Truth Functions — or Not?

Core idea

Some inferences are valid because of how whole sentences are joined by words like "and", "or", and "it is not the case that". To study them, logicians abbreviate sentences as single letters — p, q — and treat the connectives as operations on truth values, where T stands for truth and F for falsity.

The key claim is that these connectives are truth functions: the truth value of a compound is fixed entirely by the truth values of its parts. Negation flips: ¬a is T exactly when a is F. A disjunction a or b is T when at least one disjunct is T. A conjunction a & b is T only when both conjuncts are T. Tabulate every combination of inputs and you get a truth table — a complete, mechanical specification of the connective.

Why it matters

A puzzle of chained inferences

Two inferences each look obviously valid. From "the Queen is rich" you may infer "either the Queen is rich or pigs can fly" — adding a disjunct never loses truth. And from "either the Queen is rich or pigs can fly" together with "the Queen is not rich" you may infer "pigs can fly" — if one disjunct fails, the other must hold. Yet chaining these two valid steps yields the manifestly absurd "the Queen is rich, the Queen is not rich; so pigs can fly."

Priest's argument: Chaining valid inferences cannot manufacture an invalid one — so if the chain looks invalid, our intuition about one of the links must be mistaken.

The truth-table verdict

A situation assigns a truth value to every basic sentence; truth tables then compute the value of any compound. An inference is valid if no situation makes every premise T and the conclusion not-T. Run the table on the troubling chain and the premises q and ¬q are never both true together. There is no row to worry about — so the inference is vacuously valid. Our intuition misfired because everyday inferences are almost never vacuous, and our instincts were trained only on the ordinary case.

Where the machinery strains

The clean story has a crack. Truth-functional "and" makes "a and b" interchangeable with "b and a". But "John hit his head and fell down" and "John fell down and hit his head" describe different events — order, and implied causation, matter. Truth-functional "or" says a disjunction is true if either side is; yet a friend who says "either you come now or we will be late", knowing full well you had thirty minutes' slack, has lied. English connectives often carry a connection that truth values alone cannot capture.

Key takeaways

Mental model

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Practical application

When checking a connective-based argument, build the truth table rather than trusting instinct. List every basic sentence, enumerate every combination of T and F, compute each premise and the conclusion, and scan for a "bad row" where the premises hold but the conclusion fails. No bad row means valid.

Example

Suppose a deployment rule says: "Either tests passed or the build is blocked." You also know the build is not blocked. Truth-functionally, you may infer "tests passed" — there is no row where both premises are true and that conclusion is false.

But notice the non-truth-functional residue. If a teammate says "either you merge now or the release slips" while knowing the release window is wide open, the disjunction is technically true the moment you merge — yet you would rightly feel misled. The sentence promised a real connection between the alternatives, and truth tables alone never recorded that promise.

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