The Timpson Lyceum · History of Philosophy (2024)

Crash Course on Logic: Propositions, Predicates, Identity – Lecture 1 (History of Philosophy)

Overview

This opening lecture sets up the whole course by answering three questions: What is philosophy?, How do we reason?, and Why is logic the philosopher's main tool? Philosophy literally means the love of wisdom ($\phi\iota\lambda\epsilon\tilde{\iota}\nu$, to love + $\sigma o\phi\acute{\iota}\alpha$, wisdom), and historically every field of study — mathematics, science, psychology — began under its umbrella before splitting off on its own. What remains distinctively philosophical is a method: reason inductively from common human experience to premises, then reason deductively from those premises to conclusions in a systematic way. Because logic is the only reliable instrument for checking such arguments, the lecture is a crash course in it: propositional logic first, then predicate logic, then logic with identity.

A guiding attitude runs through the lecture: if an argument leads to an absurd conclusion, reject the argument — even if you can't yet say where it went wrong. A clever one-line "proof" that $-1 = 1$ should not convince you that $-1 = 1$; it should convince you that a step was cheated.

Key Concepts

• Philosophy — Inductively reasoning from experiences common to all people to some premises, then applying deductive reasoning to those premises to reach conclusions systematically.
• Science vs. philosophy — Science uses the same method but starts from observable, quantifiable phenomena rather than common experience, which lets it encode premises in mathematics and reason far faster and more reliably.
• Deductive reasoning — From general premises to a guaranteed conclusion; the only reasoning allowed in logic and mathematics. It is certain but says nothing about our actual universe (it establishes the implication, e.g. "if all men are mortal and Socrates is a man, then Socrates is mortal").
• Inductive reasoning — From many particular cases to a generalization; how we learn about the actual world, but only with probability, never certainty.
• Abductive reasoning — From generalizations back to the best explanation of a single event (used in history and law).
• Axiomatic system — Four parts: undefined terms, axioms (starting assumptions that give meaning to the undefined terms), definitions (built from prior/undefined terms), and theorems (proved from the axioms). All of mathematics presupposes logic plus a system's axioms.
• Truth with a capital T — Attainable only in logic and mathematics (an implication that cannot be false). Science, history, and philosophy trade in weaker, probabilistic notions of "true."
• Statement (proposition) — A sentence that is either true or false, but not both. Represented by Greek letters ($\alpha, \beta$).
• Sentential connective — Anything that joins sentences into a sentence. Logic keeps only those that are univocal (one exact meaning — so "but," though a connective, is excluded) and truth-functional (the truth of the whole is fixed by the truth of the parts — so "Jack believes …" is excluded).
• Conjunction $\alpha \land \beta$ — True only when both are true.
• Disjunction $\alpha \lor \beta$ — The inclusive or: false only when both are false. (Exclusive or, $\veebar$, is true when exactly one holds.)
• Negation $\neg\alpha$ — True exactly when $\alpha$ is false. Distinct from a contrary or "opposite": "all swans are white" is negated by "some swan is not white," not by "all swans are black" (contrary statements can both be false).
• De Morgan's laws — $\neg(\alpha \land \beta) \equiv \neg\alpha \lor \neg\beta$ and $\neg(\alpha \lor \beta) \equiv \neg\alpha \land \neg\beta$.
• Implication $\alpha \rightarrow \beta$ — "if $\alpha$ then $\beta$." $\alpha$ is the antecedent (a sufficient condition), $\beta$ the consequent (a necessary condition). Note "$\alpha$ only if $\beta$" means $\alpha \rightarrow \beta$, while "$\alpha$ if $\beta$" means $\beta \rightarrow \alpha$.
• Biconditional $\alpha \leftrightarrow \beta$ — Logical equivalence; both directions hold.
• Predicate — An open sentence such as $x + 2 = 5$ that becomes a proposition only once its variable is specified; the bridge from propositional to predicate logic.

Worked Example

A one-line "proof" that $2 = 0$ — and how to locate the cheat by refusing the absurd conclusion.

1. Start from a true statement: $2 = 1 + 1$.
2. Rewrite each $1$ as $\sqrt{1}$: $\;2 = \sqrt{1} + \sqrt{1}$.
3. Rewrite $1$ under one radical as $(-1)(-1)$: $\;2 = \sqrt{1} + \sqrt{(-1)(-1)}$.
4. Split the radical: $\;2 = \sqrt{1} + \sqrt{-1}\,\sqrt{-1}$.
5. Since $\sqrt{-1} = i$: $\;2 = 1 + i \cdot i = 1 + (-1) = 0$.
6. Conclusion reached: $2 = 0$ — which is absurd, so the argument must be rejected even before we diagnose it.
7. The cheat is at step 4: $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ is not valid for negative numbers, so the manipulation is illegitimate. The lesson: a slick argument on familiar ground can still be wrong; an absurd conclusion is sufficient reason to reject the reasoning.

Summary

• Philosophy is the systematic pursuit of wisdom: induction to premises, deduction to conclusions; science is the same method restricted to measurable phenomena.
• There are three modes of reasoning — deductive (certain, contentless about the world), inductive (about the world, but probabilistic), and abductive (inference to the best explanation).
• Certainty ("capital-T truth") lives only in logic and mathematics, which rest on axiomatic systems (undefined terms, axioms, definitions, theorems); every higher field may always fall back on logic.
• Propositional logic studies statements joined by connectives that are univocal and truth-functional: $\land$, $\lor$, $\neg$, $\rightarrow$, $\leftrightarrow$, each pinned down by a truth table.
• Watch the traps: negation is not the same as a contrary/opposite; "only if" reverses the direction you might expect; De Morgan's laws convert a negated and into an or (and vice versa).
• Reject absurd conclusions. If every step doesn't genuinely compel you, don't be talked into nonsense — a rule that will matter as later philosophers push arguments to extreme ends.