The Timpson Lyceum · First-Order Predicate Logic (2024)

Example of an Axiomatic System - Lecture 7 (Logic)

Overview

This lecture works through a second, richer example of an axiomatic system to build fluency with the machinery of formal logic. After recapping the grammar of our formal language, the four components of any axiomatic system, and the five rules of derivation, we meet a new system built from a single predicate $C$, a two-term operator $+$, and a one-term operator $'$ (prime). The whole point of the exercise is to reason about uninterpreted symbols: we are deliberately given no intuition for what the axioms "mean," so every theorem must follow from the axioms alone. Working this way trains the habit of deriving conclusions strictly from stated premises, which is the heart of axiomatic reasoning. (The intended interpretation — that this is a fragment of propositional logic — is revealed only at the very end.)

Key Concepts

The grammar. Every symbol we write falls into exactly one of these categories (plus parentheses):

• Sentential connectives — $\land$ (and), $\lor$ (or), $\lnot$ (not), $\rightarrow$ (implies), $\leftrightarrow$ (equivalence).
• Terms — constants, variables, or operators combining them into new terms.
• Predicates — applied to terms, they return true or false.
• Quantifiers — only two: $\forall$ (for all) and $\exists$ (there exists).

A sentence is a formula with no free variables (every variable is bound by a quantifier).

The four components of an axiomatic system.

• Undefined terms — the predicates, operators, and constants appearing in the axioms; the axioms are what give them meaning.
• Axioms — the statements assumed without proof.
• Definitions — convenient shorthand introducing a new symbol, defined via logical equivalence (or, later, the identity symbol) in terms of already-known symbols. The new symbol goes on the left, only previously-seen symbols on the right.
• Theorems — everything logically entailed by the axioms.

Predicate vs. operator. A predicate takes terms and returns true or false (e.g. "is blue," or a two-term relation like $<$). An operator takes terms and returns a term (e.g. $+$ or $\sqrt{\;}$). Both can be one-term, two-term, etc. For a two-term predicate we often write the name between its arguments ($xQy$ means $Q(x,y)$).

The new axiomatic system. Undefined terms: the one-term predicate $C$, the two-term operator $+$, and the one-term operator $'$. No definitions; five axioms:

• A1. $\forall x\,\forall y\,\big(C(x+y)\rightarrow C(y+x)\big)$
• A2. $\forall x\,\forall y\,\big(C(x)\rightarrow C(x+y)\big)$
• A3. $\forall x\,\big(C(x')\rightarrow \lnot C(x)\big)$
• A4. $\forall x\,\big(\lnot C(x')\rightarrow C(x)\big)$
• A5. $\forall x\,\forall y\,\big((C(x+y)\land C(x'))\rightarrow C(y)\big)$

The five rules of derivation. (Distinct from rules of logic — the tautologies you can verify with a truth table.)

• Rule P — introduce a premise at any point.
• Rule CP (conditional proof) — if assuming $\alpha$ leads to $\beta$, discharge $\alpha$ and conclude $\alpha\rightarrow\beta$.
• Rule RAA (reductio ad absurdum) — if assuming $\alpha$ leads to a contradiction, conclude $\lnot\alpha$.
• Rule T — you may use any tautology (rule of logic) inside a derivation; this is where contraposition, double negation, etc. enter.
• Universal Specification (US) / Universal Generalization (UG) — strip a $\forall$ to reason about an arbitrary instance, or reattach a $\forall$ once done. Under US you may substitute any term for the variable, provided you do so at every occurrence.

Flag variables. A variable that is free in an assumed premise is flagged. You may not universally generalize on a flagged variable while it is still active — doing so would let you "prove" absurdities like "if anything is human, everything is human." A discharged premise (via CP or RAA) removes its flag.

Worked Example

Theorem 1: $\forall x\,\big(\lnot C(x)\rightarrow C(x')\big)$.

The strategy: assume the antecedent, derive the consequent, discharge with CP, then generalize. The key move is applying the contrapositive (Rule T) of A4, $\lnot C(x')\rightarrow C(x)$, which gives $\lnot C(x)\rightarrow C(x')$.

Once a theorem is proved it becomes a reusable block: later proofs cite Theorem 1 directly rather than re-deriving it. For the "always true" (non-implication) theorems, the standard tactic is RAA — assume the negation, break the sum apart with results like Theorem 5 ($\lnot C(x+y)\rightarrow(\lnot C(x)\land\lnot C(y))$), then drive both $C(x)$ and $\lnot C(x)$ to reach a contradiction.

Summary

• An axiomatic system has four parts: undefined terms, axioms, definitions, and theorems.
• This system's undefined terms are the predicate $C$, the two-term operator $+$, and the one-term operator $'$; it has five axioms and no definitions.
• The exercise's purpose is to derive theorems from the bare axioms with no appeal to intuition about what the symbols mean.
• The five derivation rules — P, CP, RAA, T, and US/UG — are the only tools used; choose CP for implications and RAA for unconditional ("always true") statements.
• US permits substituting any term (consistently) for a quantified variable; never generalize on a flagged (free-in-a-premise) variable until it is discharged.
• Proven theorems are reusable as single steps, so later proofs lean heavily on Theorem 5 to split sums and on Theorems 1–4 to convert between primed and unprimed terms.
• Under the intended interpretation revealed at the end, $C$ reads "is true," $+$ reads "or," and $'$ reads "not" — making the system a fragment of propositional logic in the symbolism of Russell and Whitehead.