Topics Logic

Question Recently I asked a question about logic, and the answer directed me to an SEP entry, which then took me to two other SEP entries, on Russell's paradox and on the Liar's paradox. Frankly, after having read through those explanations, there was a glaring omission from every cited philosopher, and I wondered if everyone was overcomplicating things: I don't see how there is any "paradox" at all. Consider the concept of a "round square" or a "six-sided pentagon." Those are nonsensical terms, because of the structural nature of the underlying grammar. They are neither logical nor illogical, they are merely grammatically inconsistent at the fundamental level of linguistic definition. The so-called "paradox" of Russell and the Liar seem to me to be exactly the same kind of nonsensical formulations: the so-called "paradox" is merely a feature of the language, these concepts also are grammatically inconsistent at the fundamental level of linguistic definition. Russell's "paradox" is just as "paradoxical" as a seven-sided hexagon: it's not a "logic" problem at all, it is a grammar problem. I suppose then the panel's response will then be: "suppose it is 'merely' a grammar problem: that merely leads us to another, related conundrum: what 'rules' do we need so that we can identify when it is not a logic problem and it is a grammar problem?"

Accepted:
August 7, 2015

Comments

If I may, I think you're

If I may, I think you're being a bit too dismissive of Russell's paradox.

We start with the observation that some sets aren't members of themselves: the set of stars in the Milky Way galaxy isn't itself a star in the Milky Way galaxy; the set of regular polyhedra isn't itself a regular polyhedron; and so on. It seems that we've easily found two items that answer to the well-defined predicate

S: is a set that isn't a member of itself.

Naively, we might assume that a set exists for every well-defined predicate. (For some of those predicates, it will be the empty set.) But what about the set corresponding to the predicate S? This question doesn't seem, on the face of things, to be nonsensical or ungrammatical. But the question shows that our naive assumption implies a contradiction, and therefore our naive assumption can't possibly be true.