All dogs are mammals. All cows are mammals. Therefore, all cows are dogs.

Yeah, not so much.

If that logic sounds funny to you, good. It's an invalid argument, so the logical structure doesn't work.

The argument's based on a pretty common template: "all x are y and all z are y, so all z are x." You can try to plug in just about any word or phrase for x, y, and z. Key word: try. If you fill in something for x, y, and z in your template then—ta-da—you’ve created an instantiation, a.k.a. a specific version of an argument.

Instantiations are never valid or invalid: only the templates are. The beginning part of the argument (all x are y and all z are y) are called premises and they're what the conclusion is based on (all z are x). Premises and conclusions are all types of propositions—statements with definitive truth values. They can be proven true or false. So a statement like, "how awesome is the latest YA dystopian novel?" is

• a leading question.
• not a proposition.

Because awesome isn't a true-or-false thing, it can't be a proposition. The movie adaptation's probably going to be a summer blockbuster, though.

What if you used a different template than the all cows are dogs argument? Like:

Taylor Swift wears red lipstick. People who wear read lipstick like the color red. Therefore, Taylor Swift likes the color red.

See the difference? The first template goes: "all x are y and all z are y, so all z are x." Whenever you try to stick some arguments into x, y, and z, you need to keep in mind how they're related. Dogs (x) and cows (z) are both subsets of the animal set. Just because they're both subsets, though, doesn't mean they're both in the same subset, making this template invalid.

Saying that Taylor Swift wears red lipstick allows us to apply something we know about red lipstick-wearers to T-Swizzle because we already know Tay-Tay's a red lipstick wearer. To generalize, if we use the template, "all x are y and all y are z, so all x are z," as long as the premises are true, the statement's going to be valid. (If you want more funny examples of valid and invalid arguments, check this page out).

Now for the million dollar question: How can you tell the difference between valid and invalid arguments? As long as

• an argument is valid
• that argument's premises are true

the conclusion must also be true.

Always. No matter what. No exceptions. Find some instantiation of a template that has true premises—but a false conclusion—and you'll have an invalid argument. It’s that easy.

Okay, so it's a bit stickier than that. Technically you never know if an argument is valid forever and some day some instantiation will prove that it's invalid. Imagine the devastation it'd cause if Taylor Swift released an album called No More Red. Obviously our argument that she must love red would now be invalid. Logicians have two options. They could

• tell Taylor Swift that she was never allowed to change her mind about loving red.
• accept that, at least for the moment, the argument's valid.

As tempting as the first option is (imagine the dystopic universe where no one could change their mind about their favorite color), they generally just accept some arguments to be valid and others to be invalid.

Without further ado, we're going to go through some of the generally valid arguments, but look out for traps. They're similar—but deceptively different. You've been warned.
(Source)

All x are y. All y are z. Therefore, all x are z.

A valid argument for this is going to look like this: All Flying Purple People Eaters can fly. All flying things have wings. Therefore, all Flying Purple People Eaters have wings.

Be careful, though. You can't do this: All x are y. All x are z. Therefore, all y are z. That's like saying that because all Flying Purple People Eaters are purple and all Flying Purple People Eaters can fly, then all purple things can fly.

We can explain all that's happening using a bit of set theory. In the first example, the group of Purple People Eaters, x, is a subset (or a smaller group) of a larger set of things that can fly—y. Y, in turn, is a subset of z, which includes everything with wings (including the ones that can't fly, like ostriches and emus).

In the second example, x (Purple People Eaters) is a subset of y (everything purple) and it's also a subset of z (all things that can fly). But we know nothing—count 'em, nothing—about the relation between everything purple and everything that flies (sets y and z). We can't say that everything purple flies because there are plenty of purple things that can't fly, like grapes.

Or at least any grapes outside of our worst nightmares.

If a, then b. A. Therefore, b.

If you watch every summer blockbuster one after another, then you'll be really tired. We watched all the summer blockbusters back to back. Therefore, we're really tired.

Also, we're strangely energized by the dark, frantic movie soundtracks.

Be careful, though, Shmooper: this isn't the same as: if a, then b. B. Therefore, a.

If you watch every summer blockbuster one after another, then you'll be really tired. We're really tired. Therefore, we watched all the summer blockbusters back to back.

That's…less than true. We could be really tired for other reasons, like having to battles hordes of mindless zombies or staying up to watch all this year's rom coms. Being tired is a superset (the opposite of a subset) of watching all the summer blockbusters, so it doesn't automatically mean that we watched a billion movies about people driving sports cars and blowing things up. This relationship is a one-way street.

P or q. Not P. Therefore, Q.

Shmoop buys Half Life 4 (just waiting on you, Valve) for a nondenominational Winter Holiday or Shmoop buys the thematically similar Gossip Girl. Shmoop didn’t buy Half Life 3. Therefore, Shmoop bought Gossip Girl.

This isn't a two-way road, though. It won't work if you try: q. Therefore, not p.

Shmoop buys Half Life 4 for Christmahanukwanzali-stice or Shmoop buys Gossip Girl: Director's Cut. Shmoop bought Gossip Girl. Therefore, Shmoop didn't buy Half Life 4.

Despite the fact that Half Life 4 will probably never come out, there's nothing in our statement that says we can't buy both. That's the difference between inclusive and exclusive or, so keep that in mind while you shop for presents.