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

Common Tautologies and Indirect Proof - Lecture 3 (Logic)

Overview

This lecture consolidates the machinery for justifying valid rules of logic and then introduces a powerful new proof technique. We first review how a truth table certifies that an implication is a tautological implication (and therefore a valid rule of logic), and why a counterexample certifies that a rule is invalid. We then catalog the most common tautologies (modus ponens, modus tollens, the syllogisms, De Morgan's laws, contraposition, and more) and discuss why $F \to F$ and $F \to T$ are vacuously true. Finally we introduce the notion of consistent vs. inconsistent premises and the rule of reductio ad absurdum — indirect proof — reproving an earlier result by contradiction. The big theme: truth tables discover simple rules, but derivations are what let us prove things tables cannot.

Key Concepts

Valid rule of logic. A rule $\alpha \to \beta$ is valid exactly when it is a tautological implication — true under every sentential interpretation. Equivalently: every atomic interpretation is true, which holds iff the rule's truth table is true in every row.

Sentential interpretation — replace each atomic sentence with any sentence (you may add substructure), keeping the overall connective structure fixed.
Atomic interpretation — replace each atomic sentence only with another atomic sentence (a sentence with no connectives, true or false). A truth table tests exactly these: for $n$ atomic sentences there are $2^n$ rows covering every true/false assignment.
The test: if you can build a truth table that is true in every row, the rule is valid. The converse fails — some valid rules (e.g. "if $a = b$ then $a + c = b + c$ ") range over infinitely many values and have no finite truth table; they must be established by derivation.

Invalid rule (counterexample). A rule is invalid if one interpretation makes the premises true and the conclusion false. A single $F$ in the result column is such a counterexample. For instance $\alpha \to \beta$ is not a valid rule of logic, because the row $\alpha = T,\ \beta = F$ gives $F$ . Warning: finding one interpretation that comes out true does not establish a rule — validity requires truth in every row.

Vacuous truth. An implication is vacuously true when its antecedent cannot be satisfied; by definition $F \to T = T$ and $F \to F = T$ . Three ways to make peace with this:

It is simply the definition of the connective $\to$ (the English "implies" is just the word we attach to it).
$\alpha \to \beta$ is logically equivalent to $\lnot\alpha \lor \beta$ , which is automatically true whenever $\alpha$ is false.
From one false premise you can derive anything (e.g. "if $1 = 0$ then I am the Pope"), so a false antecedent licenses any conclusion.
Note the distinction: "if $P$ then $Q$ " (no notion of time/physics) is different from the subjunctive "if it were the case that $P$ , then $Q$ would hold."

Rules of derivation (these govern proofs; they are not themselves rules of logic — Rule T lets us use rules of logic):

Rule P — introduce a premise at any line.
Rule T — introduce any sentence $S$ that is tautologically implied by previous lines.
Rule CP (conditional proof) — if $S$ can be derived from $R$ together with the premises, then $R \to S$ can be derived from the premises alone. (Discharges the temporary assumption $R$ .)
Rule RAA (reductio ad absurdum / indirect proof) — if a contradiction can be derived from the premises $\mathcal{P}$ together with $\lnot S$ , then $S$ can be derived from $\mathcal{P}$ alone.

Common tautologies (each provable by truth table; internalize them rather than memorizing names):

Modus ponens: $P,\ P \to Q \models Q$
Modus tollens: $P \to Q,\ \lnot Q \models \lnot P$
Disjunctive syllogism: $P \lor Q,\ \lnot P \models Q$
Simplification: $P \land Q \models P$ (and $\models Q$ )
Addition: $P \models P \lor Q$
Hypothetical syllogism (transitivity): $P \to Q,\ Q \to R \models P \to R$
Contraposition: $P \to Q \;\equiv\; \lnot Q \to \lnot P$
Material implication: $P \to Q \;\equiv\; \lnot P \lor Q$
Negated implication: $\lnot(P \to Q) \;\equiv\; P \land \lnot Q$
De Morgan's laws: $\lnot(P \lor Q) \equiv \lnot P \land \lnot Q$ and $\lnot(P \land Q) \equiv \lnot P \lor \lnot Q$
Double negation: $P \equiv \lnot\lnot P$
Commutativity / associativity of $\land$ and $\lor$
Exportation: $(P \land Q) \to R \;\equiv\; P \to (Q \to R)$ — the principle behind Rule CP
Law of excluded middle: $P \lor \lnot P$ (always true)
Law of contradiction: $\lnot(P \land \lnot P)$ (always true)

The last two are the only rules stated as something other than implications; they may be inserted into a derivation out of nowhere (via Rule T, implied by nothing).

Consistency. A set of premises is consistent if they can all be true together; inconsistent if it is impossible for all of them to be true at once. Consistent does not mean true, only that some interpretation satisfies them all. Show inconsistency two ways: (1) the conjunction of all premises is false in every truth-table row, or (2) derive a contradiction from them. From inconsistent premises you can prove anything (a contradiction is always false, and $F$ implies everything), so logic only has something to say about consistent premise sets.

Worked Example

Prove $A \to \lnot D$ from the premises $A \to (B \lor C)$ , $B \to \lnot A$ , and $D \to \lnot C$ , using indirect proof (RAA).

We assume the negation of the goal and derive a contradiction. Since $\lnot(A \to \lnot D) \equiv A \land D$ , we assume $A \land D$ and aim for some $X \land \lnot X$ :

\begin{aligned} &1.\ A \to (B \lor C) &&\text{Rule P} \\ &2.\ B \to \lnot A &&\text{Rule P} \\ &3.\ D \to \lnot C &&\text{Rule P} \\ &4.\ A \land D &&\text{assume } \lnot(\text{goal}),\ \text{since } \lnot(A \to \lnot D)\equiv A\land D \\ &5.\ A &&\text{simplification, } 4 \\ &6.\ B \lor C &&\text{modus ponens, } 1,5 \\ &7.\ D &&\text{simplification, } 4 \\ &8.\ \lnot C &&\text{modus ponens, } 3,7 \\ &9.\ B &&\text{disjunctive syllogism, } 6,8 \\ &10.\ \lnot A &&\text{modus ponens, } 2,9 \\ &11.\ A \land \lnot A &&\text{contradiction, } 5,10 \\ &12.\ \lnot(A \land D) &&\text{Rule RAA, discharging } 4\ (\text{lines } 4\text{–}11) \\ &13.\ A \to \lnot D &&\text{since } \lnot(A \land D)\equiv \lnot A \lor \lnot D \equiv A \to \lnot D \end{aligned}

Assuming $A \land D$ forced both $A$ and $\lnot A$ — an absurdity — so the assumption is rejected and the goal $A \to \lnot D$ follows from premises 1–3 alone.

Summary

A rule of logic is valid iff it is a tautological implication; a truth table that is true in every row proves it, and a single false row gives a counterexample that refutes it.
Truth tables only discover the simplest rules — for anything ranging over infinitely many values you need a derivation. Tables are mechanical but explode in size ( $2^n$ rows); derivations require insight but are far more powerful.
$F \to T$ and $F \to F$ are vacuously true by definition; this matches $P \to Q \equiv \lnot P \lor Q$ and the fact that a false premise entails anything.
The four derivation rules are P (introduce a premise), T (introduce a tautological consequence), CP (discharge an assumption to get an implication), and RAA (derive a contradiction from $\lnot S$ to conclude $S$ ).
Premises are consistent if some interpretation makes them all true; inconsistent premises (provably so via an all-false conjunction or a derived contradiction) entail everything and so are useless to reason from.
Indirect proof works by assuming the negation of the goal, deriving a contradiction, and discharging the assumption — exactly mirroring how a false conclusion signals a false premise.