We’ve covered how to deal with the simplest formal logic statement: If X, then Y. But what happens when our necessary or sufficient factors become more complicated? Let’s look at a couple of examples, using the idea of a vegetable salad. The simplest statement and its contrapositive might look like this:
If the stir-fry has carrots, then it has peapods.
If the stir-fry has no peapods, then it has no carrots.
Now let’s add more vegetables (and more complicated logic):
If the stir-fry has carrots or spinach, then it has peapods and peppers.
Here’s an important idea: when you are forming a contrapositive, you already know that the necessary and sufficient factors are switched around and negated. But now you also have to remember that “and” becomes “or,” and vice versa. So the statement above becomes:
If the stir-fry has no peapods or no peppers, then it has no carrots and no spinach.
It can be extremely helpful to individually negate each element of the statement; otherwise, it’s easy to get confused. For instance, if you only negate the first part of the statement above and tell someone, “If the stir-fry has no peapods or peppers…” they might interpret that as meaning that neither of those vegetables should be in the stir-fry. But in formal logic terms, it would technically mean that you either want peppers or no carrots. Neither of those ideas, though, is what you mean to say in the contrapositive; the intended meaning is that I want no carrots or no peppers.
Advanced Formal Logic Issues
Neither and Nor
The pairing of “neither” and “nor” can also cause some consternation. The easiest way to deal with that is to remember that “neither X nor Y” is the same thing as “no X and no Y.” The example above can be rephrased as follows:
If the stirfry has no carrots or no peppers, then it has neither peapods nor spinach.
So if you need to negate a “neither/nor” statement, the “nor” becomes “or” just as it would if the statement said “and.”
What if the sentence isn’t written in the order in which we expect to find the elements?
For instance, how do we interpret a sentence that says:
The burger has mustard if it has onions.
Here we can take the word “if” and read the statement that follows it as the sufficient element. We can turn that sentence into this:
If the burger has onions, then it has mustard.
A final issue is the phrase “only if.” Let’s go back to our vegetable stir-fry, and look at the following sentence:
The stir-fry has broccoli only if it has mushrooms.
Here, you can’t interpret “if” as signaling the sufficient element. “Only if” statements are interpreted differently than regular “if” statements. The “only if” statement above means the same thing as this:
If the salad has broccoli, then it has mushrooms.
- When forming a contrapositive, turn “and” into “or” and vice versa.
- “Neither X nor Y” means “No X and no Y.”
- “X only if Y” means “If X then Y.”
Formal Logic Practice Question
Craig will not take organic chemistry unless Paula lends him her notes from when she took the class.
Which of the following is/are consistent with the statement above? [Choose all that apply]
(i) Paula does not lend her notes and Craig does not take organic chemistry.
(ii) Craig takes organic chemistry using only Linda’s notes.
(iii) Craig does not take organic chemistry even though Paula lends him her notes.
(iv) Craig receives a B+ in organic chemistry.
Correct Response: (i), (iii), and (iv) only.
“Consistent” statements are statements that can both be true. (i) represents the contrapositive of the original statement, so not only is (i) consistent with the original statement, it follows from it.
In (iii), Paula has provided the necessary condition for Craig’s taking Orgo – that is, she lends her notes – but necessity is not the same thing as sufficiency. It’s therefore quite consistent with the statement that, even with Paula’s notes in hand, Craig elects not to take Orgo. (This is a very important concept.)
As for (iv), Craig’s grade is outside the scope of the statement, so it’s quite consistent that he get a B+. “Consistent” doesn’t mean “inferable” or “logically equivalent,” and now you know that even if you didn’t before!
(ii) is the only Impossible statement, since Paula’s notes are a necessary condition for Craig’s taking Orgo. For him to take the course without Paula’s notes would be inconsistent with the statement.
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