Combining Philosophers

Ideas for H.Putnam/P.Oppenheim, Frank P. Ramsey and Michal Walicki

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7 ideas

5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic

17741	To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
	Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination.
	From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro)
	A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything.

5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic

13427	Either 'a = b' vacuously names the same thing, or absurdly names different things [Ramsey]
	Full Idea: In 'a = b' either 'a' and 'b' are names of the same thing, in which case the proposition says nothing, or of different things, in which case it is absurd. In neither case is it an assertion of a fact; it only asserts when a or b are descriptions.
	From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1)
	A reaction: This is essentially Frege's problem with Hesperus and Phosphorus. How can identities be informative? So 2+2=4 is extensionally vacuous, but informative because they are different descriptions.

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models

17747	A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]
	Full Idea: A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory.
	From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3)
	A reaction: The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going.

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

17748	The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
	Full Idea: The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more.
	From: Michal Walicki (Introduction to Mathematical Logic [2012], History E.2)
	A reaction: Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh?????

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

17761	A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
	Full Idea: Having such a compact [axiomatic] presentation of a complicated field [such as Euclid's], makes it possible to relate not only to particular theorems but also to the whole field as such.
	From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)

17763	Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
	Full Idea: Axiomatic systems, their primitive terms and proofs, are purely syntactic, that is, do not presuppose any interpretation. ...[142] They never address the world directly, but address a possible semantic model which formally represents the world.
	From: Michal Walicki (Introduction to Mathematical Logic [2012], 4.1)

5. Theory of Logic / L. Paradox / 1. Paradox

13334	Contradictions are either purely logical or mathematical, or they involved thought and language [Ramsey]
	Full Idea: Group A consists of contradictions which would occur in a logical or mathematical system, involving terms such as class or number. Group B contradictions are not purely logical, and contain some reference to thought, language or symbolism.
	From: Frank P. Ramsey (The Foundations of Mathematics [1925], p.171), quoted by Graham Priest - The Structure of Paradoxes of Self-Reference 1
	A reaction: This has become the orthodox division of all paradoxes, but the division is challenged by Priest (Idea 13373). He suggests that we now realise (post-Tarski?) that language is more involved in logic and mathematics than we thought.