Equivocation
Also known as: equiv
Using a word in two different senses within the same argument, as if they were the same.
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In plain terms
Equivocation trades on a word that has more than one meaning. The argument uses the word in one sense in the first step and a different sense in the second, then treats the two as if they were the same thing. The conclusion only follows if you don't notice the word changed jobs partway through.
The slipperiness is the whole mechanism. Pin the word to a single meaning and the argument stops working.
Why it's fallacious
A valid argument needs its terms to stay fixed. If "free" means "no cost" in the premise and "at liberty" in the conclusion, the two statements aren't really connected; they just share a sound. The logical link is an illusion created by the shared spelling, not by any shared meaning.
Equivocation is especially common with words that carry both a technical and an everyday sense: "theory," "natural," "rights," "belief." The everyday meaning sneaks in to do work the technical meaning can't, or the reverse.
Canonical example
"Evolution is just a theory. And a theory is just a guess. So evolution is just a guess."
The word "theory" changes meaning between the first sentence and the second. In science, a theory is a well-substantiated explanation supported by a large body of evidence. In casual speech, "just a theory" means a hunch. The argument swaps the casual sense in for the scientific one, and the conclusion rides entirely on that swap. Hold "theory" to its scientific meaning throughout and there's no argument left.
Counter-example (not a fallacy)
"He's a free man now, and the software is free too, though in very different senses, one legal and one financial."
This isn't equivocation. The speaker uses "free" in two senses but flags that they're different and never builds an argument that depends on treating them as the same. Using an ambiguous word more than once is fine. The fallacy requires the argument's logic to rely on the shift going unnoticed.
The line: does the conclusion depend on a word quietly meaning two things at once?
How to fix it
If you've been linked here, pick one meaning for the key word and check that your argument still holds with that meaning locked in from start to finish. The fix is precision: define the term once, use it consistently, and if you genuinely need two senses, use two different words. An argument that only works while a word is allowed to drift isn't an argument; it's wordplay, and a careful reader will catch it.
If you're on the receiving end, ask for a definition and hold the word to it. "When you say X, do you mean it in sense A or sense B? Let's use it the same way in both halves." The fallacy rarely survives the term being pinned down.