Equivocation
The fallacy of equivocation occurs when an important term in an argument is used in two (or sometimes more) senses. An example might be:
Why is it okay to kill time but not to kill people?
Here the word "kill" is being used in two different ways: the first time it is employed as a figure of speech, where "killing time" means to use up some spare moments in one way or another; in the second it takes on a more specific meaning, the kind we normally associate with it. The person asking the question has confused these, so that something else we could ask with the word would mean different things depending on which sense we adopted. For instance, we could inquire, "how did you kill time?" and "how did you kill the person?" The first would give us a reply that describes an action and could be all manner of things; the second, though, would have to specifically be about the way in which someone was murdered. Asking the question, then, shows a misunderstanding in the use of the word.
In general, we can tell if someone has equivocated by finding a term used in two or more contexts, such that its meaning in one is different than in the other(s). Take another instance:
My school is supposed to provide free tuition but I've seen restrictions in the lessons I've attended.
This time the word "free" has been implicitly equivocated, with it meaning "free
of charge" in the first instance but "free of
restrictions" in the second, resulting in a confused argument. If we set it out again, this time removing the problematic term and replacing it with synonyms, we might get the following:
P1: Tuition at my school does not cost students any money;
P2: There are restrictions on course content, etc;
C: Therefore, the tuition does cost money after all.
The conclusion does not follow and the error is plain to see. Rewriting an argument in this way is sometimes the best way to note (or to demonstrate) that an equivocation has occurred.