The Calculus in Frege's Work
Frege declared nine of his propositions to be axioms, and justified them by arguing informally that, given their intended meanings, they express intuitive truths. Re-expressed in contemporary notation, these axioms are:
These are propositions 1, 2, 8, 28, 31, 41, 52, 54, and 58 in the Begriffschrifft. (1)-(3) govern material implication, (4)-(6) negation, (7) and (8) identity, and (9) the universal quantifier. (7) expresses Leibniz's indiscernibility of identicals, and (8) asserts that identity is a reflexive relation.
All other propositions are deduced from (1)-(9) by invoking any of the following inference rules:
- Modus ponens allows us to infer from and ;
- The rule of generalization allows us to infer from if x does not occur in P;
- The rule of substitution, which Frege does not state explicitly. This rule is much harder to articulate precisely than the two preceding rules, and Frege invokes it in ways that are not obviously legitimate.
The main results of the third chapter, titled "Parts from a general series theory," concern what is now called the ancestral of a relation R. "a is an R-ancestor of b" is written "aR*b".
Frege applied the results from the Begriffsschrifft, including those on the ancestral of a relation, in his later work The Foundations of Arithmetic. Thus, if we take xRy to be the relation y=x+1, then 0R*y is the predicate "y is a natural number." (133) says that if x, y, and z are natural numbers, then one of the following must hold: x<y, x=y, or y<x. This is the so-called "law of trichotomy".
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