WPM: ...First you want to be able to make a distinction between a sentence’s literal meaning and its non-literal meaning. Its literal meaning is just what you think it is. The non-literal meaning can take two forms—which I’ll call idiomatic and metaphorical. The literal meaning is much easier to characterize than the non-literal meaning, but underlying both is the notion of compositionality.
WPM: Language works because it’s compositional. The meaning of a larger linguistic unit is a function of the smaller units that make it up. You comprehend a conversation by stringing together individual utterances. Utterances have meaning by virtue of the words they contain. The meaning of words is a product of their constituent parts, which we call “morphemes”. Sometimes a morpheme is a whole word. Sometimes it’s something smaller. For example, “unhappy” is composed of the morphemes “un-” meaning “not” plus “happy” meaning “happy”. At some point the compositionality has to bottom out in a set of unanalyzable symbol/meaning pairings—this is the famous “arbitrariness of the sign” which I’ll bet you’re all familiar with since Ferdinand de Saussure is the granddaddy of us all. Typically this bottoming out happens at the morpheme level, but sometimes it can happen higher up.
CF: Can you give an example of a higher up case?
WPM: Idioms are sequences of words whose meaning is not compositional. They’re one kind of non-literal meaning. The classic example is “Mark kicked the bucket” meaning “Mark died”. Dying has nothing to do with kicking or buckets. If you were a non-native speaker of English, you wouldn’t be able to reason out what this means. Presumably there’s a historical connection between the concepts, but it may be completely opaque to present-day speakers, who just memorize the meaning of “kicked the bucket” en bloc. Linguists say that it has become lexicalized—“kicked the bucket” is like a synonym for “died”. The other kind of non-literal meaning is basically everything else. I’m going to call this category “metaphorical”, and while I doubt most linguists would object, it’s telling that “metaphor” does not have a clear technical definition. Crucially, though, it’s a case where the meaning of a phrase is non-literal, but nevertheless compositional.
CF: I think I’m following you.
WPM: I could say to you, “Mark is a cyclone in a sweater vest”, reasonably expecting you to interpret this to mean that Mark is a person whose dynamic personality is belied by an unremarkable exterior.
NM: That’s not a bad description of Post Brothers!
WPM: This is non-literal—neither Mark nor Post Brothers is actually a conical configuration of high-speed winds—but it is also compositional. The metaphor turns on the interlocutor's familiarity with the power of cyclones and the nebbishy connotations of sweater vests. The meaning of the whole is a function of the meaning of its parts. It’s just that the function is a highly complex, contingent, culturally determined one. Characterizing these connections is a very difficult task that often moves beyond the purview of linguistics. So linguists sometimes choose examples that don’t lend themselves to metaphorical interpretation. This is actually more what I’m getting at when I talk about “clean-room sentences”, rather than simple, culturally portable ones. It’s all just a roundabout way of saying that I think your speculation—that “the cat is on the table” may have already become an idiom—correctly uses “idiom” in its technical linguistic sense.