Lecture: Deductive and Inductive Reasoning
Lecture script · about 15–20 minutes spoken · Week 3
Opening
Picture a soccer team's group chat on the morning of a match. One teammate writes, "It rained overnight, so the field is wet. Bring your long studs." A few minutes later, a weather app chimes in: rain is likely again by noon. Both messages give reasons. But they promise very different things.
That difference matters well beyond soccer. A nursing student weighs what a lab result guarantees against what it only suggests. A business major reads a sales forecast. A fan judges a sports prediction. In every case, the first question is whether the reasons are meant to prove the conclusion or only to make it likely, because that tells you which test to run.
In Week 1, you learned to find an argument. In Week 2, you wrote it in standard form, supplied its missing premise, and drew its map. In this lecture, we start judging those maps. I'll show you how to sort deductive from inductive arguments, how to test a deductive argument for validity and soundness, how to rate an inductive argument's strength, and how to break a bad pattern with one counterexample. Then we'll run every test on the running question. Next, in Week 4, these patterns get their names.
1. Deductive versus inductive
Let's start with two friends on a road trip. One says, "Every gas station on this highway takes cards. The next stop is a gas station on this highway. So it takes cards." The other says, "The last four stations took cards, so the next one probably will too." Both are arguments. But the first is trying to settle the question, and the second is only placing a good bet.
Think of it like the difference between a key and a weather vane. A key is supposed to open the lock every time. A weather vane only tells you which way things are likely heading.
Here are the terms. A deductive argument is an argument whose premises are meant to guarantee its conclusion, so that if the premises were true, the conclusion could not be false. An inductive argument is an argument whose premises are meant to make its conclusion probable rather than to guarantee it, so that true premises still leave some chance the conclusion is false.
Notice the phrase "meant to." You sort an argument by the promise it makes, not by whether it keeps that promise. A deductive argument can fail. It's still deductive, because a guarantee is what it was going for.
Now the example this week keeps coming back to. "If it rained overnight, the soccer field is wet. It rained overnight. So the soccer field is wet." If both premises are true, there's no room for a dry field. That's deductive. Compare a forecast: "The weather service has called most rainy mornings this month correctly, and it says rain is likely by noon. So it will probably rain by noon." That word "probably" admits a miss. That's inductive.
Hedge words like "probably" and "likely" often signal induction. Words like "must" and "it follows that" often signal deduction. They're clues, not proof, the same way indicator words were clues in Week 1.
And here's an honest admission. Philosophers disagree about how best to draw this line, and some everyday passages sit right on it. So in this course, you're graded on the test you apply, not on the label you pick for a borderline case. A borderline passage is something to discuss.
Pause and try this. Is the gas-station friend who says "the next one probably will too" making a deductive or an inductive argument?
Find the hedge word in what the friend said.
Here's the answer. It's inductive, because the word "probably" shows the premises are meant only to make the conclusion likely, not to guarantee it.
Ask what the premises promise: a guarantee, or a good chance.
2. Validity and soundness
Picture a rule taped to a restaurant's walk-in cooler. "Everything on the top shelf is for tonight's dinner service. This tray of salmon is on the top shelf. So this tray is for tonight's dinner service." Before anyone opens the cooler, you can already see something. If both premises were true, there'd be no way for the conclusion to be false.
That's the first test for any deductive argument. An argument is valid when it is impossible for all its premises to be true and its conclusion false; validity depends on the argument's form, not on whether the premises are in fact true. And an argument is invalid when at least one possible case exists in which all its premises are true and its conclusion is false.
Validity checks the link. It doesn't check the lines. For the lines, you need a second standard. An argument is sound when it is valid and every premise is true; a sound argument's conclusion must be true.
Take the field argument on two different mornings. On a morning when it rained, both premises are true. The argument is valid with true premises, so it's sound. On a morning when it didn't rain, the premise "it rained overnight" is false. The link hasn't changed, so the argument is still valid. But it's not sound anymore, and it no longer settles whether the field is wet.
Here's where people get tripped up. A valid argument can have a false conclusion. A transit rider says, "Every bus on this route is electric. The early bus runs on this route. So the early bus is electric." Suppose some buses on that route actually run on diesel, and the early bus is one of them. The conclusion is false. But the link is fine. If every bus really were electric, the early bus would have to be. The trouble is a false premise.
It also runs the other way. An invalid argument can have a true conclusion. A shopper says, "This grocery store sells fresh bread. So it's open on weekends." Maybe it is open on weekends. But selling bread guarantees nothing about the hours. The conclusion is true by luck.
So when someone outside this course says "that's a valid point," they usually mean "that's right." In here, valid means only that true premises would force the conclusion. Sound and valid aren't synonyms either. Every sound argument is valid, but a valid argument is sound only when all its premises are true.
Pause and try this. The bus argument has a false conclusion. Is it invalid, unsound, or both?
Check the link first, then check the lines.
Here's the answer. It's valid but unsound, because true premises would force the conclusion, and the false premise about every bus is what let the conclusion go wrong.
Valid checks the link; sound checks the link and the lines.
3. Strength and cogency
Inductive arguments need their own test, because they never promised a guarantee. Calling a weather forecast "invalid" misses the point. The real question is how much the premises raise the chance of the conclusion.
An inductive argument is strong when its premises, if true, would make its conclusion likely; strength comes in degrees, and an argument that falls short is weak. A cogent argument is a strong inductive argument whose premises are all true; even a cogent argument can have a false conclusion.
The phrase to hold onto is "in degrees." Validity works like a light switch: on or off. Strength works like a dimmer. New information can turn it up or down.
Let me tell you a short story. A resident of an apartment building comes home, and her package isn't at her door, even though the tracker says "delivered." She thinks it was left at the wrong door. With only that one fact, her argument is weak. The package could have been stolen, or held in the mailroom, or marked delivered by mistake. Then she checks the mailroom, and there's nothing for her. Now the argument is stronger. Then she remembers that last month her neighbor upstairs, in the unit with the same number, got a box with her name on it. Now a wrong-door delivery is the best explanation, and the argument is fairly strong.
If all three of her premises are true, the argument is cogent. And it still doesn't guarantee anything. This one time, the package could have been taken from the right door.
Here's another from sports radio. A host says the home team has won 9 of its last 10 home games, so it will probably win at home this weekend. That's 90 percent of recent home games. If the record is accurate, the argument is fairly strong and cogent. But one injury to a key player, which the host never mentioned, could weaken it at once.
So be careful about two mix-ups. One is treating strength as all-or-nothing, as if an inductive argument is either proved or worthless. The other is reading "cogent" as a fancy word for "valid." Cogent belongs to inductive arguments, and a cogent argument can still miss. A sound argument never can.
Pause and try this. After the resident checks the mailroom, does her argument become valid, or only stronger?
Ask whether any true premise could still sit beside a false conclusion.
Here's the answer. Only stronger, because even with the mailroom checked, her premises could all be true while the package was stolen from the right door.
Deductive arguments pass or fail; inductive arguments come in degrees.
4. The counterexample method
Suppose someone hands you a deductive argument and you suspect the link is broken. How do you show it? You don't have to prove the conclusion false. You need one case, real or imagined, where the premises are true and the conclusion is false.
The method works on the pattern. An argument form is the pattern of an argument's reasoning once its particular subject matter is set aside, so that two arguments about different things can share one pattern. A counterexample is a case, real or imagined, with the same form as an argument in which every premise is true and the conclusion is false. And the counterexample method is the test that shows an argument form is invalid by describing one case with the same form in which every premise is true and the conclusion is false.
Here's the field argument run in reverse. "If it rained overnight, the soccer field is wet. The soccer field is wet. So it rained overnight." It sounds fine. But can the field be wet when it didn't rain? Yes. The sprinklers ran overnight and it did not rain. In that case, "if it rained, the field is wet" is still true. "The field is wet" is true. And "it rained overnight" is false. One case like that is enough. The reversed argument is invalid, even though on many mornings its conclusion happens to be true.
Think of it like a stress test on a bridge design. One collapse under ordinary weight shows the design is flawed, no matter how many sunny days it stood.
Because the method targets the form, it works on any argument with the same pattern. A phone owner says, "If my battery is dead, the screen stays dark. The screen stays dark. So the battery is dead." The counterexample: the screen is cracked and broken while the battery is fully charged. Same pattern, same verdict.
Now try the original field argument instead. Can you describe a morning when it rained, rain wets the field, and the field is dry? You can't. No counterexample exists, because that form is valid.
One warning. A single example with true premises and a true conclusion proves nothing about validity. A valid argument must hold in every possible case. And the method only refutes deductive arguments. An inductive argument already admits it could miss.
Pause and try this. For the reversed field argument, why does the sprinkler case count as a counterexample?
Check each premise and the conclusion in that case, one at a time.
Here's the answer. Both premises stay true while the conclusion, that it rained overnight, is false, so the argument's form cannot guarantee its conclusion.
One true-premises, false-conclusion case breaks the whole pattern.
5. The running question, link by link
Now let's run every test on the running question, the illustrative campus-newsletter argument about keeping the library open around the clock during finals week. The course takes no side.
Here's the standard form from Week 2. P1: during finals week many students need a quiet, safe place to study late at night. P2: the library is the only quiet, safe, well-lit study space on campus large enough to meet that need. P3, which we supplied: the campus should provide study space students need during finals when it can do so at modest cost. P4: keeping the library open overnight for the five nights of finals week costs about 1,600 dollars in staff wages. And C: the campus library should stay open around the clock during finals week.
The map has two support links. P1, P2, and P3 work together to support C. And P4 supports P3's "modest cost" condition.
Start with the first link as it stands. Could P1, P2, and P3 all be true while C is false? They could. If keeping the library open overnight didn't cost a modest amount, P3 wouldn't require it. So the link isn't valid as it stands.
Now grant one more claim, the cost condition: the claim that keeping the library open overnight during finals week costs a modest amount, which the supplied premise needs and the cost premise is offered to support. None of P1, P2, or P3 says it. But with all of them true, plus the cost condition, C can't be false. So the first link is deductive, and valid once the cost is granted to be modest.
Now the figure behind P4. Two staff times eight hours a night is 16 staff hours a night. Then 16 times five nights is 80 staff hours. Then 80 hours times 20 dollars an hour is 1,600 dollars. Spread across five nights, that's 320 dollars a night.
So test the second link. Could P4 be true while "the cost is modest" is false? It could. A campus that has already spent its whole study-space budget might not count 1,600 dollars as modest. No figure guarantees "modest" on its own. P4 makes the cost condition more or less likely, depending on what else the campus spends. That makes the second link inductive, with strength in degrees.
Count them up. There are 2 support links: 1 deductive and 1 inductive.
So here's the verdict in words. The step from P1, P2, and P3 to C is deductive and valid once the cost is granted to be modest. The step from P4 to "modest" is inductive. So the whole argument can be no better than cogent, and the place to look hardest is that inductive step and the truth of P3.
Notice what we didn't decide. We didn't decide whether 320 dollars a night is a lot, or whether the library should stay open. A supporter and an opponent should find the same two links. Testing a link isn't taking a side.
Pause and try this. In the running question's map, which link is inductive, and why can't it be a guarantee?
Think about whether any dollar figure could prove the word "modest."
Here's the answer. The link from P4 to the cost condition is inductive, because 1,600 dollars could be true while a campus with a spent budget still finds it too much.
One link guarantees once the cost is granted; the other only makes "modest" likely.
Closing
Let me pull it together. A deductive argument is meant to guarantee its conclusion, and an inductive argument is meant to make it probable, so you sort by the promise. Valid means true premises could never come with a false conclusion, and sound means valid with every premise true. Strong means true premises would make the conclusion likely, in degrees, and cogent means strong with true premises. The counterexample method breaks an invalid pattern with one case of true premises and a false conclusion, like the sprinklers. And the running question now has one deductive link and one inductive link.
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 4, Conditionals and Deductive Forms, the patterns you tested here get their names, and the counterexample method becomes your test for each one.
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