The conditional

One column of the previous lesson's table was handed over without justification, and it is the column almost every argument in the world runs through.

The conditional is the connective of rules, laws, predictions, promises, contracts and proofs. It is also the only one whose truth table offends people on first sight, because it counts "if the moon is made of cheese then Paris is in Spain" as true. This lesson shows that the table is forced rather than chosen, sets out honestly what it fails to capture, and then works through the practical consequences: necessary and sufficient conditions, converse and contrapositive, and the two fallacies that account for most bad reasoning about evidence.

The row nobody disputes and the two that follow

Start from what is not controversial. "If you pass the exam, you get the certificate" is broken by exactly one situation: you passed and got nothing. Passing and receiving is the promise kept. So $P \to Q$ must be false when $P$ is true and $Q$ false, and true when both are true. Those two rows are fixed by the meaning of "if" in any reading.

The dispute is over the two rows where $P$ is false. You did not pass. Was the promise kept or broken? The natural answer is that it was neither, that the question does not arise. But the language of the second lesson has no third value, and every connective in it is a function from the values of the parts to a value for the whole. Something must go in those rows.

There are only four ways to fill two rows, so enumerate them and see which survives. Filling both with false gives a connective true only when $P$ and $Q$ are both true, which is $P \land Q$ : asserting "if you pass you get the certificate" would then assert that you passed. Filling the false-true row with false and the false-false row with true gives the biconditional, so "if" would mean "if and only if", making "if it rains the match is cancelled" claim that nothing else could cancel it. Filling the false-true row with true and the false-false row with false gives a connective whose value is just $Q$ , so the antecedent would be idle and "if $P$ then $Q$ " would say no more than $Q$ .

Each of the three is plainly wrong. The remaining option, true in both rows where the antecedent is false, is the material conditional, and it is the only truth-functional candidate left standing. A conditional with a false antecedent is called vacuously true, and the name is a fair description: nothing has been claimed, so nothing has been broken.

What the material conditional gets wrong

Being the only candidate does not make it a good model of English "if", and pretending otherwise is how logic gets a reputation for sophistry. Three consequences are genuinely uncomfortable, and all three are tautologies you can verify in four rows.

First, $\neg P \to (P \to Q)$ : a false statement implies anything. Second, $Q \to (P \to Q)$ : a true statement is implied by anything. Third, $(P \to Q) \lor (Q \to P)$ : for any two statements whatever, one implies the other. Take "the reactor is scrammed" and "the price of tin is rising", and one of the two conditionals joining them is true, which no English speaker would accept.

The reason is that the material conditional records only a pattern of truth values, and English "if" claims a connection. Counterfactuals show the gap most sharply. "If this bridge had been built in steel, it would have survived" and "if this bridge had been built in steel, it would have collapsed anyway" have the same false antecedent, so both are vacuously true as material conditionals, while engineers can disagree about which is right for good reasons. Whole logics exist for that, and the honest position is that $\to$ is a deliberately crude tool: it is exactly right for the conditionals of mathematics and formal rules, and approximate for conditionals about causes, times and possibilities.

Necessary and sufficient

The vocabulary that goes with the conditional is worth getting exactly right, because it is used constantly in law, engineering and mathematics and confused just as often.

$P$ is a sufficient condition for $Q$ when $P \to Q$ : having $P$ is enough for $Q$ . $Q$ is a necessary condition for $P$ in the same situation: without $Q$ you cannot have $P$ . A single conditional therefore states both facts at once, and which word you reach for depends only on which end you look from.

Being a square is sufficient for being a rectangle, and being a rectangle is necessary for being a square. Passing the medical is necessary for a pilot's licence and nowhere near sufficient. When a condition is both necessary and sufficient, the biconditional holds, and that is what a definition asserts: an integer is even if and only if it is divisible by two, which is two conditionals in one symbol.

Example. "A valid contract requires consideration." Which is consideration, necessary or sufficient, and what conditional does the sentence assert?

"Requires" marks a necessary condition, so with $V$ for validity and $C$ for consideration the sentence is $V \to C$ : if the contract is valid, consideration was present. Equivalently, no consideration means no valid contract. It plainly does not say $C \to V$ , since a contract can have consideration and fail for a dozen other reasons, and a lawyer who read it that way would be badly wrong.

Now you. "Only members may use the pool." Express it as a conditional, and say which condition is which.

Answer

With $U$ for using the pool and $M$ for membership, it is $U \to M$ . Membership is necessary for use and not sufficient, since the pool can be closed. "Only" always introduces the consequent, which is the same rule as "only if" from the second lesson.

Converse, contrapositive and inverse

Three formulas can be built from $P \to Q$ by swapping and negating, and exactly one of them is equivalent to it.

The converse is $Q \to P$ , which is a different claim. The inverse is $\neg P \to \neg Q$ , also different, and in fact the converse of the converse. The contrapositive is $\neg Q \to \neg P$ , and it is equivalent to the original: check the four rows and the two columns match everywhere. The converse and the inverse are equivalent to each other and to neither of the other two.

The equivalence of a conditional with its contrapositive is a working tool rather than a curiosity. "If the sample contains chloride, the silver nitrate test goes cloudy" and "if the test stays clear, there is no chloride" are the same statement, and the second is often the easier one to use, since it is the one a negative test licenses. A whole proof technique is built on this and arrives in the thirteenth lesson.

The converse error is so common it has a name in every field. "All fraudulent accounts look smooth" does not give "all smooth accounts are fraudulent". "Every case of the disease shows this marker" does not give "everyone with the marker has the disease", and the gap between those two is the entire subject of diagnostic testing.

The two fallacies, seen in the table

The first lesson named affirming the consequent and denying the antecedent as invalid. The table now shows exactly why, and shows that they are the same mistake twice.

Affirming the consequent argues $P \to Q$ , $Q$ , therefore $P$ . Look for a row with both premises true and the conclusion false: $P$ false and $Q$ true does it, since a false antecedent makes the conditional vacuously true and the second premise is satisfied. One row is all it takes.

Denying the antecedent argues $P \to Q$ , $\neg P$ , therefore $\neg Q$ . The same row refutes it: $P$ false, $Q$ true makes both premises true and the conclusion false. The two fallacies are refuted by one valuation, which is a hint that they are the same error, namely reading $\to$ as $\leftrightarrow$ . Both would be valid if the conditional really were a biconditional, and people commit them because ordinary conversation often does mean a biconditional. "If you tidy your room you can have ice cream" is normally understood to promise no ice cream otherwise, which is precisely the reading logic refuses.

Example. A test for a condition is positive in every case of the condition. A patient tests positive. What follows?

With $C$ for having the condition and $T$ for a positive test, the premise is $C \to T$ and the fact is $T$ . That is affirming the consequent, and nothing follows about $C$ . What would follow from a negative test is $\neg C$ , by modus tollens, which is why a highly sensitive test is used to rule a condition out rather than in. Numbers make the point sharper: if the condition affects 1 in 1000 people and the test has a 5 per cent false positive rate, then among 100,000 people about 100 have the condition and test positive, while about 4995 of the rest test positive anyway, so a positive result is wrong roughly 98 times in 100.

Now you. "If the shipment cleared customs, the tracking page shows a customs event. The tracking page shows no customs event." What follows, and by which form?

Answer

That the shipment did not clear customs, by modus tollens: $C \to E$ with $\neg E$ gives $\neg C$ . This is valid, unlike the previous example, and the difference is only which of the two the premise denies.

Why this is hard for people

There is direct evidence about how badly untrained reasoners handle the conditional, and it is one of the most replicated results in psychology.

Peter Wason's selection task, from 1966, lays four cards on a table showing D, K, 3 and 7. Each card has a letter on one side and a number on the other. The rule is: if a card has a D on one side, it has a 3 on the other. Which cards must be turned over to find out whether the rule is broken?

The correct answer is D and 7. The rule is $D \to 3$ , and only one combination breaks it, a D with a non-3 on the back. Turning D can reveal exactly that. Turning 7 can too, since a D behind the 7 is a D without a 3. Turning 3 is useless, because whatever is behind it, no rule is broken: the rule never said only D cards carry 3s, and thinking otherwise is affirming the consequent. Turning K is useless, because the rule says nothing about non-D cards.

Fewer than one in ten participants get this right. Most turn D alone, or D and 3, choosing the card that could confirm the rule rather than the card that could break it. The interesting part is what fixes it: Griggs and Cox in 1982 posed the same task as a bar checking that anyone drinking beer is over 18, with cards reading "beer", "cola", "22" and "16". About three quarters of participants then chose correctly, which is "beer" and "16", the exact same logical form. The structure was never the problem; the abstraction was.

Example. In the drinking version, why is the "22" card the one that matches the useless "3" card?

The rule is: drinking beer implies over 18. The "22" card is a person over 18, so it satisfies the consequent, and whatever they are drinking the rule stands, since over-18s may drink anything. Turning it can produce no violation, which is exactly the position of the "3" card. Checking it is affirming the consequent dressed as diligence.

Now you. The rule is "if a package is marked fragile, it must be shipped by air". Four packages are on the bench: one marked fragile, one not marked, one already loaded on a plane, one already loaded on a truck. Which two must be checked?

Answer

The one marked fragile, to see how it is shipped, and the one on the truck, to see whether it is marked fragile. The unmarked package is unconstrained by the rule and the one on the plane cannot break it, whatever its marking.

Where this leaves the language

The conditional is now fully understood as a truth function, with its uses and its known distortions. The same four rows license modus ponens and modus tollens, refute their two impostors, and turn "necessary", "sufficient" and "only" into placements of an arrow.

The next question is what else the tables license. A conditional and its contrapositive have identical columns, which means either can replace the other anywhere without changing any truth value, and that is a far more powerful observation than it looks. If formulas with identical columns are interchangeable, logic can be done by substitution and calculation rather than by row-counting, and long formulas can be pushed into standard shapes. That is the next lesson.