Argument Forms

Most logic courses teach argument forms with invented examples: if it rains the match is cancelled, all men are mortal, some such. This is a catalogue of the other kind. Every form here has been seen in the wild — the criterion for inclusion is not that an argument is instructive in the abstract, but that somebody, somewhere in the philosophical literature, actually used it, argued about it, or got caught out by it.

So it is really a catalogue of episodes in philosophy that happen to have a shape. The Dutch book argument for probabilism, and the exact line where it crosses from is to ought. Pollock’s pink elephant, and why a defeater is not a denial. The ratio/obiter distinction as courts actually deploy it. Curry’s paradox, and which structural rule you have to give up. The Axiom of Choice, in its finite shadow. Each entry says where it was found and quotes the source.

Every verdict is machine-verified. The truth tables, countermodels, trees and metrics are computed from the formulas rather than asserted on anybody’s authority, and the natural-deduction proofs are checked by a Fitch checker. Where a form is invalid, the entry does not merely say so — it shows you the countermodel, names the premise you were supplying without noticing, and links to the repaired form that adding it produces.

What is in an entry

Each form is presented three ways, because a first-order logic course teaches three methods and they illuminate different things:

• The truth table settles the question by brute force, and shows you exactly which row breaks an invalid argument. Some of them are needles: the Dutch book form used in lecture has one countermodel in sixty-four rows.
• The truth tree shows you why. A branch that closes is a contradiction you were forced into; a branch that stays open is a countermodel the tree built for you.
• The natural-deduction analysis asks whether you can derive it. Where no proof exists, the entry says where the attempt breaks down — and that is usually the most interesting sentence on the page. “o occurs in no premise, so no rule can ever introduce it” is about as clean a statement of Hume’s gap as a formal system can offer.

Alongside those: the provenance, with quotations marked as verbatim, paraphrase or our own reconstruction; the tags, including what exactly has gone wrong with an invalid form; and a deletion test showing which premises were doing work and which were idle all along.

For students in PHIL 1115

The practice page can restrict itself to what we have covered. Each form records the earliest lecture at which each method becomes available — separately per method, because the same argument can be a Lecture 4 truth table and a Lecture 11 proof — so you can ask for problems you actually have the tools for.

A word on notation. Formulas are set in the course’s symbols: ∼ for negation, & for conjunction, ∨ for disjunction, ⊃ for the conditional, ≡ for the biconditional, and ⊥ for falsum. The turnstile changes with the method: ⊨ for semantic consequence, ⊢ for the tree, ⊢ND for a derivation — and each gets a slash through it where the argument fails.

Scope

This is a pilot slice — thirty-five forms out of a much larger inventory still being mined, and the collection will grow. It is propositional logic only for now; predicate-logic forms need a different treatment, since a truth table is no longer the right kind of evidence for them.

Corrections and suggestions are welcome, particularly a form you have seen in print that ought to be here.