Also called the Politician's Syllogism or Equivocation.
A standard three-step syllogism uses three terms — the things that are being linked by the line of reasoning. If A, then B. If B, then C. Therefore if A, then C. The fallacy of four terms occurs when, exactly like it says, four terms are used instead of three. In most cases, a single term (B) is used two (or more) times, in differing contexts with different meanings; and yet the argument treats the two usages as exactly the same, since the same term was used.
It's best explained by this example from Land of the Blind.
A dry crust of bread is better than nothing But nothing is better than a big juicy steak. Therefore, a dry crust of bread is better than a big juicy steak.
This uses two different meanings of the word "nothing." The first line uses "nothing" to mean "a lack of food", while the second line uses "nothing" as "no such thing exists."
An episode of Yes, Minister called this by name, as "The Politician's Syllogism", specifically the form: "Something must be done. This is something. Therefore we must do this." The two different meanings of "something": "A solution to this problem" and "A thing" are mixed and said to be the same.
Why, if thou never wast at court, thou never sawest good manners; if thou never sawest good manners, then thy manners must be wicked; and wickedness is sin, and sin is damnation. Thou art in a parlous state, shepherd.
In thisOrder of the Stick, the Empress is committing the Four Terms fallacy by using "grown larger" in two different senses: Dragons that are more powerful have grown larger (as a result of age), so she thinks that if she grows larger (by eating more and getting fat) that she will also be more powerful. That's not the only fallacy involved in her reasoning, either.
The Heys in The Tick worship Nothing, as outlined in translation from their Pokémon Speak: "Nothing lasts forever. Nothing is worth fighting for."
Not really a Four Terms Fallacy, just a normal syllogism with an implicit proposition of "Only things that last forever are worth fighting for" and it really hinges on the acceptance of that proposition. Note that the two times "nothing" appears it means the same thing.
In a museum, an employee sees a patron tapping on one of the replicas with his fist.
Employee: Sir, please don't touch that.
Patron: But it's a replica, isn't it?
Employee: Yes, it is, but we still ask for you not to touch it.
Patron: Well, it's not under a glass case, which means it's not valuable. I have every right to touch it.
Employer: No, actually-
Patron: Yes, if I see something that's not cased, it means I can touch it, AND I WILL TOUCH IT!
The four terms used in the fallacy here are
"It is not in a glass case"
"I can (in the sense of "am physically able to") touch it",
"I can (in the sense of "am permitted to") touch it" and
"I will touch it".
If 1 is true, then 2 is true; if 3 is true, then 4 is true. What's missing is the necessary step establishing either that 2 and 3 are the same (they aren't), or, that if 2 is true, 3 is true as well (it isn't).
There's an old joke revolving around the word nothing, similar to the example above, where a bar patron turns down a beer from the bartender because "nothing is better than a cold drink".
Garfield has a similar philosophy. "If nobody is perfect, I must be nobody."
Similarly, Anadin pain pills used to be marketed with the slogan "Nothing acts faster than Anadin", prompting the zinger "So take nothing - it's cheaper".
The most famous version is probably this:
All the world loves a lover
I love you, therefore I am a lover
You are all the world to me, therefore you love me
Comedian Richard Jeni talks about when he learned about the syllogism in college, giving a shortened version of the page quote as an example (Ray Charles instead of Stevie Wonder), in his Crazy from the Heat stand-up routine.
Commonly used in LSAT testing for law school admissions in the United States. For example, using the word interest to mean financial stake, concern, or welfare. "The judge has no financial ties to the settlement with ABC Company. He is completely disinterested." This is not the same thing as saying he is "uninterested."
Mathmatics easily falls prey to this if you ignore a few key rules, e.g.:
An old joke: When you drink, you get drunk. When you get drunk, you go to sleep. When you go to sleep, you commit no sin. When you commit no sin, you go to Heaven. So, let's all get drunk and go to Heaven!
This joke: A kid asks his teacher "You wouldn't punish me for something I didn't do, would you?" The teacher says "Of course I wouldn't." The kid says, "Good, because I didn't do my homework!" (He's changing the meaning of "something I didn't do" from "something wrong that was not done by me" to "something I should have done and failed to.")