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

Logical Operators and Truth Tables - Lecture 1 (Logic)

Logical Operators and Truth Tables – Lecture 1

Overview

This lecture lays the foundation for propositional logic, the logic of things that are either true or false. We begin by locating logic beneath every systematic body of knowledge and isolating the one kind of reasoning logic studies — deductive reasoning — as distinct from inductive and abductive reasoning. From there we strip the unambiguous core out of ordinary English: we define what counts as a statement, identify the truth-functional connectives worth keeping, and build the truth tables for the five operators used throughout the course — negation, conjunction, disjunction, the conditional, and the biconditional. We pay special attention to the slippery cases: negating quantified sentences, reading "if" versus "only if," vacuous truth, and the difference between an implication and its converse. The payoff is a precise vocabulary and the first logical equivalences (De Morgan's laws and the contrapositive) that later proofs rely on.

Key Concepts

Logic and language. Logic is the study of the rules of deductive reasoning — the rules for building correct statements from given statements. Because we must communicate it in English, an imprecise language, we deliberately strip away the ambiguous parts and keep only the univocal parts (those with exactly one interpretation).

Three kinds of reasoning.

• Deductive — the conclusion is guaranteed by the premises (e.g. "If all men are mortal and Socrates is a man, then Socrates is mortal"). This whole implication is always true; such always-true statements are tautologies. Logic studies only this kind.
• Inductive — generalizing from many particular cases ("the sun has always risen, so it will rise tomorrow"). Yields probability, not certainty. Not studied here.
• Abductive — reconstructing a particular event from your stock of generalizations (a detective, a historian). Not studied here.

Statement (proposition). A statement is a sentence that is either true or false, but not both. Commands ("Go back home"), fragments ("Thursday up paper"), and questions are not statements. We denote statements by Greek letters $\alpha, \beta, \dots$ The subject is called propositional logic; we treat "statement" and "proposition" as the same thing.

Predicate vs. statement. A sentence with a free variable, like $x + 2 = 5$, is not a statement — its truth depends on $x$. It is a predicate, handled later with quantifiers.

Sentential connective. Any word or phrase that can be attached to sentences so the result is still a meaningful sentence (still a sentence — not necessarily true). There are infinitely many ("and," "or," "not," "Jack believes that," "it is probable that," "if … then," "if and only if"). "Cal the yesterday $2+2=4$" is nonsense, so "Cal the yesterday" is not a connective.

Truth-functional connective. A connective for which the truth value of the result is completely determined by the truth values of the component statements — equivalently, one for which a truth table can be built. We keep only these.

• Truth-functional: "not," "and," "or," "if … then," "if and only if."
• NOT truth-functional: "Jack believes that $\alpha$" and "it is probable that $\beta$" — knowing $\alpha$ is true does not fix whether Jack believes it, or whether it is probable. Their value depends on content, not just truth value.

The five operators and their truth tables.

• Negation $\lnot\alpha$ ("not $\alpha$"): true exactly when $\alpha$ is false.
• Conjunction $\alpha \land \beta$ ("$\alpha$ and $\beta$"): true only when both are true. (The English "but" is logically identical to "and" but carries extra, ambiguous connotation, so we use "and.")
• Disjunction $\alpha \lor \beta$ ("$\alpha$ or $\beta$"): the inclusive or — false only when both are false, true otherwise. (Exclusive or, $\veebar$, true when exactly one holds, is valid but rarely needed, so we omit it.)
• Conditional $\alpha \to \beta$ ("if $\alpha$ then $\beta$"): false only in the single case where $\alpha$ is true and $\beta$ is false; true in all three other cases. $\alpha$ is the antecedent, $\beta$ the consequent.
• Biconditional $\alpha \leftrightarrow \beta$ ("$\alpha$ if and only if $\beta$"): true exactly when $\alpha$ and $\beta$ share the same truth value.

Vacuous truth. When the antecedent of a conditional is false, the conditional is vacuously true — there are no cases to check, so it is not false, hence true. This is the definition of $\to$, not a claim about the physical world: there is no notion of time or "what if things changed." This is why "if $1 = 0$ then I'm the pope" counts as true. (We avoid words like "never," which smuggle in time/physics, just as we avoid "but.")

Reading "if" and "only if."

• "$\alpha$ if $\beta$" means $\beta \to \alpha$ (e.g. "there's oxygen if there's fire" means fire $\to$ oxygen).
• "$\alpha$ only if $\beta$" means $\alpha \to \beta$ (e.g. "there's fire only if there's oxygen" means fire $\to$ oxygen).
• "$\alpha$ if and only if $\beta$" asserts both: $(\alpha \to \beta) \land (\beta \to \alpha)$ — hence biconditional.

Converse and contrapositive.

• Converse of $\alpha \to \beta$ is $\beta \to \alpha$. It is not equivalent to the original (fire $\to$ oxygen is true; oxygen $\to$ fire is false). Confusing a statement with its converse is among the most common reasoning errors.
• Contrapositive of $\alpha \to \beta$ is $\lnot\beta \to \lnot\alpha$, and it is logically equivalent ("if fire then oxygen" $\equiv$ "if no oxygen then no fire").

Negation is subtle. $\lnot\alpha$ must be true exactly when $\alpha$ is false. So the negation of "all swans are white" is "some swans are not white" (some = at least one) — not "some swans are black" or "no swans are white" (those are merely contrary: they cannot both be true with $\alpha$, but can both be false). Likewise the negation of "some men play golf" is "no men play golf," and the negation of "there is at least one $x$ with $x+3=7$" is "there is no such $x$."

De Morgan's laws (the first theorems). Pushing a negation through a connective flips $\land \leftrightarrow \lor$ and negates each part:

Precedence. Apply operators in the order: $\lnot$, then $\land$, then $\lor$, then $\to$, then $\leftrightarrow$ (analogous to negative sign, $\times$, $+$, and $=$ in arithmetic). When in doubt, parenthesize $\land$ and $\lor$ explicitly.

Tautology and contradiction. A tautology is a statement that is always true (e.g. $\alpha \lor \lnot\alpha$); proving a theorem means showing the implication is a tautology. A contradiction is always false (e.g. $\alpha \land \lnot\alpha$).

Worked Example

Verify the first De Morgan law, $\lnot(\alpha \land \beta) \equiv \lnot\alpha \lor \lnot\beta$, by showing both sides agree in every row. A logical equivalence holds exactly when the two columns are identical for all truth assignments.

The column for $\lnot(\alpha \land \beta)$ ($F,T,T,T$) matches the column for $\lnot\alpha \lor \lnot\beta$ ($F,T,T,T$) in every row, so the two are logically equivalent. In words: denying "today is Monday and the weather is cold" is the same as asserting "today is not Monday, or the weather is not cold."

Summary

• Logic is the study of deductive reasoning; inductive and abductive reasoning are set aside.
• A statement is a sentence that is true or false (not both); a sentence with a free variable is a predicate, not a statement.
• We keep only truth-functional connectives — those whose result is fixed by the components' truth values — namely $\lnot$, $\land$, $\lor$, $\to$, and $\leftrightarrow$, each defined by its truth table.
• A conditional $\alpha \to \beta$ is false only when $\alpha$ is true and $\beta$ false; with a false antecedent it is vacuously true. "$\alpha$ only if $\beta$" means $\alpha \to \beta$; "$\alpha$ if $\beta$" means $\beta \to \alpha$.
• An implication is equivalent to its contrapositive $\lnot\beta \to \lnot\alpha$ but not to its converse $\beta \to \alpha$; negate quantified sentences carefully ("all are" $\to$ "at least one is not").
• De Morgan's laws — $\lnot(\alpha\land\beta)\equiv\lnot\alpha\lor\lnot\beta$ and $\lnot(\alpha\lor\beta)\equiv\lnot\alpha\land\lnot\beta$ — are the first proven equivalences; always-true statements are tautologies (theorems), always-false ones contradictions.