A form is refuted by an argument of that form whose premises are plainly true and whose conclusion is plainly false. The method sorts the valid forms, hypothetical syllogism, disjunctive syllogism and constructive dilemma, from the fallacies that resemble them, and it is the course's table put to use on sentences.
Core ideas
Showing that a form is bad takes one argument of that form that obviously fails. The table says which row to imitate.
Take a counterexample row from the table, and choose a real sentence for each letter so that it has the value the row gives it. The result is an argument of the same shape with true premises and a false conclusion.
Hypothetical syllogism, disjunctive syllogism and constructive dilemma have none, so no substitution can work. That is a theorem about the form, not a failure of imagination.
A valid argument may have a false conclusion, and an invalid one a true conclusion. The verdict comes from the rows. What the world supplies is only the sentences with which to illustrate them.
The material
The lab names each shape from a catalogue; the table decides the verdict, and the method turns a row into a sentence.
Three forms are valid and are worth knowing by name. Hypothetical syllogism chains two conditionals: p → q, q → r, so p → r. Disjunctive syllogism rules out one alternative: p ∨ q, ¬p, so q. Constructive dilemma combines two conditionals with a disjunction: p → q, r → s, p ∨ r, so q ∨ s. Each has no row in which the premises hold and the conclusion fails.
Two look similar and are not. Affirming a disjunct treats an inclusive or as exclusive: p ∨ q, p, so ¬q. Denying the antecedent, from “Validity by Truth Table”, is p → q, ¬p, so ¬q. In the first the troublesome row is p true and q true.
To refute a form, find a row of its table with every premise true and the conclusion false, then replace each letter with a sentence that has the value the row gives it. The result is an argument of the same form whose premises are plainly true and whose conclusion is plainly false.
“Either Paris is in France or Rome is in Italy. Paris is in France. So Rome is not in Italy.” The row is p true and q true, and both sentences are true. Both premises are true and the conclusion is false, so the shape cannot be valid, and the argument needed no detective story to show it.
The method is also how a weak argument is identified in a conversation without a table. Say the form in letters, ask which row would break it, and offer a case of that kind. For denying the antecedent the row is p false and q true: “If 6 is divisible by 4, then 6 is even; 6 is not divisible by 4; so 6 is not even.” Both premises are true and the conclusion is false.
What the method does not do is refute the conclusion of the original argument. An invalid form shows that the premises do not guarantee the conclusion. The conclusion may still be true, and may be established by another route. “If 9 is divisible by 4, then 9 is even; 9 is not divisible by 4; so 9 is not even” has the same shape, true premises and a true conclusion, and is invalid still.
Valid forms behave differently when their conclusion is false. Disjunctive syllogism with “the moon is cheese or Paris is in Germany; the moon is not cheese; so Paris is in Germany” is valid and the conclusion is false. A false conclusion forces a false premise, and here it is the first.
If a valid argument has a false conclusion, at least one premise is false. If an argument has true premises and a false conclusion, it is invalid. A false conclusion alone settles nothing about validity.
In the lab, choose each of the five presets in turn. The lab names the form from the shape of the formulas, and the verdict comes from the rows. Hypothetical syllogism has three letters and eight rows, disjunctive syllogism two and four, and the dilemma four letters and sixteen rows.
Interactive
Every row, every counterexample, and the form
Method
Five steps; the third is the craft.
Replace each simple sentence with a letter, one letter per sentence, and keep the connectives.
Use the table, or the lab. If there is none, the form is valid and the method has nothing to offer.
For each letter pick a sentence whose actual truth value is the row's. Arithmetic and geography are good sources, because nobody disputes them.
Every premise must be true, and the conclusion false. If a premise is false, the instance refutes nothing.
The form is invalid. Say nothing about the original conclusion, which may be true.
“Either the butler or the maid did it. The butler did it. So the maid did not.” Let p be the butler and q the maid.
In the row p true and q true the premises hold and the conclusion fails: they did it together. The pattern is audible with the same form on a rainy Tuesday: it is raining or it is Tuesday, it is raining, so it is not Tuesday. The argument moves from one alternative to the denial of the other and treats or as exclusive. It does not show that the maid is guilty; it shows that this reason does not clear her.
Given an argument in sentences, write its form in letters, say whether it is valid, and if not produce a counterexample row and an instance of the same form with true premises and a false conclusion.
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