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

Interpretations, Validity, & Logic's Limitations - Lecture 8 (Logic)

Interpretations, Validity, & Logic's Limitations – Lecture 8

Overview

This lecture upgrades two ideas from sentential logic into the richer setting of first-order predicate logic. In sentential logic we replaced whole sentences with other sentences (a sentential interpretation) and called a sentence always-true-under-every-such-replacement a tautology. With a full grammar of terms, predicates, operation symbols, and quantifiers, we now define a stricter notion of interpretation and its always-true counterpart, universal validity. We see how interpretations let us prove arguments invalid, prove premise sets consistent, and prove axioms independent. The lecture closes with a tour of logic's hard limits: the undecidability of proof-search (via the halting problem and Church's theorem) and Gödel's completeness and incompleteness theorems.

Key Concepts

Sentential interpretation (review). A substitution that replaces each sentence letter with some sentence, preserving only the top-level sentential connectives. A tautology is a sentence true under every sentential interpretation — checkable with a truth table.

Interpretation. A sentence $s$ is an interpretation of a formula $F$ over a non-empty domain of individuals $D$ if and only if $s$ is obtained from $F$ by:

• substituting predicates and operation symbols defined on $D$ for the predicates and operation symbols in $F$, and
• substituting constants in $D$ for the constants and free variables in $F$.

Quantifiers and sentential connectives are never replaced — all of $F$'s logical structure is preserved. Because $s$ must be a sentence (a formula with no free variables), every free variable of $F$ must become a constant.

The three kinds of symbols.

• Terms / constants — name individuals in $D$.
• Operation symbols (e.g. $+$, $\times$) — take terms, return a term. Must be closed on $D$: every output of an input drawn from $D$ must also lie in $D$ (e.g. subtraction is not closed on $\mathbb{N}$; division is not closed on $\mathbb{Q}$ because of zero).
• Predicates / relations (e.g. $\le$, $=$) — take terms, return a proposition (true or false).

Universal validity. A formula is universally valid if and only if every interpretation of it, over every non-empty domain, is true. A tautology is true by its sentential connectives alone; a universally valid formula is true by virtue of all its logical structure (quantifiers and predicates included). Hence every tautology is universally valid, but not conversely — tautologies are a strict subset. Example: $\forall x\,(Rx \lor \lnot Rx)$ is universally valid but is not a tautology, because it cannot be reduced to $r \lor \lnot r$ and has no finite truth table.

Why non-empty domains. Empty domains are excluded because (1) we never study "nothing," and (2) two pieces of reasoning we rely on fail on the empty domain:

• $\forall x\,Px \Rightarrow \exists x\,Px$ (universal-to-existential) holds only if something exists.
• $\forall x\,Px$ and $\forall x\,\lnot Px$ should be contraries (jointly impossible); on an empty domain both are vacuously true.

What proofs actually show. Proving "axioms $\Rightarrow$ theorem" demonstrates that the implication is universally valid — not merely a tautology. Any interpretation making the premises true makes the conclusion true; any interpretation making the premises false makes the implication vacuously true.

Three uses of interpretations.

• Showing an argument is invalid — exhibit one interpretation (a counterexample) on some non-empty domain making all premises true and the conclusion false.
• Showing premises are consistent — exhibit one interpretation making the conjunction of all premises true. Consistent means "possibly all true together," not "true."
• Showing axioms are independent — to show axiom $A_2$ cannot be derived from $A_1$, exhibit an interpretation making $A_1$ true and $A_2$ false; then $A_1 \Rightarrow A_2$ is invalid, so $A_2$ adds genuinely new information.

Limits of proof-search. There is no general procedure that, given a formula, decides whether it is universally valid — proof-generation in first-order logic is undecidable (Church). Since every system of mathematics presupposes first-order logic, no such universal proof-generating algorithm exists for any of them. (Checking a given proof is easy; finding one is not.)

The halting problem. Suppose a machine $H$ could decide, for any program-and-input, whether it halts. Build $M$ that runs $H$ then negates the answer, and feed $M$ its own description: if $H$ says "halts," $M$ does the opposite, and vice versa. The contradiction proves no such $H$ exists — undecidability is a logical impossibility, not a matter of insufficient technology.

Gödel's theorems.

• Completeness: in first-order logic, if $Q$ logically follows from $P$, then $Q$ is derivable from $P$ — everything true is provable.
• Incompleteness: any consistent system rich enough to encode basic arithmetic contains true statements that cannot be proved within it. The construction encodes symbols, formulas, and whole proofs as unique numbers (via prime factorization), then finds a true sentence whose code is not the code of any provable statement — intuitively, "this statement is not provable."
• Consistency: a consistent system strong enough to encode arithmetic cannot prove its own consistency.

These collectively answered Hilbert's program (seeking systems that are complete, consistent, and decidable) — in the negative.

Worked Example

Goal. Show the argument "Some men are liars; Adam is a man; therefore Adam is a liar" is invalid by exhibiting a counterexample interpretation.

First symbolize, then interpret over the non-empty domain $D = \mathbb{R}$, choosing predicates so the premises come out true and the conclusion false:

Premises true, conclusion false on a non-empty domain — so the argument is invalid. Translating into arithmetic, where validity is easy to judge, is a standard tactic for spotting counterexamples.

Summary

• An interpretation replaces a formula's predicates, operation symbols, constants, and free variables with items defined on a chosen non-empty domain, while preserving all quantifiers and connectives; the result must be a sentence.
• Operation symbols must be closed on the domain (every output stays inside $D$); predicates return propositions, operations return terms.
• A formula is universally valid when every interpretation over every non-empty domain is true; every tautology is universally valid but not vice versa.
• Interpretations prove arguments invalid (a counterexample), premise sets consistent (one all-true interpretation), and axioms independent (true premises, false candidate).
• Church: universal validity in first-order logic is undecidable — no algorithm generates proofs in general; the halting problem illustrates the same kind of self-reference impossibility.
• Gödel: first-order logic is complete (truth implies provability), but any consistent arithmetic-capable system is incomplete (some truths are unprovable) and cannot prove its own consistency.