The Timpson Lyceum · Discrete Math (2025)

Propositional Logic & Introduction – Lecture 1 (Discrete Math)

Propositional Logic & Introduction – Lecture 1

Overview

This lecture sets up what discrete mathematics is really about — constructing proofs and understanding axiomatic systems — and then begins building the logical foundation that all of mathematics rests on. Every body of knowledge, from Euclidean geometry to Newtonian mechanics, is an axiomatic system: a small set of starting assumptions from which everything else is derived. Logic sits beneath math, math beneath science, so we begin with logic. The rest of the lecture is a crash course in propositional logic: what counts as a statement, which connectives we care about, their truth tables, negation rules, implications, and logical equivalence. These tools let us read English precisely and say exactly what we mean — the prerequisite for writing real proofs.

Key Concepts

Axiomatic systems. Every theoretical body of knowledge is (or is built from) an axiomatic system with four components:

Axioms — initial assumptions we accept without proof (e.g., "All men are mortal"; "Socrates is a man").
Undefined terms — you cannot define everything, or the dictionary becomes circular or infinite, so some terms are left undefined and get their meaning from the axioms.
Definitions — built up from undefined terms; once defined, a term can be used to define further terms.
Theorems — statements proved by combining axioms (and previously proved theorems). This is the heart of mathematics. Euclid derives all of geometry from just five postulates.

The dependency hierarchy. You use logic to do math; you use math to do science — never the reverse. If a real-world count contradicts arithmetic, the counting is wrong, not the math.

Statement (proposition). A sentence that is either true or false, but not both. Statements are denoted by Greek letters such as $\alpha, \beta$ .

" $2+2=5$ " is a (false) statement; "New York is a suburb of Los Angeles" is a (false) statement.
"Go back home" is a command — not a statement.
"This statement is false" is a paradox — neither true nor false, so not a statement.
" $x + 2 = 5$ " contains a free variable, so it is not yet a statement — it is a predicate (covered later).

Sentential connectives. Words or phrases that attach to statements and still yield something meaningful (e.g., not, and, or, if…then, Jack believes that…). There are infinitely many.

Truth-functional connectives. A connective is truth-functional when the truth value of the whole can be determined purely from the truth values of its components.

and, or, not are truth-functional.
"Jack believes that $\alpha$ " and "It is probable that $\alpha$ " are meaningful connectives but not truth-functional — knowing $\alpha$ 's truth value doesn't fix the result.

Univocal language. Logic uses only words with exactly one interpretation. and is univocal and welcome; but and never are ambiguous (they smuggle in contrast or time) and are rejected — good for poetry, bad for logic.

The five connectives we keep (with truth tables):

Negation $\neg\alpha$ ("not $\alpha$ " / " $\alpha$ is false"). $\alpha$ and $\neg\alpha$ are contradictions: exactly one is true.

$\alpha$	$\neg\alpha$
T	F
F	T

Conjunction $\alpha \wedge \beta$ (" $\alpha$ and $\beta$ "). True only when both are true.

$\alpha$	$\beta$	$\alpha\wedge\beta$
T	T	T
T	F	F
F	T	F
F	F	F

Disjunction $\alpha \vee \beta$ (" $\alpha$ or $\beta$ "). This is the inclusive or — true when at least one holds, including both. (The exclusive or, $\oplus$ , true only when exactly one holds, exists but is not one of our five.)

$\alpha$	$\beta$	$\alpha\vee\beta$
T	T	T
T	F	T
F	T	T
F	F	F

Implication $\alpha \rightarrow \beta$ ("if $\alpha$ then $\beta$ "). $\alpha$ is the antecedent (condition), $\beta$ the conclusion. Equivalent English phrasings: "if $\alpha$ then $\beta$ ", " $\beta$ if $\alpha$ ", " $\alpha$ only if $\beta$ ", " $\alpha$ is sufficient for $\beta$ ", " $\beta$ is necessary for $\alpha$ ". Crucially, when the antecedent is false the implication is vacuously true — a false premise can prove anything.

$\alpha$	$\beta$	$\alpha\rightarrow\beta$
T	T	T
T	F	F
F	T	T
F	F	T

Equivalence / biconditional $\alpha \leftrightarrow \beta$ (" $\alpha$ if and only if $\beta$ ", abbreviated iff). Means $\alpha\rightarrow\beta$ and $\beta\rightarrow\alpha$ . True exactly when $\alpha$ and $\beta$ share the same truth value. Two such statements are logically equivalent.

$\alpha$	$\beta$	$\alpha\leftrightarrow\beta$
T	T	T
T	F	F
F	T	F
F	F	T

Negating quantified statements. To negate, swap "all" with "some" (and insert a "not"), and vice versa — do not merely prefix "not."

$\neg$ ("All men in Georgia are at least 6 ft tall") = "Some man in Georgia is not at least 6 ft tall."
$\neg$ ("Some men play golf") = "No men play golf" (i.e., all men do not play golf).
$\neg$ ("There is at least one $x$ with $x+3=7$ ") = "For all $x$ , $x+3 \neq 7$ ."

De Morgan's laws. Distributing a negation flips the connective:

\begin{aligned} \neg(\alpha \wedge \beta) &\equiv \neg\alpha \vee \neg\beta \\ \neg(\alpha \vee \beta) &\equiv \neg\alpha \wedge \neg\beta \end{aligned}

Converse and contrapositive of $\alpha \rightarrow \beta$ .

Converse $\beta \rightarrow \alpha$ — flips the arrow. Not logically equivalent to the original (a true implication may have a false converse).
Contrapositive $\neg\beta \rightarrow \neg\alpha$ — flips the arrow and negates both sides. Logically equivalent to the original.

Precedence (order of operations). Apply in this order: $\neg$ , then $\wedge$ , then $\vee$ , then $\rightarrow$ , then $\leftrightarrow$ — analogous to negative sign, $\times$ , $+$ , and $=$ .

Worked Example

Claim: The contrapositive $\neg\beta \rightarrow \neg\alpha$ is logically equivalent to the implication $\alpha \rightarrow \beta$ . Build a truth table for $\neg\beta \rightarrow \neg\alpha$ and compare it to $\alpha \rightarrow \beta$ .

State the rule first, then evaluate each row (recall a false antecedent makes an implication vacuously true):

\begin{aligned} &\alpha \rightarrow \beta \;\equiv\; \neg\beta \rightarrow \neg\alpha \\ &\text{Row } (\alpha,\beta)=(T,T):\quad \neg\beta\rightarrow\neg\alpha = F\rightarrow F = T,\qquad \alpha\rightarrow\beta = T \\ &\text{Row } (\alpha,\beta)=(T,F):\quad \neg\beta\rightarrow\neg\alpha = T\rightarrow F = F,\qquad \alpha\rightarrow\beta = F \\ &\text{Row } (\alpha,\beta)=(F,T):\quad \neg\beta\rightarrow\neg\alpha = F\rightarrow T = T,\qquad \alpha\rightarrow\beta = T \\ &\text{Row } (\alpha,\beta)=(F,F):\quad \neg\beta\rightarrow\neg\alpha = T\rightarrow T = T,\qquad \alpha\rightarrow\beta = T \end{aligned}

Both columns read $T, F, T, T$ down the rows — identical truth values — so the two statements are logically equivalent. This is why proving the contrapositive is a valid way to prove the original implication.

Summary

Discrete math is fundamentally about proofs and getting comfortable with axiomatic systems (axioms, undefined terms, definitions, theorems); logic underlies math, which underlies science.
A statement (proposition) is a sentence that is either true or false but not both; sentences with free variables are predicates, not statements.
Logic keeps only truth-functional, univocal connectives — the five core ones are $\neg$ , $\wedge$ , $\vee$ , $\rightarrow$ , $\leftrightarrow$ — each fully defined by its truth table.
Implication $\alpha\rightarrow\beta$ is false only when a true antecedent yields a false conclusion; a false antecedent makes it vacuously true.
To negate, swap all $\leftrightarrow$ some with a "not," and use De Morgan's laws to distribute negation across $\wedge$ and $\vee$ .
The converse is not equivalent to an implication, but the contrapositive always is — confirmed by matching truth tables.