Lecture: Conditionals and Deductive Forms
Lecture script · about 15–20 minutes spoken · Week 4
Opening
Picture a parking lot on campus with a sign at the entrance: "You may park in Lot C only if you display a permit." A student with a permit pulls in and thinks, "I have a permit, so I can park here." Then a second sign comes into view. The lot is closed after 10 p.m. for sweeping. The permit was needed. It was not enough.
Sentences like that sign run a lot of your life. A nursing student reads a dosing rule with two conditions in it. A business major reads a refund policy. A coder reads an if-then line in a program. In every case, knowing exactly what an "if" sentence promises keeps you from drawing a conclusion it never gave you.
In Week 1, you learned to find an argument. In Week 2, you wrote it in standard form. In Week 3, you learned validity and the counterexample method, and you met a soccer field that was wet because the sprinklers ran. In this lecture, I'll show you how to read an "if" sentence part by part, which patterns built on it always hold, and which two only look like they do. Then we'll turn a conditional around three ways, sort out necessary from sufficient, and check one verdict with a truth table. Next, in Week 5, the tests move to arguments about "all" and "some".
1. Reading the rule
Let's start in a restaurant kitchen. Above the fryer, a rule is taped to the wall: "If the oil light is red, then the oil is too hot to use." Notice what the rule does not say. It doesn't say the light is red right now. It says what follows whenever the light is red. That's the job of an "if" sentence. It links two claims without asserting either one.
Here's the term. A conditional is a claim of the form if P, then Q, which says the second claim is true whenever the first is; it is false in exactly one situation, when P is true and Q is false. The letters P and Q are sentence letters. Each one stands for a whole claim, so you can see the pattern without the subject getting in the way.
Each half has a name. The antecedent is the if-part, the claim P. The consequent is the then-part, the claim Q. In the fryer rule, "the oil light is red" is the antecedent, and "the oil is too hot to use" is the consequent.
So when is the fryer rule false? Only when the light is red and the oil is fine to use. A cold fryer with the light off breaks no promise. Neither does a hot fryer with a broken light. The rule only promised something about red-light moments.
Now the tricky phrase. "P only if Q" is rewritten as "if P, then Q" before anything is tested. Take the parking sign. "You may park in Lot C only if you display a permit" becomes "if you may park in Lot C, then you display a permit." Think of "only if" as naming a requirement, the way a bouncer checks for a wristband. The wristband is required to get in. It doesn't promise you a spot inside if the room is full.
One more thing about real arguments. People rarely say "therefore." Someone might say, "The field's wet. It rained overnight, didn't it?" That still has a premise and a conclusion. Find what the speaker wants you to accept, and what's offered for it, before you look for a pattern.
Pause and try this. In the parking sign, rewritten as "if you may park in Lot C, then you display a permit," which claim is the consequent?
Find the then-part of the rewritten sentence.
Here's the answer. The consequent is "you display a permit," because "only if" puts the claim after it into the then-part.
The if-part is the trigger; "only if" names the requirement.
2. Four patterns that never fail
Picture a friend texting before a morning game: "If it rained overnight, the soccer field is wet. It rained overnight." You don't need to see the field to finish that thought. The field is wet. And you didn't need to know anything about soccer. The pattern did the work.
The course's argument-form set holds six patterns, and four of them are valid. Here they are.
Modus ponens is the valid form if P, then Q; P; so Q, which moves from a conditional and its if-part to its then-part. That's the field argument you heard a moment ago.
Modus tollens is the valid form if P, then Q; not Q; so not P, which moves from a conditional and the denial of its then-part to the denial of its if-part. "If it rained overnight, the soccer field is wet. The soccer field is not wet. So it did not rain overnight." Had it rained, the field would be wet, and it isn't.
Hypothetical syllogism is the valid form if P, then Q; if Q, then R; so if P, then R, which chains two conditionals that share a middle claim. "If the bus is late, I miss the 8:00 a.m. class. If I miss the 8:00 a.m. class, I miss the quiz. So if the bus is late, I miss the quiz." It works like dominoes. Knock over the first, and the last one falls.
Disjunctive syllogism is the valid form P or Q; not P; so Q, which rules out one of two options to conclude the other. "Either the outlet is dead or the charger is broken. The outlet is not dead, because a lamp works in it. So the charger is broken."
How do you know these four are valid? You run the counterexample method from Week 3 and watch it fail. Try to imagine a morning when it rained overnight, rain wets the field, and the field is dry. You can't. No case gives these forms true premises and a false conclusion.
But be careful about what valid means. Suppose the charger is fine and the phone's own port is broken. Then "either the outlet is dead or the charger is broken" is false. The form is still valid. The argument is unsound, because a premise is false. Valid checks the link; true checks the lines.
Pause and try this. "If it rained overnight, the soccer field is wet. The soccer field is not wet. So it did not rain overnight." Which of the four valid forms is that?
Look at which part of the conditional the second premise denies.
Here's the answer. It's modus tollens, because the second premise denies the then-part and the conclusion denies the if-part.
Four forms never fail, as long as the lines are true.
3. Two look-alikes that fail
Here's the soccer field again, with one change. "If it rained overnight, the soccer field is wet. The soccer field is wet. So it rained overnight." Say it fast and it sounds like modus ponens. It isn't. The second premise affirms the then-part instead of the if-part, and that one swap breaks the pattern.
A formal fallacy is an argument form that is invalid, so that some argument with that pattern has true premises and a false conclusion whatever its subject matter. The course names two.
Affirming the consequent is the invalid form if P, then Q; Q; so P, which treats the then-part being true as proof that the if-part is true. That's the field argument I read a moment ago.
Denying the antecedent is the invalid form if P, then Q; not P; so not Q, which treats the if-part being false as proof that the then-part is false. "If it rained overnight, the soccer field is wet. It did not rain overnight. So the soccer field is not wet."
One case knocks out both. The sprinklers ran overnight and it did not rain. Check affirming the consequent on that morning. "If it rained, the field is wet" is still true. "The field is wet" is true. But "it rained overnight" is false. Now check denying the antecedent on the same morning. "If it rained, the field is wet" is true. "It did not rain" is true. But "the field is not wet" is false. True premises, false conclusion, twice over.
So count the set. Four valid forms plus two invalid forms makes six forms.
Here's a story that shows why the fault is in the pattern, not the topic. A gym runs an ad: if you train with us, you'll get stronger. Your neighbor got stronger this year, so your neighbor must have trained with them. But maybe your neighbor lifted weights at home all year. Same pattern as the field, same verdict.
Now the mix-up almost everyone makes. People think denying the antecedent is the same as modus tollens, because both have a "not" in them. Look at what gets denied. Modus tollens denies the then-part, the wet field, and that's valid. Denying the antecedent denies the if-part, the rain, and the sprinklers refute it.
And one honest note. Real speakers sometimes say "if" when they mean "if and only if." A manager might say, "If you cover the weekend shift, you can leave early on the next workday," and mean that's the only way to leave early. Reading a speaker charitably is a judgment about meaning. The form you test is the one the words state.
Pause and try this. On the sprinkler morning, which premise of affirming the consequent is true even though it never rained?
Picture the field at dawn with the sprinklers shutting off.
Here's the answer. "The soccer field is wet" is true, because the sprinklers wet it, while the conclusion "it rained overnight" is false.
A true then-part does not prove the if-part.
4. Turning it around, and reading conditions
Picture the front desk of a clinic with a rule: "If a visitor has a fever, the visitor waits in the side room." Flip it: "If a visitor waits in the side room, the visitor has a fever." It sounds similar, but it says something new. Someone could be in the side room because the main room was full.
There are three ways to turn a conditional around. The converse swaps the two parts, so "if P, then Q" becomes "if Q, then P". The inverse denies both parts, so it becomes "if not P, then not Q". The contrapositive swaps and denies both parts, so it becomes "if not Q, then not P". Only the contrapositive is equivalent to the original. Equivalent means two claims cannot differ in truth value, so in every possible case both are true or both are false.
Run all three on the field. The converse says, "If the soccer field is wet, it rained overnight." The sprinkler morning makes that false. The inverse says, "If it did not rain overnight, the soccer field is not wet." The same morning makes that false too. The contrapositive says, "If the soccer field is not wet, it did not rain overnight." That one holds whenever the original holds, because a dry field means no rain fell on it.
Turning a rule around also tells you what it requires and what it guarantees. A sufficient condition is a condition whose truth is enough to bring another with it; in if P, then Q, the if-part P is sufficient for Q. A necessary condition is a condition that must hold for another to hold, though it may not be enough on its own; in if P, then Q, the then-part Q is necessary for P.
Here's the example the course uses. "You can register for the course only if you passed the prerequisite." Rewrite it: if you can register, you passed the prerequisite. Passing the prerequisite is necessary for registering. Nobody registers without it. But it's not sufficient. You also need an open seat. The parking sign works the same way. The permit is necessary, and the sweeping hours show it isn't sufficient.
Think of a necessary condition like a ticket to a concert. You can't get in without one. But a ticket for a canceled show gets you nowhere.
Pause and try this. Is "if the soccer field is wet, it rained overnight" the converse, the inverse, or the contrapositive of the original?
Check whether the parts were swapped, denied, or both.
Here's the answer. It's the converse, because the two parts were swapped and neither was denied, and the sprinklers make it false.
Flip and deny both parts, or you've changed the claim.
5. One truth table, as a check
You've already settled every verdict in this lecture with a counterexample. A truth table is a second way to look at one of them, the way you might check a long sum by adding it again from the bottom. It shows up once, as a check, and it's never something you're graded on building.
A truth table is a small table that lists every combination of true and false for the sentence letters and shows whether a claim built from them is true or false in each row. Let me describe the one for "if P, then Q" in words. There are 2 sentence letters, and each can be true or false, so 2 times 2 gives 4 rows.
Row one: P true, Q true, and the conditional is true. Row two: P true, Q false, and the conditional is false. Row three: P false, Q true, and the conditional is true. Row four: P false, Q false, and the conditional is true. So the conditional is false in 1 row, and 4 minus 1 leaves 3 rows where it's true.
Now look at row three. P is false and Q is true. That's the sprinkler morning: "it rained overnight" is false, and "the field is wet" is true. In that row, both premises of affirming the consequent are true, "if P, then Q" and "Q". Its conclusion, "P", is false. That's a counterexample row. The table confirms the verdict the sprinklers already gave us, without adding a new one.
People often think rows three and four "should" be false, since nothing in them confirms the rule. But think back to the fryer. A cold fryer with the light off doesn't break the rule. A conditional only promises something when its if-part is true.
Pause and try this. In the truth table for "if P, then Q", which single row makes the conditional false?
Remember the one situation the fryer rule couldn't survive.
Here's the answer. Row two, where P is true and Q is false, because that's the only case where the if-part holds and the then-part fails.
One false row: the if-part true and the then-part false.
Closing
Let me pull it together. A conditional, "if P, then Q", has an antecedent and a consequent, and it's false only when P is true and Q is false; "only if" gets rewritten before you test it. Modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism are valid, because no case gives them true premises and a false conclusion. Affirming the consequent and denying the antecedent are formal fallacies, and one sprinkler morning refutes both. Only the contrapositive is equivalent to the original, and in "if P, then Q", P is sufficient and Q is necessary. And the truth table's one false row confirms what the counterexample already showed.
Next in this module come the slide deck, the Readings and Resources page, the lecture tutorial, the practice set, and the quiz. Then, in Week 5, Categorical Reasoning, the Venn diagram becomes your test for arguments built on "all" and "some".
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