11_Logic — what does "if A then B" actually claim?¶
Level: 101 · for anyone reading theorems, from the first page of a precalculus book on
Every theorem in the other chapters has the shape "if A then B", and the book uses words like converse and if and only if as if everyone already knew what they mean. This chapter says what they mean, and checks each claim on numbers and triangles by brute force. It needs nothing but school arithmetic.
| # | Lesson | The question it answers |
|---|---|---|
| 1 | If A then B: converse, contrapositive and inverse | Which rewordings of a theorem say the same thing, which need their own proof, and what is "if and only if"? |
| 2 | Truth tables and the laws of logic | What makes a "law of logic" true, and what is the difference between ↔ and ⇔? |
| 3 | Predicates and quantifiers | What do ∀ and ∃ mean, which pairs of them can be swapped, and what does "not everyone" say? |
| 4 | What a proof is | What does "prove it" ask for, why is a loop over cases not it, and where does a chain of "why?" stop? |
| 5 | Induction | Why is "assume P(n), prove P(n + 1)" allowed, and where does the proof that all cars are the same colour break? |
The through-line¶
An "if A then B" breaks in one case only, A true and B false. Everything in lesson 1 follows from that: the contrapositive breaks in the same case and so is the same claim; the converse breaks in a different case and so needs its own proof; and a theorem whose converse is also true is an "if and only if", like Pythagoras.
Lesson 2 widens the view from one connective to all five. A sentence built from letters and connectives is a function of the letters' truth values, a truth table lists it on every row, and a law of logic is two sentences whose columns agree on every row; that is why a finite table proves a law, and why ↔ inside a sentence and ⇔ between sentences are different things. Lesson 3 adds the two symbols a table cannot handle, ∀ and ∃, as loops over a universe of discourse: ¬ flips one into the other, same-type pairs swap and mixed pairs do not, and "everyone likes someone" is not "someone is liked by everyone". The three lessons together are the language the axioms of set theory are written in. Lesson 4 says what a proof is at all: a chain from the givens to the claim, a reason beside every line, argued with letters so that it covers every case at once where a loop covers only the cases it reaches; the distance formula is written out as ten such lines, and a program checks the chain's shape and breaks it three ways. Lesson 5 is the first proof method: induction, valid because a false claim about the natural numbers has a least counterexample and a step that always works forbids one; the famous wrong proof that all cars are the same colour breaks at exactly one rung, which the program finds. Proof by contradiction appears inside it; construction is still on the roadmap.
Po polsku, w skrócie¶
Prawie każde twierdzenie ma postać „jeśli A, to B". Ten rozdział wyjaśnia, co to zdanie naprawdę twierdzi: jest fałszywe tylko wtedy, gdy A jest prawdziwe, a B fałszywe. Z tego wynika, że transpozycja („jeśli nie B, to nie A") mówi to samo, a twierdzenie odwrotne („jeśli B, to A") trzeba udowodnić osobno. Gdy oba kierunki są prawdziwe, mówimy „wtedy i tylko wtedy", jak w twierdzeniu Pitagorasa z jego odwrotnym. Druga lekcja pokazuje, że prawo logiki to dwie kolumny tabeli prawdy, które zgadzają się w każdym wierszu, więc skończona tabela jest dowodem, a ↔ wewnątrz zdania i ⇔ między zdaniami to dwie różne rzeczy. Trzecia dodaje kwantyfikatory ∀ i ∃ jako pętle po uniwersum: negacja odwraca kwantyfikator, kwantyfikatory tego samego rodzaju można zamieniać, mieszanych nie. W tym języku zapisane są aksjomaty teorii mnogości z rozdziału 13. Czwarta lekcja mówi, czym w ogóle jest dowód: łańcuchem zdań od założeń do tezy, z powodem przy każdej linii, zapisanym literami, więc obejmującym wszystkie przypadki naraz, czego pętla po przypadkach nie potrafi; wzór na odległość jest w niej rozpisany na dziesięć takich linii. Piąta lekcja to indukcja: pierwsza metoda dowodu, poprawna dzięki zasadzie dobrego uporządkowania, z błędnym dowodem o samochodach, który zawodzi na jednym szczeblu.
Auf Deutsch: Stichwörter¶
Was „wenn A, dann B“ behauptet: Umkehrung, Kontraposition, Wahrheitstafeln, Quantoren, was ein Beweis ist, Induktion.
Stichwörter: Logik, Implikation, Umkehrung (converse), Kontraposition, Wahrheitstafel, Aussage, Prädikat, Quantor, Beweis, vollständige Induktion.
A note on the code¶
Each "if A then B" is a pair of Python functions, and the program lists every case in a finite range where A holds and B fails. An empty list is evidence, not proof; the lesson says so, and shows a claim that survives four tests and fails the fifth. Lesson 4 turns that into a checker: a proof is data, a list of lines with their reasons, and the program verifies the shape of the chain, then tries each line on numbers, which can refute a line and never confirm one. Lessons 2 and 3 are different: a truth table over n letters, or a quantifier over a universe of four points, has finitely many cases and the programs run all of them, so there the check is the proof, and the pages say why.