3. Proof Theory and Direct Derivations

Historical Materialist Hook

[RESEARCH REQUIRED: Frame the transition to the rigorous rules of the Fitch system around the bureaucratic necessity for standardized, interchangeable intellectual labor. Utilize Theodore M. Porter’s Trust in Numbers to explain how mechanical objectivity functions as a technology of distance, and incorporate Dirk J. Struik’s A Concise History of Mathematics.]

Core Technical Synthesis

[RESEARCH REQUIRED: Explain the Fitch system of natural deduction. Discuss how to manipulate logical strings independently of semantic meaning. Detail the strict introduction and elimination rules required to rigorously transform premises into verified conclusions.]

Guided Example: Direct Derivation

A direct derivation using Modus Ponens (Conditional Elimination) and Conjunction Introduction:

1. P -> Q      : PR
2. P           : PR
3. Q           : ->E 1, 2
4. P /\ Q      : /\I 2, 3

Problem Set

Exercise 3.1

Execute a direct derivation in the Carnap checker.

1. (P /\ Q) : PR
2. P -> R : PR
3. P : /\E 1
4. R : ->E 2, 3

Dictionary Reference Map

For Fitch deduction mechanics: forall x Calgary (Magnus et al. 2025, Ch. 16–17)
For Carnap proof checking: The Carnap Book (2025, Ch. 3)

Next: Hypothetical Reasoning and Deduction Architecture ➔