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  • Die Logik ist keine Lehre, sondern ein Spiegelbild der Welt.
    Logik ist transzendental.

    - Ludwig Wittgenstein's (1921, 6.13) Tractatus Logico-Philosophicus in the original German

    Logic is not a body of doctrine, but a mirror-reflection of the world.
    Logic is transcendental.
    - Pears/McGuinness translation

    Proof Theory

    Crockett Johnson's (1965) Painting - Proof of the Pythagorean Theorem

    1. A well-defined proof theory makes precise how reasoning in that system is to proceed by manipulating formulae in accordance with certain rules
    2. Whereas the truth table test is used to illustrate the validity of an argument, proof theory is used to demonstrate the validity of an argument

    3. Proof theory is the argument theory that is employed in deductive reasoning
    4. The argument theory specifies the correct inference from a premise set φ to the conclusion set ψ

    5. NOTE: A wider-ranging logicism can encompass the non-deductive forms of reasoning:

    φMX ψ

    1. The superscript M denotes the mode of inference: deductive, abductive, inductive, analogical, etc
    2. The subscript X denotes particular inferential mechanisms

    1. In the context of deductive reasoning and its formalization in proof theory, X could be:
    2. Proof by natural deduction, which employs rules of inference that closely relate to the natural way of reasoning; or
    3. Proof by resolution, which is employed in automated theorem proving