Valid inference

A pink title card reads "Logical methods for ∀I, Lecture 3". The A of AI is drawn as the upside-down A of the universal quantifier, ∀, which is the course mascot's name.

Below the card: "Chapter 3: Valid inference", and the course address, logicalmethods.ai. At the right the mascot ∀I holds up a green board carrying the inference from all humans being mortal and Socrates being human to Socrates being mortal.

Heading: "What you'll be able to do". Introduction: "After this lecture and chapter 3, you can:". Five goals follow, each ticked off with a green check mark and tagged on the right with its level in Bloom's revised taxonomy.

One: explain what makes an inference valid. Understand. Two: tell deductive from inductive, logical from material. Understand and analyse. Three: test an inference with a pattern or a counterexample. Apply. Four: read validity off sets of models. Understand and apply. Five: measure inductive support with probabilities. Apply.

Closing line: "Read chapter 3 first; exercise sheet 3 comes after the lecture." The mascot ∀I stands at the right wearing a lit lightbulb.

Heading: "Valid and invalid". A green box states that validity is a standard of correctness, and that an invalid inference has made a mistake. At the top right the mascot cycles twice: once under the sun, once under a rain cloud.

Two inferences stand side by side. The green one, labelled Valid: from "If it is sunny, then ∀I is cycling" and "It is sunny", infer "∀I is cycling". The red one, labelled Invalid: from the same conditional and "∀I is cycling", infer "It is sunny".

Under the green one: the other premise is about the weather. Under the red one: ∀I also cycles in the rain.

Closing lines: "They look alike. Logic wants a test that does not depend on a second look," and "This mistake has a name: affirming the consequent."

Heading: "Truth preservation". A green box gives the definition: an inference is deductively valid if and only if the conclusion is true under the hypothetical assumption that all the premises are true.

Three points: the test is hypothetical, so we never check the premises; a valid inference can therefore have false premises; imagine a world where cats fly, and there the cat flies.

A yellow box holds the inference: from "All cats can fly" and "∃I is a cat", infer "∃I can fly". Beside it: cats do not fly, and the reasoning is correct anyway. At the right, the drawing of a winged cat chasing a mouse.

Closing line: "Validity relates truth-values. It does not report the facts."

Heading: "Valid, sound, true". A green box defines a sound argument as a deductively valid argument whose premises are actually true, so its conclusion must be true as well.

Three panels. First, green: all cats can fly, ∃I is a cat, so ∃I can fly — false premises, valid but not sound. Second, green: all humans are mortal, Socrates is human, so Socrates is mortal — true premises, valid and sound. Third, red: if sunny then ∀I cycles, ∀I is cycling, so it is sunny — the premises may well be true, but this is not valid and not sound.

Closing lines: "Truth and validity come apart in every direction," and "Logic checks the connection. Whether the premises hold is another question."

Heading: "Logical form". Two points: swapping Socrates for Turing and human for mathematician leaves the inference valid; and since it is valid for every such swap, it is valid in virtue of form.

Two grey panels show the pair: all humans are mortal and Socrates is human, so Socrates is mortal; all mathematicians are smart and Alan Turing is a mathematician, so Alan Turing is smart. At the right the mascot presents the shared schema on a blackboard: all As are Bs, c is an A, so c is B.

A blue box separates two notations. "A, (A → B) ∴ B" is the inference. "A, (A → B) ⊨ B" is the claim that it is valid. Beside it: A and B stand in for any formulas at all, and some systems make the second line false.

Heading: "Material validity". A green box contrasts the two: logically valid means truth preservation depends on the form; materially valid means it depends on what the words mean.

In green: from "Sam is the brother of ∀I's mother" infer "Sam is ∀I's uncle". In red, with "brother" replaced by "sister": the same conclusion no longer follows, because then Sam is ∀I's aunt.

A blue box, headed "Carnap: add a meaning postulate", adds the premise "The brother of a person's mother is their uncle" to the first inference. Beside it: now it follows by form alone; and mathematics is material too, since n divisible by 4 makes n even.

Closing lines: "Logical validity is domain-general; material validity is domain-specific," and "Language models do better when familiar content points at the answer."

Heading: "Always: deductive validity". A green box: deductively valid means that necessarily, in every situation where the premises are true, the conclusion is true as well.

Three points. No exceptions: one bad situation breaks it. It is characteristic of mathematics, and of physics when it calculates. And it is indefeasible: extra premises never undo it. At the right, the mascot works an integral out at a blackboard.

Closing lines: "So we can keep adding to what we know: belief accumulation," and "The prediction is only as certain as the laws and observations behind it."

Heading: "Deduction at work". Two points: vixens are female foxes, so a vixen here means a fox here; and programming languages are formal languages with deductive rules.

A grey panel holds a five-line Python program. It reads a whole number with int(input(...)), and prints that the number is positive if num is greater than 0, and that it is not positive otherwise.

Beside it: enter 1 and modus ponens applies, so it prints. One that only mostly obeyed would sometimes say nothing.

Closing line: "Expert systems and reasoning LLMs should draw these inferences too."

Heading: "Schemas and fallacies", in two columns.

Valid, in green: modus ponens, from (A → B) and A infer B; modus tollens, from (A → B) and ¬B infer ¬A; disjunctive syllogism, from (A ∨ B) and ¬A infer B.

Fallacies, in red: affirming the consequent, from (A → B) and B to A; denying the antecedent, from (A → B) and ¬A to ¬B; affirming a disjunct, from (A ∨ B) and A to ¬B.

Closing line: "A countermodel each: A false with B true; for the third, both true."

Heading: "Models". A green box: a model represents a possible reasoning scenario, and every formula is true or false in it.

Three points: necessarily means in every model; the logical space is the models we consider; a proposition is a set of models. Below them, the definition "[A] = { M : M is a model in which A is true }".

At the right, the chapter's set diagram: a yellow logical space, labelled below, holding twelve little worlds. A green region labelled [A] gathers five of them.

Closing lines: "We have not said what a model is. Every language supplies its own," and "Modelling assumption: each formula is true or false in each model."

Heading: "Sets again". Two green boxes: "S ⊆ T", every member of S is one of T; and "S ∩ T = { a : a ∈ S and a ∈ T }".

Two rows of picture sets follow. First: the set containing Little Jimmy and Mr. Sir is a subset of the set containing Little Jimmy, the set containing 1 and a beer, and Mr. Sir. Second: the set containing Little Jimmy and Mr. Sir, intersected with the set containing Little Jimmy and 1, is the set containing Little Jimmy.

Closing lines: "The beer is not an element; the set containing it is.", "Several premises: the models where all of them are true at once," and "That is an intersection, and validity is about to become a subset claim."

Heading: "Deductive consequence". A blue box states the definition: P₁, …, Pₙ ⊨ C if and only if [P₁] ∩ … ∩ [Pₙ] ⊆ [C].

Two points: every model of all the premises is a model of C; and test the intersection, not one premise at a time.

At the right, the chapter's valid-consequence diagram. Eight little worlds, M₁ to M₈, sit in a yellow logical space. The regions [P] and [Q] overlap, and the tall region [C] covers the whole overlap: M₁ and M₂ lie in all three. M₃ is in [P] only, M₄ in [Q] only, M₆ in [C] only, and M₅, M₇ and M₈ outside everything.

Closing lines: "Neither premise alone guarantees C," and "Unsatisfiable premises entail everything; with no premises, ⊨ C says C holds always."

Heading: "Countermodels". A red box: a countermodel makes every premise true and the conclusion false.

Two points: one is enough to refute an inference; validity is about every model, invalidity about a single one.

A grey panel shows both jobs in notation. Valid: [(A ∨ B)] ∩ [¬A] ⊆ [B]. Invalid: [(A → B)] ∩ [¬A] is not a subset of [¬B], the subset sign struck through.

At the right, the chapter's countermodel diagram: the same eight worlds, but [C] now sits higher. M₂ lies in the overlap of [P] and [Q] and outside [C] — both premises true, the conclusion false.

Closing line: "Denying the antecedent falls to a single model: A false and B true."

Heading: "A blueprint". Three points: we still have not said what a model is; supply the models of a language and say when a formula is true in one, and validity follows; that is the work of the next chapters. At the right, the mascot ∀I looks on, wondering.

A yellow box headed "Still to supply" lists three things: what a model for a given language is; when ¬, ∧, ∨, → and ↔ hold in one; and a way to survey all of them at once.

Closing line: "Next chapter: models for the language of propositional logic."

Heading: "Mostly: inductive validity". A green box: inductively valid means the premises make the conclusion likely, without guaranteeing it.

A grey box quotes A Study in Scarlet, 1887: "From a drop of water a logician could infer the possibility of an Atlantic or a Niagara." Below it: Holmes calls it deduction; almost all of it is induction.

Two points. How probable is C given the premises, and how much do the premises raise it? And: induction is defeasible — smoke means fire, until you hear that Mr. Sir smokes his pipe here. At the right, the drawing of smoke rising behind the trees while the mascot imagines a camp fire.

Closing lines: "Deduction accumulates beliefs; induction modulates them," and "The evidence did not go away. It was supplemented, and that changed the conclusion."

Heading: "Enumerative induction". Two yellow panels: from "All swans we have observed so far were white" infer "All swans are white"; and from "All marbles we have drawn so far were white" infer "All marbles are white". At the right, a swan on a pond, and the mascot at a jar of marbles announcing "They're all white!!".

A blue box, headed "the shared form", gives the schema: from "All As we have observed so far were B" infer "All As are B".

One point: the observations cannot lower the probability of C.

Closing lines: "They do not make it likely either: size alone does not remove selection bias," and "Careful: an inductive definition and an inductive inference are different things."

Heading: "Probabilities over models". Two points: in inductive logic a model is a way the world could turn out; one roll of a die gives six models, and a proposition is a set of them.

A yellow band shows the six dice faces. The faces 2, 4 and 6 are ringed in green, and labelled "[EVEN] = the models 2, 4 and 6".

A green box: a probability function gives each proposition a number between 0 and 1, and the outcomes sum to 1.

Two panels give distributions. Fair die: Pr(1) = … = Pr(6) = 1/6, so Pr([EVEN]) = 1/2. Loaded die: Pr(1) = … = Pr(5) = 1/10 and Pr(6) = 1/2, so Pr([EVEN]) = 7/10.

Closing line: "Fixing the distribution is statistics; saying what it supports is logic."

Heading: "Conditional probability". A blue box gives the definition: Pr([A] | [B]) = Pr([A] ∩ [B]) / Pr([B]), provided Pr([B]) > 0. Beside it: the part of B that is also A, measured against B.

A table with four columns — distribution, Pr(2), Pr([EVEN]) and Pr(2 | [EVEN]). Fair die: 1/6, 3/6, and (1/6) / (3/6) = 1/3. Loaded die: 1/10, 7/10, and (1/10) / (7/10) = 1/7.

Closing lines: "Either way, learning that the roll is even raises the probability of a two.", "In that sense the hypothesis supports the conclusion," and "Only division by zero is ruled out: we test premises with positive probability."

Heading: "Support, and logical support". At the top right: "Next: Boolean algebra (ch. 4)".

A blue box, headed "Probability raising", defines d(C, E) = Pr([C] | E) − Pr([C]), and works it out: fair die, 1/3 − 1/6 = 1/6; loaded die, 1/7 − 1/10 = 3/70. Positive means support; zero means the premises are irrelevant.

A green box, headed "Weak logical inductive consequence", gives the definition: P₁, …, Pₙ inductively support C if and only if Pr([C] | E) ≥ Pr([C]) for every probability function Pr with Pr(E) > 0. The logical relations stay put; the numbers vary.

Below the boxes, E is defined as [P₁] ∩ … ∩ [Pₙ], the joint premise region. Then two lines. Logical: [2] ⊆ [EVEN], so EVEN supports 2 under every distribution. Material: EVEN raises HIGH on a fair die and lowers it on another — same models, different probabilities. At the right, the mascot ∀I looks suitably impressed.

Closing lines: "More likely than before does not mean likely: a two is still only 1/3," and "'Stronger' depends on the measure: the log-likelihood ratio ranks cases differently."