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

Tautologies and Theory of Inference - Lecture 2 (Logic)

Overview

This lecture builds the bridge from individual logical connectives to full deductive arguments. After reviewing the five truth-functional connectives and their truth tables, it introduces tautologies (sentences that are always true regardless of how their atomic parts are interpreted) and explains why truth tables alone cannot establish every truth we care about. From there it develops the machinery of tautological implication, sentential interpretation, and what it means for one sentence to logically follow from others. The payoff is the theory of inference: a small set of derivation rules (Rule P, Rule T, and Rule CP) that let us construct formal proofs without writing out exponentially large truth tables.

Key Concepts

The five connectives (review). A statement (used interchangeably with sentence) is something that is either true or false, not both. The five truth-functional connectives are $\land$ (and), $\lor$ (or, inclusive), $\lnot$ (not), $\to$ (implies), and $\leftrightarrow$ (equivalence). "Truth-functional" means the truth value of the compound is fully determined by the truth values of its parts.

• Conjunction $\alpha \land \beta$: true only when both are true.
• Disjunction $\alpha \lor \beta$: true unless both are false (inclusive or).
• Implication $\alpha \to \beta$: false only in the case $T \to F$; $F \to T$ and $F \to F$ are both vacuously true.
• Equivalence $\alpha \leftrightarrow \beta$: true exactly when the two share the same truth value.
• Negation $\lnot\alpha$: flips the truth value.

Precedence. Evaluate in this order: $\lnot$, then $\land$, then $\lor$, then $\to$, then $\leftrightarrow$. Think of $\lnot$ like a negative sign, $\land$ like multiplication, $\lor$ like addition, and $\leftrightarrow$ like equals.

Key equivalences (provable by truth table).

• De Morgan's laws: $\lnot(\alpha \land \beta) \equiv \lnot\alpha \lor \lnot\beta$ and $\lnot(\alpha \lor \beta) \equiv \lnot\alpha \land \lnot\beta$.
• Contrapositive: $\alpha \to \beta \equiv \lnot\beta \to \lnot\alpha$ ("if fire then oxygen" says the same as "if no oxygen then no fire").
• A common error is confusing $\alpha \to \beta$ with its converse $\beta \to \alpha$ — they are different.
• Biconditional as two implications: $\alpha \leftrightarrow \beta \equiv (\alpha \to \beta) \land (\beta \to \alpha)$, which is why the symbol is a double arrow.

Atomic sentence. A sentence with no sentential connectives — the simplest possible statement (e.g. "the sky is blue"). "The sky is not blue" is not atomic (you can strip the not); "$1+1=2$ and $9<3$" is not atomic (it's a conjunction of two atomic pieces).

Tautology. A sentence is a tautology if and only if replacing each of its component atomic sentences — in all occurrences — by any other atomic sentences always yields a truth. Equivalently, every atomic interpretation evaluates to true. Examples: $P \lor \lnot P$, $(P \land Q) \to P$, and $P \to (P \lor Q)$. The "in all occurrences" restriction matters: you must substitute the same sentence for every copy of a given variable.

Why truth tables aren't enough. A sentence with $n$ atomic parts needs $2^n$ rows, so verification blows up exponentially. Worse, some theorems (e.g. the Pythagorean theorem) are not provable by any truth table at all. Truth tables are a fine starting point for simple cases, but we need rules of derivation to go further.

Two useful observations about tautologies.

• A sentence that is not a tautology may become one by substituting a compound sentence for an atomic one (or the same sentence for two distinct ones): $\alpha \to \beta$ is not a tautology, but $\alpha \to (\gamma \lor \lnot\gamma)$ is, since the consequent is always true.
• A sentence that is a tautology stays one under any substitution of sentences for its atomic parts (done in all occurrences) — adding structure never breaks a tautology, because every replacement still evaluates to true or false.

Sentential interpretation. A sentence $P$ is a sentential interpretation of $Q$ if and only if $P$ can be obtained from $Q$ by replacing $Q$'s atomic sentences with any other sentences. An interpretation must preserve the structure of $Q$: you may add structure, never remove it, and the same atomic sentence must be replaced consistently everywhere.

Logically follows. $Q$ logically follows from $P$ if and only if every sentential interpretation of $P \to Q$ is true. Equivalently — and this is the working criterion — if $P \to Q$ is a tautology (a tautological implication), then $Q$ logically follows from $P$. A tautological equivalence is a biconditional that is a tautology.

The two criteria for inference rules. Given a set of premises, the rules of logical derivation must let us infer (1) only conclusions that logically follow from the premises (soundness — reject any rule that derives a false conclusion from true premises), and (2) all conclusions that logically follow (completeness — don't make the rule set too weak).

The three derivation rules (theory of inference).

• Rule P — a premise may be introduced at any point in a derivation.
• Rule T — a sentence may be introduced if it is tautologically implied by any preceding sentence or by the conjunction of preceding sentences.
• Rule CP (conditional proof) — if $S$ can be derived from $R$ together with a set of premises, then $R \to S$ may be concluded from the premises alone. (To prove an implication, assume its antecedent, derive the consequent, then discharge the assumption.)

A worked sanity check on Rule P: introducing extra premises does not let you prove anything you like. From $P \lor Q$ and $\lnot Q$ you can derive $P$ — but that proves $P$ follows from both premises together, not from $P \lor Q$ alone.

Worked Example

Prove: from premises $\lnot S \to C$, $\;C \to \lnot D$, $\;D \lor O$, and $\lnot O$, conclude $S$.

Each line is justified by a rule, with the premise lines it depends on tracked in brackets. The key tautological implications used are disjunctive syllogism, $(\alpha \lor \beta) \land \lnot\beta \Rightarrow \alpha$, and modus tollens via the contrapositive.

Line 5 applies disjunctive syllogism to $D \lor O$ and $\lnot O$. Line 6 reads $C \to \lnot D$ as $D \to \lnot C$ and discharges $D$. Line 7 reads $\lnot S \to C$ as $\lnot C \to S$. Every step is a simple tautological implication that could, if doubted, be confirmed with a small truth table — which is exactly the point of the inference rules: they replace one giant truth table with a chain of tiny, verifiable ones.

Summary

• The five truth-functional connectives ($\lnot, \land, \lor, \to, \leftrightarrow$) and their precedence let any compound statement be evaluated mechanically, just like an arithmetic expression.
• A tautology is true under every interpretation of its atomic parts; substitutions must be made in all occurrences of a variable.
• Truth tables verify tautologies in simple cases but grow as $2^n$ and cannot establish every theorem — motivating formal proof.
• $Q$ logically follows from $P$ exactly when $P \to Q$ is a tautology (a tautological implication); a sentential interpretation may add structure but never remove it.
• Inference rules must be sound (Criterion 1: derive only what follows) and complete (Criterion 2: derive all that follows).
• The theory of inference rests on three rules — P (introduce a premise), T (introduce a tautological consequence), and CP (conditional proof: assume the antecedent, derive the consequent, discharge) — which together build formal proofs from true premises to true conclusions.