable at that time, but implausible now). Nevertheless, his general point is well made. Your theory of what mathematics is had better make it knowable.
19. Language / E. Analyticity / 1. Analytic Propositions
Sentences are 'analytical' if every sequence of objects models them [Tarski]
     Full Idea: A class of sentences can be called 'analytical' if every sequence of objects is a model of it.
     From: Alfred Tarski (The Concept of Logical Consequence [1936], p.418)
     A reaction: See Idea 13344 and Idea 13343 for the context of this assertion.