Basic Truth Tables for Compound Propositions

The first answer is what the table is for. The connective is defined to be false in exactly one case, where the antecedent is true and the consequent false, because that is the only case in which a promise of the form "if p then q" has been broken. Consider an undertaking to the court: "if the appeal is dismissed, my client will vacate within thirty days." When has the undertaking been violated? Only if the appeal is dismissed and the client does not vacate. If the appeal is allowed, the undertaking has not been violated, whatever the client does. Rows three and four record exactly that.

The second answer is that any other table would be worse. Suppose rows three and four were both made false. Then p ⊃ q would mean the same as p • q, and the connective would be useless. Suppose row three were true and row four false. Then the truth of a conditional would depend on the truth of its consequent when the antecedent is false, which would make "if p then q" behave differently from "if p then q" with the same parts in a different context, and truth-functionality would be lost.

The third answer is the honest one. Material implication is not an analysis of the English "if". It is a deliberately weakened substitute, chosen because it is truth-functional and because it validates the inferences deduction needs, namely affirming the antecedent and denying the consequent. Ordinary conditionals carry more, usually a connection of meaning or cause, and modern logic simply declines to represent that surplus. Saying this in an examination is worth more than reciting the table.

Constructing a table for a longer compound

Method. Count the distinct simple components. Write 2 to that power rows. Fill the component columns in the standard order. Then build the compound up in stages, one connective at a time, giving each stage its own column. The last column is the answer.

Example: ~(p • q) ≡ (~p ∨ ~q), which is one of De Morgan's rules.

The last column is true in every row.

Tautology, contradiction and contingency

A tautology is a compound proposition true in every row of its table. The example above is one. So is p ∨ ~p, which is the law of excluded middle written as a formula.

A contradiction is false in every row. p • ~p is the standard case, and it is the law of contradiction written as a formula.

A contingent proposition is true in some rows and false in others. Almost every proposition anybody actually asserts is contingent.

Why this matters. A tautology says nothing about the world, since it is true whatever the facts are. That is not a criticism: the rules of logic are tautologies, and they are useful precisely because they hold whatever happens. But an argument whose conclusion is a tautology has told you nothing, and a pleading whose case is a tautology has pleaded nothing.

De Morgan's rules

Two transformations, both provable by table, and both used constantly in reading statutes.

~(p • q) is equivalent to ~p ∨ ~q. The denial of a conjunction is the disjunction of the denials. To deny that both conditions were satisfied is to say that at least one was not.

~(p ∨ q) is equivalent to ~p • ~q. The denial of a disjunction is the conjunction of the denials. To deny that either happened is to say that neither did.

Where a lawyer meets them. Section 14 of the Indian Contract Act 1872 says consent is free when it is not caused by coercion, undue influence, fraud, misrepresentation or mistake. That is ~(c ∨ u ∨ d ∨ m ∨ k), and by the second rule it is equivalent to ~c • ~u • ~d • ~m • ~k. The section can therefore be read as "none of these five", which is how a court states it, or as "not this, and not this, and not this, and not this, and not this", which is how a pleading has to meet it.

Testing an argument by table

An argument is valid when there is no row in which every premise is true and the conclusion is false. That is the definition of validity from sequence 110, applied mechanically.

Test: affirming the antecedent. Premises p ⊃ q and p; conclusion q.

Look for a row where both premises are true. Only row one: p ⊃ q is true and p is true. In that row the conclusion q is true. There is no row with true premises and a false conclusion, so the argument is valid.

Test: affirming the consequent. Premises p ⊃ q and q; conclusion p.

Rows where both premises are true: row one, where p ⊃ q is T and q is T; and row three, where p ⊃ q is T and q is T. In row three the conclusion p is false. There is a row with true premises and a false conclusion, so the argument is invalid, which is what sequence 290 asserted and this proves.

That is the whole method, and it is mechanical: no judgment is exercised anywhere in it, which is what "the test can be computed" meant at sequence 330.

A worked example

A section provides: "A licence shall be cancelled if the holder has been convicted of an offence under this Act and has failed to pay the penalty within thirty days."

Symbolise. c: the holder has been convicted. f: the holder has failed to pay within thirty days. x: the licence shall be cancelled. The section is (c • f) ⊃ x.

Question one. The holder was convicted but paid within thirty days. May the licence be cancelled under this section?

Build the antecedent. c is true, f is false, so c • f is false. With a false antecedent the implication is true whatever x is, which means the section is satisfied and says nothing about x. The section supplies no power to cancel. That is rows three and four of the implication table doing real work: a false antecedent leaves the consequent entirely open.

Question two. The licence was cancelled. Does it follow that the holder was convicted?

This is affirming the consequent, and it is invalid. The licence may have been cancelled under some other provision.

Question three. The licence was not cancelled. What follows?

Denying the consequent, which is valid: ~x gives ~(c • f), and by De Morgan that is ~c ∨ ~f. Either he was not convicted or he did not fail to pay, and the section does not tell us which. A conclusion in the alternative is a real conclusion, and it is often all a provision will yield.

Distinctions that carry marks

What this does not mean

A true conditional is not a good argument. p ⊃ q being true in a row where p is false does not mean anything follows; it means the conditional has not been falsified.

Truth tables do not test the premises. They test the connection. Whether p is in fact true is not a question a table can answer.

Not every "if" is material implication. Conditionals about what would have happened, and conditionals expressing a causal connection, are not truth-functional, and a table cannot represent them.

Quick revision

Truth-functional: the truth of the whole is fixed by the truth of the parts. This is why tables are possible.

Rows: 2 to the power n, in the standard order TT, TF, FT, FF.

Tables: ~p true when p false; p • q true only when both true; p ∨ q false only when both false; p ⊃ q false only when p true and q false; p ≡ q true when they agree.

The two odd rows: a false antecedent makes an implication true, because the only way to break "if p then q" is to have p and not q.

Tautology true in every row; contradiction false in every row; contingent neither.

De Morgan: ~(p • q) is ~p ∨ ~q; ~(p ∨ q) is ~p • ~q.

Validity test: no row where every premise is true and the conclusion is false.

Test yourself

1. What makes a truth table possible, and how many rows does one have?

That the connectives of modern logic are truth-functional, meaning that the truth value of the compound is completely determined by the truth values of its components and by nothing else. Since each simple component is either true or false, a compound containing n distinct simple components has 2 to the power n rows, so two components give four rows, three give eight and four give sixteen.

2. Give the table for implication and explain the last two rows.

p ⊃ q is true when p and q are both true, false when p is true and q false, and true in both cases where p is false. The last two rows record that a conditional is not broken when its antecedent fails: an undertaking that "if the appeal is dismissed my client will vacate" is violated only if the appeal is dismissed and the client does not vacate, and it is not violated at all if the appeal is allowed, whatever the client then does.

3. Why is material implication not an analysis of the English "if"?

Because ordinary conditionals usually assert a connection of meaning, cause or law between the parts, and material implication asserts nothing of the kind: it requires only that the antecedent not be true while the consequent is false. It is adopted because it is truth-functional, so a table can handle it, and because it validates the two inferences deduction needs. The oddities that result are accepted as the price, not defended as a discovery about conditionals.

4. Define tautology, contradiction and contingent proposition, with an example of each.

A tautology is true in every row of its truth table, such as p ∨ ~p. A contradiction is false in every row, such as p • ~p. A contingent proposition is true in some rows and false in others, such as p ⊃ q. A tautology and a contradiction each say nothing about the world, since their truth value does not depend on the facts, while a contingent proposition does.

5. State De Morgan's rules and give a statutory application.

The denial of a conjunction is the disjunction of the denials, ~(p • q) being equivalent to ~p ∨ ~q; and the denial of a disjunction is the conjunction of the denials, ~(p ∨ q) being equivalent to ~p • ~q. Section 14 of the Indian Contract Act 1872, which makes consent free when it is not caused by any of five factors, is a negated disjunction and is therefore equivalent to the conjunction of five denials, which is the form a pleading has to meet.

6. How is an argument tested for validity by a truth table, and what does the test show about affirming the consequent?

By constructing the table and looking for a row in which every premise is true and the conclusion is false; if there is no such row the argument is valid, and if there is one it is invalid. For affirming the consequent, with premises p ⊃ q and q and conclusion p, there is a row in which p is false and q is true: both premises are true there and the conclusion is false, so the argument is invalid.