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

Solving Exercise Problems - Lecture 4 (Logic)

Overview

This session is a problem-solving workshop that applies the machinery built up so far in the course: the rules of derivation, the notion of a truth-functional connective, tautologies, tautological implication and equivalence, sentential interpretation, and indirect (reductio) proof. Rather than introducing new theory, it works a long string of textbook exercises and shows how to attack each type. The goal is to make the foundations automatic — recognizing when a connective is truth-functional, translating English into symbolic form, deciding whether a sentence is a tautology, telling a valid argument from a fallacy, and constructing both direct and indirect derivations. A recurring lesson is that when an argument is not valid, the right move is to produce a concrete sentential interpretation making every premise true and the conclusion false.

Key Concepts

The four rules of derivation (recap). Every derivation keeps track of which premises each line depends on.

• Rule P (premise): introduce a premise on any line; that line depends on itself.
• Rule T (tautological inference): introduce a statement $S$ if it is tautologically implied by previous lines. This is the rule that lets us use the laws of logic.
• Rule CP (conditional premise): to prove $R \to S$, introduce $R$ as a temporary premise, derive $S$, then discharge $R$. Justified by the equivalence $(P \wedge R \to S) \equiv (P \to (R \to S))$.
• Rule RAA (reductio ad absurdum): to prove $S$, assume $\neg S$, derive a contradiction, then discharge. Justified by $(P \wedge \neg S \to \mathbf{F}) \equiv (P \to S)$.

Truth-functional connectives. A connective is truth-functional when the truth value of the compound depends only on the truth values of its components — nothing else. Test: if feeding in components of the same truth value can yield different outputs, the connective is not truth-functional. Non-truth-functional examples:

• Possibly ($\Diamond$): $\Diamond(\text{life on Europa})$ is true while $\Diamond(2+2=5)$ is false, yet both inputs may be false — output depends on content, not truth value.
• Necessarily ($\Box$), "X believes that ..." — likewise depend on more than the input's truth value.

English-to-symbolic translation rules.

• "$\alpha$ unless $\beta$" $\equiv \alpha \vee \beta$ (inclusive or) — equivalently $\neg\beta \to \alpha$. ("You will fail unless you do your homework" is false only when you neither fail nor do homework.)
• "but" behaves logically like and ($\wedge$).
• Commas in the English sentence signal where parentheses (the main connective grouping) go.

Tautology / contradiction.

• A tautology is true under every interpretation; a contradiction is false under every interpretation. The canonical contradiction form is $\alpha \wedge \neg\alpha$.
• Useful simplifications for testing: $\alpha \wedge \neg\alpha \equiv \mathbf{F}$; $\alpha \vee \neg\alpha \equiv \mathbf{T}$; $\mathbf{F} \to \text{anything} \equiv \mathbf{T}$; $\text{anything} \to \mathbf{T} \equiv \mathbf{T}$; conjunction with $\mathbf{T}$ leaves a value unchanged. Often faster than a full truth table.

Tautological implication vs. validity. $A$ tautologically implies $B$ iff every interpretation that makes $A$ true also makes $B$ true. An argument is valid iff its premises tautologically imply its conclusion. To show invalidity, exhibit a sentential interpretation making all premises true and the conclusion false.

Sentential interpretation. Replace atomic sentences with sentences (the same sentence at every occurrence), possibly adding substructure. Rules: the main/major connectives of the original must be preserved (you may add structure, never remove it); the number of distinct atomics may increase or decrease; the count of connective occurrences may change; but the number of (not necessarily distinct) atomic occurrences can only stay the same or grow.

Common fallacies.

• Affirming the consequent: from $Q$ and $P \to Q$, concluding $P$ — invalid.
• Denying the antecedent: from $\neg P$ and $P \to Q$, concluding $\neg Q$ — invalid. (These are contrapositives of each other.)

Two criteria for a rule of logic. Criterion 1: the rule may yield only true conclusions from true premises (soundness). Criterion 2: the rules together must be able to derive everything that follows from the premises (completeness). "From $P$ derive $Q \vee \neg Q$" satisfies Criterion 1 but violates Criterion 2; "from $P$ derive $Q$" ($P \to Q$ as a rule) satisfies Criterion 2 but violates Criterion 1.

Worked Example

Direct derivation. Premises: $P \to W$, $\;P \vee C$, $\;C \to \neg I$, $\;I$. Prove $W$.

Indirect (RAA) derivation. Premises: $D \to W$, $\;A \vee \neg W$, $\;\neg(D \wedge A)$. Prove $\neg D$.

Showing an argument is invalid. For the bad argument $F \to S$, $\;S \to N$, $\;N$, $\;\therefore F$, interpret $F$ as "$2+2=5$", $S$ as "$2+2=4$", $N$ as "$1+1=2$". Then $N$ is true, both conditionals are true (true $\to$ true, false $\to$ true), yet $F$ is false — so the conclusion does not follow.

Summary

• The four derivation rules (P, T, CP, RAA) and the bookkeeping of which premises each line depends on are the backbone of every proof.
• A connective is truth-functional only if its output depends solely on its inputs' truth values; possibly, necessarily, and believes that fail this test.
• Translate carefully: "unless" is inclusive or ($\alpha \vee \beta \equiv \neg\beta \to \alpha$), "but" is and, and commas mark the parenthesization.
• Test tautologies and contradictions by algebraic simplification ($\alpha \wedge \neg\alpha \equiv \mathbf{F}$, $\mathbf{F}\to X \equiv \mathbf{T}$, etc.) rather than always writing the full truth table.
• Validity is tautological implication; to disprove it, build a sentential interpretation making all premises true and the conclusion false — and watch for affirming the consequent / denying the antecedent.
• Use a direct derivation (modus ponens, disjunctive syllogism, contraposition) when the conclusion follows; use RAA — assume the negation, derive a contradiction, discharge — when a direct route is awkward or when showing premises are inconsistent.