1
A promise is broken in one way
“If it rains, the match is off” is broken when it rains and the match goes ahead. If it does not rain, nothing has been violated whether the match goes ahead or not. So the conditional is true in those rows.
Lesson · Arguments and Validity · Propositional logic
The Conditional
A conditional is false in exactly one row: the antecedent true and the consequent false. The other three rows are true, including both where the antecedent is false, and that is what a promise requires. The converse, which swaps the parts, is a different sentence.
p → q is false in one row only
p true, q false: the promise is broken
p → q ≡ ¬p ∨ q
p only if q: p → q
p if q: q → p
p unless q: ¬q → p
The conditional, and what to do with it
Core ideas
What this lesson establishes
If-then is the connective arguments are built from, and the one that feels wrong in the table. The feeling is worth examining.
2
The antecedent is the condition
In p → q, p is the antecedent and q the consequent. The conditional says nothing about what happens when the antecedent is false, and a sentence that says nothing there cannot be false there.
3
The direction matters
p → q and q → p are different sentences. The second is the converse, and the two disagree in the two rows where p and q have different values.
The material
The table, and the words that point to it
One table, then the English phrases that translate into it and the ones that translate into its converse.
Definition · The conditional
p → q (“if p then q”) is false exactly when p is true and q is false, and true in the other three rows. It is equivalent to ¬p ∨ q.
p q | p → q q → p ¬p ∨ q
-----------------------------------
T T | T T T
T F | F T F
F T | T F T
F F | T T T
The last column shows why the third and fourth rows, where p is false, are T. ¬p ∨ q is false only when p is true and q is false, and the conditional was introduced to say exactly that: it is ruled out that p holds without q. A sentence that rules out one case and is silent about the rest is true in the rest.
An independent check on the two rows where p is false: take the claim that every number divisible by four is even. As a conditional about a number n, it reads: if n is divisible by four, then n is even. For n = 6 the antecedent is false and the consequent true; for n = 3 both are false. The claim is plainly true of all numbers, so it cannot be false at 6 or 3. If the conditional were false, or had no value, whenever its antecedent is false, every such claim would fail.
Translation is mostly a matter of which part is which. These phrases all say p → q:
- if p, then q; q if p
- p only if q
- p is sufficient for q
- q is necessary for p
“If” and “only if” run opposite ways. “You may vote only if you are registered” says that voting requires registration, so v → r. “You may vote if you are registered” says that registration is enough, so r → v. “p unless q” says that if q fails then p, so ¬q → p.
In the lab the first formula is the conditional, compared with its form using or. The columns agree. Change B to the converse and the lab marks the two rows that differ: p true with q false, and p false with q true. A conditional and its converse disagree in two rows, not one. The menu's last two formulas, which negate both parts, are taken up in “Equivalence, De Morgan and Contraposition”; for now, only count the rows in which each differs from A.
Interactive
Truth table
Every assignment, evaluated
Method
Translating an if-then sentence
Decide which part is the condition before you decide the word.
1
Name the two parts
Give each simple sentence a letter: p for the part that would have to hold, q for the part that is promised.
2
Find the part that is required
Ask which part the sentence says cannot fail while the other holds. That part is the consequent; the other part is the antecedent.
3
Choose the form from the word
“If” introduces the antecedent. “Only if” introduces the consequent. “Sufficient” goes with the antecedent, “necessary” with the consequent.
4
Check the one false row
A conditional is false only when the antecedent holds and the consequent fails. Describe that case in English and see whether it is the case the sentence forbids.
Rows and translations
Score 0 / 0Choose the answer you can justify.
The match is off
Let r be “it rains” and m be “the match is off”. The promise: if it rains, the match is off.
rain off | r → m what happened
T T | T rain, match off: kept
T F | F rain, match on: broken
F T | T no rain, match off: kept
F F | T no rain, match on: kept
The third row is the one that looks odd. It did not rain and the match was called off anyway, perhaps because a ground was flooded. The promise was about what follows rain; it did not say the match would be on whenever it was dry. It was kept, so the sentence is true there.
Common mistakes
1
Thinking a conditional with a false antecedent is false, or has no valueThe table says T in both rows where the antecedent is false. Check it against a promise: if it does not rain, “if it rains the match is off” has not been broken, whatever happens to the match. And a general claim such as “if a number is divisible by four it is even” stays true at the numbers 3 and 6, where its antecedent is false.
2
Reversing “only if”“p only if q” is p → q, not q → p. “Only if” names the requirement. Voting only if registered makes registration necessary, and does not make it enough.
3
Taking the converse to say the same thingIf a shape is a square it has four sides; if it has four sides it is a square. The first is true and the second false. The lab shows two rows where the two conditionals disagree, and in each of them one is true and the other false.
Check your understanding
✓
Finish when you can fill the table and translate without guessing.
Fill in the four rows of p → q from memory, translate a sentence using only if, unless, necessary or sufficient, and give the row where a conditional and its converse have different values.
Note: This is the conditional of logic. Ordinary if-then sometimes means more (a causal link, a counterfactual), and Science, Induction and Causation returns to that. Nothing in this course needs more than the table.