Formal Logic

Mike McGarry
Lesson by Mike McGarry
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Formal logic, this lesson will introduce the ideas of formal logic, which sometimes appear on the ACT. The test may give a statement and ask us what other statement must be true. Other times the test may give us a few statements and ask us to infer what additional statement must be true. These statements are typically conditional statements.

Conditional statements are if-then statements and their logical equivalents. All of these are conditional statements, and all of these are true. If an animal is a cow, then the animal is a mammal, perfectly true. If a shape is a square, then it is a rectangle, perfectly true. If a plant is a potato plant, then it is a nightshade, perfectly true. If a piece of music is a piano trio, then it is a chamber-music composition, perfectly true.

All if-then statements can be rewritten as logically equivalent statements involving the words all or every. So for example, every cow is a mammal, all squares are rectangles. Every potato plant is a nightshade, all piano trios are chamber-music compositions. Notice that it's a little sleeker and briefer if we write it in terms of all or every.

Now the ACT will ask you to recognize that statements of this form still count as conditional statements. So you should be able to change these back into if then form and work with them in that form if need be. Notice that, as a general rule, we cannot reverse a true conditional statement to produce another true statement.

It's true that every cow is a mammal, but not every mammal is a cow. For example, there are cats and dogs and horses and pigs and even humans. We're all mammals. We're not cows. So there are many types of mammals besides cows. So it's not true that every mammal is a cow.

Similarly if a shape is a square, then it is a rectangle, 100% true. If a shape is a rectangle, then it is a square. That's a false statement. It's easy to understand this with ordinary facts, but it's important to understand that this logical relationship, that they all work even when the facts are not familiar.

So sometimes the ACT will give us kind of obscure pieces of information. Sometimes they're just going to give us fictional information. And we're going to work have to work with the fictional information. We have to trust the pure logical relationships. For example, if we're told that every baryon is a hadron, we may not know what either of those things are.

They're actually technical terms from physics. But the ACT would expect us to recognize that if the above statement is true, they could say assume that it's true, that every baryon is a hadron. They would expect us to recognize that it is not equivalent to the statement every hadron is a baryon. That even if the former statement is true, this latter statement is likely not true.

We have to be able to recognize the patterns among these statements even if we are not familiar with the specific content. And as I said, the content will sometimes be entirely fictional and imaginary anyway. If the original true statement is of the form, if P, then Q, then we can't merely switch the places of P and Q to get another true statement.

Nevertheless, we can always make a logically equivalent statement if we switch the places of P and Q and add not to each. If not Q, then not P. This is known as the contrapositive. You don't need to know that word, but you need to know this idea. It's a very important idea.

If the original statement is true, then its contrapositive is true as well. It may help to change an every statement or an all statement back to an if then statement form, for the purposes of constructing the contrapositive. So if we're told every potato plant is a nightshade, then it's better to write that as an original if-then statement. If a plant is a potato plant, then it is a nightshade.

Then to construct the contrapositive, we switch the order and add not to both sides. If a plant is not a nightshade, then it is not a potato plant. The ACT will often give you a conditional statement in some form and expect you to construct its contrapositive, very important. If we are given a statement of the form if P, then Q, and we're also told that condition P is true, then we can conclude that Q must be true.

That basic form of deduction is called a syllogism. Similarly, if we are given a statement of the form if P then Q, and we are told the condition not Q is also true. Well remember, if P then Q is equivalent to its contrapositive, if not Q then not P. That would allow us that conclude that not P also must be true.

Suppose we are given the true statement, if a car runs, then it has gas. Perfectly, common sense, logical statement. If the test also tells us that Elizabeth's car is running, then we can conclude that Elizabeth's car has gas. Alternately, if the test tells us that Richard's car has no gas right now, then we can conclude that Richard's car cannot run, at least at the moment.

At the very least we'd have to add gas, that may or may not solve the problem. Again, this conditional statement refers to common sense information, be we should be able to make the same deductions when the material is not familiar. Here's a practice problem. Pause the video, and then we'll talk about this. Okay, so they give us a conditional statement.

If a bird is a blue-tipped puffer, then it prefers the seeds of the Thomson's Mulberry tree. Incidentally, this is entirely fictional. There are no such birds out there as a blue-tipped puffer. It's totally made up. So don't worry if you never learned about this in biology.

It doesn't exist. But okay, we're gonna assume that that's a true statement. If the bird is a blue-tipped puffer, then it prefers these seeds. Well, okay, if it is one of those birds, it prefers those seeds. If it's not one of those birds, well who knows, maybe it prefers the seeds, maybe it doesn't.

Maybe there are other birds that also like those seeds, we don't know. So we certainly just can't switch it around. If a bird prefers the seeds of the Thomson Mulberry tree, then it is a blue-tipped puffer. That's the solution that A makes, simply switching the order from the p and the q, going if q then p.

And that is not correct. What we need to do is construct the contrapositive. So that means switch the order and put nots in there. If a bird does not prefer the seeds of the Thomson Mulberry tree, then it is definitely not a blue-tipped puffer. And that's exactly what E says.

E is the correct contrapositive. Here's another practice problem. Pause the video and then we'll talk about this. Consider the 3 statements below to be true. All students who have taken the ACT go to college. Well, technically that's probably not true, but we're going to assume for the purposes of this problem that it is 100% true.

All students who take the ACT go to college. L has not taken the ACT. M is not going to college. Well, which of the following can be true? Well, let's think about this. We know that the students who take the ACT go to college.

Well, are there ways to go to college without taking the ACT? Well in fact we know there's another test out there, you may have heard of it, called the SAT. Some students can take the SAT and go to college. So let's think about L. L has not taken the ACT.

Well, it could be that L is someone who is not college bound. So they haven't taken the ACT, they're not going to college at all. Or it could be that L is someone who has taken the SAT and is going to college. So we can't make any clear deduction about L. But think about M. M is not going to college.

Well, let's pause a minute. Let's go back to that first statement, all students who have taken the ACT go to college. Think about it in if then form. If a student has taken the ACT, then that student goes to college. And the contrapositive, well, if a student is not going to college, then that student has not taken the ACT.

Well, M is not going to college. So we can conclude that M has not taken the ACT. In summary, a conditional states is of the form, if P, then Q. These statements can also appear in the form every P is Q, or all P are Q. If a conditional statement is true, then the contrapositive, if not Q, then not P must also be true.

So it's very important to be able to change a statement to if then form and to construct the contrapositive.

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