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Ideas for 'fragments/reports', 'Methodological Reflections on Current Linguistic Theory' and 'Introduction to Mathematical Logic'

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5 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 / 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)