A statement is a sentence that can be false
In logic, a statement (or proposition) is a declarative sentence that is either true or false, but never both and never neither. "Lahore is in Pakistan" is a statement, and it is true. "7 is an even number" is a statement, and it is false — being false does not stop something from being a statement.
Questions, commands and opinions are not statements. "What time is it?", "Close the door" and "Mathematics is beautiful" cannot be assigned a truth value, so logic has nothing to say about them.
Statements are labelled p, q, r, and their truth value is written T or F. Everything else in this chapter is built from combining them.
| Name | Symbol | Read as | True when… |
|---|---|---|---|
| Negation | ~p | not p | p is false |
| Conjunction | p ∧ q | p and q | both are true |
| Disjunction | p ∨ q | p or q | at least one is true |
| Conditional | p → q | if p then q | except when p is true and q is false |
| Biconditional | p ↔ q | p if and only if q | both have the same truth value |
Truth tables
A truth table lists every possible combination of truth values for the simple statements and works out the compound statement for each. With two statements there are 2² = 4 rows; with three there are 2³ = 8. Set the rows out in the same order every time — TT, TF, FT, FF — and you will never miss one.
Four of the five connectives are exactly what everyday language would suggest. The conditional is the one that surprises people.
Look at p → q. Only the second row is false. When the hypothesis p is false, the implication counts as true whatever q does — the highlighted row is the only way to break a conditional.
Why a false hypothesis makes a true implication
Suppose a teacher says: "If you score above 90, I will give you a prize." When has that promise been broken? Only in one situation — you scored above 90 and got nothing. If you scored 60 and got no prize, the promise was not broken; it simply never came into play. If you scored 60 and got a prize anyway, the promise was still not broken.
So p → q is false in exactly one case: p true, q false. In every other case it is true. This is called vacuous truth when p is false, and it is the single most tested idea in the chapter.
The conditional is not the same as "causes"
In logic, p → q makes no claim that p brings q about. "If 2 + 2 = 5 then the moon is made of cheese" is a true statement, because the hypothesis is false. Everyday language rarely uses "if… then" this way, which is exactly why the exam asks about it.
Converse, inverse and contrapositive
Given a conditional p → q, three related statements can be built from it. They are easy to write and easy to confuse, so learn them by what happens to the two halves.
| Name | Form | From "If it is raining, the ground is wet" |
|---|---|---|
| Conditional | p → q | If it is raining, the ground is wet |
| Converse | q → p | If the ground is wet, it is raining (swap) |
| Inverse | ~p → ~q | If it is not raining, the ground is not wet (negate) |
| Contrapositive | ~q → ~p | If the ground is not wet, it is not raining (swap and negate) |
Read the rain example again
The original is clearly true. The converse is not — the ground could be wet because somebody washed a car. The contrapositive is true, and obviously so: dry ground means it cannot have been raining. That pattern holds in general, which is why mathematicians so often prove the contrapositive instead of the original statement.
Tautologies, and the two kinds of reasoning
A compound statement whose truth-table column is all T is called a tautology — it is true no matter what the parts are. One whose column is all F is a contradiction. Anything else is a contingency. Two statements are logically equivalent when their columns match row for row.
The chapter closes with the distinction between the two ways of arguing to a conclusion.
- Deductive reasoning goes from general to particular. "All squares have four equal sides; ABCD is a square; therefore ABCD has four equal sides." If the premises are true, the conclusion is guaranteed. This is what mathematical proof is made of.
- Inductive reasoning goes from particular to general. "3, 5, 7 and 11 are odd primes, so all primes are odd." The conclusion is only probable, and this one is false — 2 is prime and even. Induction suggests results; deduction proves them.
Show that p ∨ ~p is a tautology.
- There is only one simple statement, so the table has 2¹ = 2 rows: p = T and p = F.The number of rows is 2 raised to the number of distinct statements.
- Row 1: p = T, so ~p = F, and
T ∨ F = T.A disjunction needs only one true part. - Row 2: p = F, so ~p = T, and
F ∨ T = T. - Every entry in the final column is T, so the statement is a tautology.This particular one is the law of the excluded middle: either something is so, or it is not.
Both rows give T, so p ∨ ~p is a tautology.
Before you leave this chapter
- A statement must be capable of being true or false. Questions and commands are not statements.
- n simple statements give 2ⁿ rows in the truth table.
- p → q is false in one case only: p true and q false.
- Contrapositive ≡ original. Converse ≡ inverse. The converse is NOT equivalent to the original.
- Deduction proves; induction only suggests. A single counterexample destroys an inductive conclusion.