The Timpson Lyceum · Geometry (2024)

Propositional Logic & History – Lecture 1 (Geometry)

Propositional Logic & History – Lecture 1

Overview

This opening lecture sets the stage for the whole course by explaining why a geometry class begins with logic. Geometry ("earth measurement") was the first branch of mathematics, but mathematics proper — the practice of proving universal truths rather than collecting empirical observations — begins with the Greeks. Because every theorem is built by deductive reasoning, logic is the true foundation of geometry, set theory, and ultimately physics. The bulk of the lecture is an introduction to propositional logic: what counts as a statement, the five truth-functional connectives, truth tables, and the key relationships among an implication, its converse, and its contrapositive.

Key Concepts

A short history of proof.

• Geometry comes from Greek roots meaning earth measurement; it was the first field of mathematics to develop.
• The Babylonians and Egyptians gathered empirical data (brute observation): the angle sum of a triangle was about $180^\circ$, $\pi \approx \tfrac{22}{7}$, a 3–4–5 triangle is about a right angle. They cared whether results were good enough for use.
• Thales ($\approx 600$ BC) introduced the idea of proving a result — establishing it exactly and for all cases.
• Pythagoras and the Pythagoreans gave the Pythagorean theorem; Plato's Academy (the source of the words academy / academic) developed geometry systematically. Its motto: "Let none ignorant of geometry enter here" — geometry was prized as a training ground for correct thinking.
• Aristotle is the father of logic, which he showed underlies every field of study.
• Euclid's Elements (13 books) organized all known geometry so that every theorem follows from 5 postulates and 5 common notions — the model every later axiomatic system imitates (e.g. Newton's three laws in physics).
• This is why the course order is logic → set theory → geometry: each layer presupposes the one below it.

Three kinds of reasoning.

• Deductive reasoning — conclusions follow with certainty from premises. If all men are mortal and Socrates is a man, then Socrates is mortal. Note the whole if–then is what is asserted true; we never claimed the premises themselves are true. This is the only reasoning used in logic and mathematics, which is why every theorem can be phrased as an implication.
• Inductive reasoning — generalizing from repeated particular observations (every observed object falls when released, so all unsupported objects fall). Probable, not certain; this is the reasoning of empirical science.
• Abductive reasoning — using known generalizations to reconstruct the best explanation of a single observation (a drop on your hand → maybe rain, sprinklers, or a friend's spittle).

Statements (propositions).

• A statement is a sentence that is either true or false, but not both. "Statement" and "proposition" are used interchangeably here.
• Greek letters $\alpha, \beta$ denote statements.
• Examples that are statements: $2 \times 2 = 5$ (false), "New York is a suburb of Los Angeles." Not statements: commands ("go home"), nonsense, and the liar sentence "This statement is false" (it can be neither consistently true nor false). A sentence with a free variable such as $x + 2 = 5$ is a predicate — true or false only once $x$ is fixed (covered next week).

Sentential connectives. Words or phrases that attach to statements to form a meaningful new sentence (not, and, or, if…then, if and only if). We restrict to truth-functional connectives: ones where the truth value of the result is completely determined by the truth values of the parts. "Jack believes $\alpha$," "it is probable that $\beta$," and the vague word "but" are not truth-functional and are excluded — logic uses only univocal (one-meaning) language.

The five connectives (each justified by its truth table):

• Negation $\neg\alpha$ — true exactly when $\alpha$ is false. $\alpha$ and $\neg\alpha$ always contradict each other. Caution: "not" $\neq$ "never" (never smuggles in time, a physics notion). Negation is the subtle one — the negation of "all swans are white" is "some swans are not white," not "no swans are white."
• Conjunction $\alpha \wedge \beta$ ("and") — true only when both are true.
• Disjunction $\alpha \vee \beta$ ("or") — false only when both are false; otherwise true. This is the inclusive or (both allowed). Exclusive or is written $\alpha \veebar \beta$ but is not used in this course.
• Implication $\alpha \to \beta$ ("if $\alpha$ then $\beta$") — false only when $\alpha$ is true and $\beta$ is false; when $\alpha$ is false the implication is vacuously true. $\alpha$ is the antecedent (condition), $\beta$ the conclusion. Every theorem is an implication. Many English phrasings mean $\alpha \to \beta$: "if $\alpha$ then $\beta$," "$\beta$ if $\alpha$," "$\alpha$ only if $\beta$," "$\alpha$ is sufficient for $\beta$," "$\beta$ is necessary for $\alpha$."
• Biconditional $\alpha \leftrightarrow \beta$ ("if and only if") — true exactly when $\alpha$ and $\beta$ have the same truth value; equals $(\alpha \to \beta) \wedge (\beta \to \alpha)$. Read as logical equivalence.

Converse and contrapositive.

• The converse of $\alpha \to \beta$ is $\beta \to \alpha$. It is not equivalent to the original — confusing the two is the most common error in logic ("shoes $\to$ socks" does not give "socks $\to$ shoes").
• The contrapositive of $\alpha \to \beta$ is $\neg\beta \to \neg\alpha$. It is logically equivalent to the original (identical truth tables).

Key equivalences (De Morgan's laws + contrapositive).

Precedence of operators (evaluate in this order, like order of operations in arithmetic): $\neg$, then $\wedge$, then $\vee$, then $\to$, then $\leftrightarrow$. Parentheses are recommended around $\wedge$ and $\vee$ to avoid confusion.

Worked Example

Verify that an implication and its contrapositive are logically equivalent by building a truth table. For each row, compute $\alpha \to \beta$ (false only when $\alpha$ is true and $\beta$ false) and $\neg\beta \to \neg\alpha$:

Both columns read $T, F, T, T$ for every assignment of $\alpha$ and $\beta$, so the two statements are logically equivalent. Concretely: "if I wear shoes then I wear socks" says exactly the same thing as "if I am not wearing socks then I am not wearing shoes."

Summary

• Mathematics begins not with measurement but with proof; logic is the foundation beneath geometry, so the course runs logic → set theory → geometry.
• The three modes of reasoning are deductive (certain, used in math), inductive (generalizing, used in science), and abductive (best explanation of a single event).
• A statement is a sentence that is exactly one of true or false; logic uses only truth-functional, univocal connectives.
• The five connectives are negation $\neg$, conjunction $\wedge$, disjunction $\vee$ (inclusive), implication $\to$, and biconditional $\leftrightarrow$, each defined by a truth table; an implication is false only when its antecedent is true and conclusion false.
• An implication is not equivalent to its converse but is equivalent to its contrapositive; De Morgan's laws govern negating $\wedge$ and $\vee$.
• Operator precedence runs $\neg, \wedge, \vee, \to, \leftrightarrow$; a sentence with a free variable like $x + 2 = 5$ is a predicate, the topic of the next lecture.