Do numbers “exist”? What about properties? I know my red apple exists, but does “redness” itself exist?
The existence of abstract objects seems, at first, like a deep metaphysical question. In fact, it’s a question of the pragmatics of language.
A couple of quick definitions. The view that abstract objects do exist is called “Platonism.” The view that they don’t is “nominalism.” Those who think they do exist, but only in the mind, are “conceptualists.”
Let’s take the case of numbers. A typical nominalist argument says that while you may bump into two apples somewhere along your travels, you’re never going to bump into “2.” There is no such thing independent of our descriptions of states of affairs. And that’s all abstract objects are – descriptions. They exist only in language.
The conceptualist replies: the fact that they exist even just as descriptions demonstrates that they do exist – in the mind. Abstract objects are mental fictions, and as such, they exist.
The Platonist’s retort: how do you explain that we all come up with the same mental fictions? When you and I speak of “the number of apples here,” we’re not talking about two different fictions that each of us came up with and which we happened to give the same name to. We’re speaking about the same thing: the number 2!
The Platonist may add that science corroborates the existence of numbers. Science predicts reality, and it does so through the use of numbers. This verifies the fact that numbers aren’t just some arbitrary or socially conditioned way of interpreting the world.…
In Part 1, we disambiguated between the logic that we intuitively use (hereafter “intuitive logic”) and systems of logic. Systems of logic codify a set of rules of entailment. That set of rules may or may not accurately represent the rules involved in intuitive logic. Logic selection is the process of finding a set of rules that does.
You might think of it this way: there are many systems of geometry. There’s traditional Euclidean geometry, but there are also various non-Euclidean geometries. No one of them is any more a geometry than any other. To choose between these geometries, we need to match up their rules against what happens in the type of space we want them to represent.
So it is with logic. To choose a system of logic, we need to match its rules of entailment against the rules of intuitive logic.
Of course, the process of logic selection cannot be a merely logical one. That would presuppose what we’re looking for. We can (must) use intuitive logic in the selection process, as we use it in all our reasoning. But we can’t use any formal system of logic. So, what do we use?
The selection criteria
Logician Graham Priest presents a theory of logic selection in this video. Priest argues that the process of theory selection is always the same, whether you’re selecting a theory of logic, physics, or economics. The selection criteria are as follows:
Agreement with data
Power (how much data it accounts for)
Relevance (avoidance of the ad-hoc)
Agreement with data is always the most important criterion.…
Some philosophers, including C.S. Peirce, have argued that the Liar is demonstrably false and not true. The argument is based on the premise that all statements implicitly assert their own truth.
At first glance, this seems plausible. If I tell someone, “I did your laundry,” it carries with it an implied “it is true that I did your laundry.” This would seem to hold for all assertions. So, Peirce argues, the Liar is really saying: “It is true that this sentence is false,” which essentially comes down to saying “this sentence is true and false.” This is no longer a paradox, but a plain contradiction, and so false. It is like saying, “I’m a cat owner that doesn’t own a cat.” That’s not a mystery, just a lie. What makes the Liar a paradox is that what it says is, on the surface of it, coherent. If it just asserts a flat-out contradiction, then it poses no problem.
That a proposition automatically asserts its own truth is an interesting notion, and it is not easy to say whether it is accurate or not. Peirce later in life argued that it was incorrect. Luckily, it is not necessary to determine whether it is accurate or not because, even if it is, it does not resolve the Liar.
It is not true that, if a proposition automatically asserts its own truth, then the Liar really says: “It is true that this sentence is false.” In that sentence, “this sentence” refers to that whole sentence.…