There are different ways that contemporary symbolic logic tends to handle relations like the kinds Parmenides was discussing in the previous post. I'll walk through just one. Let's say that we wanted to express a likeness relation such as "All crows are black." It would look like this:
That is read, "For every x, if x is a crow then x is black."
Notice that this relationship does not include any actual crows, only a variable x. It is a statement that describes every object in the universe, most of which will fail to be crows. Those things that satisfy the crow condition will, if the statement is true, also satisfy the black condition.
It is possible to talk about an actual crow. Let's say object a is a crow. (Variables are from the end of the alphabet, whereas letters taken from the beginning of the alphabet are constants. That is to say that x or y could be anything, but a is a particular something.) Now you can test the proposition, because if Ca is true and Ba is false, then the proposition is false. It only takes one counterexample to falsify a universally quantified statement like the one above (the upside-down A is the universal quantifier).
So one way we can talk about relations between objects is to use the capital letter to indicate a class of things. All of those things are automatically related to each other by being members of the class.
That doesn't actually solve the problem Parmenides is raising, though, because you still need two things at work to express the relation. Let us say that a is not a crow, but the One or really any Form that is a unitary idea. Being a pure unity, it is just a and not Fa. A genuinely pure unity cannot admit of likeness in this way either.
It also, it turns out, can't admit of unlikeness on this model because you'd have to say that a was not a crow, and that requires three concepts working together: the constant, the class, and the negative operator. (¬ Ca). The pure unity does not admit of either the ¬ or the C, because if it did, it would no longer be just one idea but multiple ideas.
It actually seems like the Form of a Crow would be C, though, not a. After all, it is x's participation in C that makes it a crow. Now you might say that the One is "O," and any x might participate in it without changing it. So you could have Oa and Ob, where a and b each participate in the One without being the One. This addresses Parmenides' concerns somewhat, because whether any x does or doesn't participate in O, O remains singular and unchanged by the participation or lack thereof.
Viewed that way, contemporary symbolic logic depends upon Platonic forms. So too does mathematical logic, and therefore math itself.
I'm not sure how closely this matches the Forms.
ReplyDeleteThe "C" is an operator that gives true or false (or possibly something mixed) depending on whether "a" partakes of "crowness". The logical theories I've seen have been set-based: is "a" in the set of all things with "crowness" or not? This seems to dodge the Forms question in favor of an operational question. It lets us do calculations without inquiring too deeply into how the set of things with "crowness" gets filled.
So, that's a good point -- one that is analogous to the psychology vs. metaphysics discussion we've had going in the background to this series.
ReplyDeleteIt is possible to approach the logic from a set theory perspective, in which case C loses all meaning except to designate a set. Then ∀x(Cx⊃Bx) only means that any individual in set called "C" will also be in the set called "B." This is akin to the psychology position in the analogy, in that all that matters is the designation a human being has imposed (rightly or wrongly, and for whatever reason).
It's also possible to approach the logic from a philosophical perspective, in which case the status of the C really matters. It's not just to be treated as a designation; the question of whether a really belongs in C is not just a question about whether a was assigned to set C. It's a question about whether a is in fact a C; perhaps it was wrongly assigned, or wrongly excluded.
For that to be answerable, a and C both have to be real. This is analogous to the metaphysical position.
Mathematics is chiefly useful because it lets us simulate reality, and if the simulation is adequately useful it's possible to stop caring about just why it works. (It's even possible to mistake the simulation for the reality, as perhaps Pythagoras did.) Some contemporary philosophy, led by the late David Lewis, essentially adopts the approach that the set theory will do.
The consequence of doing that is that things like 'natural laws' become senseless, and you end up arguing that reality is just a bunch of sets; when we notice patterns in this 'mosaic,' we tend to explain them away as one thing causing the other. But really, it's just a bunch of stuff that's out there, and the assigning of the sets is just a kind of convention that misleads us from the fact that there is no law governing nature.
This is called "Humean Supervenience," though I doubt Hume would have thought much of it.
https://plato.stanford.edu/entries/david-lewis/#5
"Patterns in this 'mosaic'" reminds me of the AI/ML trend. Within a physics framework, those sorts of tools can be useful for teasing out signals, but when you do something like that "learn from scratch" approach to having a machine "learn" to play chess, plainly there's no understanding--just weighted patterns.
ReplyDeleteYes, exactly. It’s somewhat like arguing that we should leave all the chess-playing to the AIs from now on, because they have gotten “so much better at chess than we are.” Well, no, they haven’t; not if the point of chess is enjoying the game.
ReplyDeleteThis is akin to the psychology position in the analogy, in that all that matters is the designation a human being has imposed (rightly or wrongly, and for whatever reason).
ReplyDeleteThree thoughts about that.
First, you've mentioned before in another context the notion of "the domain of discourse". A claim is true, or otherwise, within the framework being considered. Introducing a new factor or breaking the frame in some fashion doesn't make the original claim either more true or more false. All crows are black... ALBINO creatures exist and paint and bleach exist but exceptions and hypothetical contrivances aren't useful to the discussion. Frequent typical observations result in defined or designated generalizations. Inferences. And these are "true" to the extent they are USEFUL in communication.
The second thought is that communication of claims isn't restricted to humanity. " MINE! " is perhaps the most general communication in existence. A wolf marking a territory with his urine, a meadowlark singing his own claimed territory, ant scent determining sisterhood with an ant of the same colony, a newborn human infant smiling in recognition of his mother... The postulated claim has three (magic number again) axiomatic components: the Cartesian recognition of self, the classification of a desirable and severable thing among all other things, and the awareness of possible future change. Look to the ant, thou sluggard! Even the ant. Mine, like me, part of my life - or not. If not then enmity with a rival-colony ant; avoidance of a diseased sister ant; walk on by the husk of a dead ant. The biologists tell us that
these sorts of signals back and forth and the resulting behaviors exist even in microscopic organisms.
Third -- there exist deliberate false claims. The plant that smells, to a bug, like rotting meat. The non-venomous snake that mimics the dangerous one. The con artist phoning me with offers to clean my air ducts, reduce my credit card payments, and extend my automobile repair warranty... It's not clear to me where in philosophy or strict mathematical logic intentional deception fits in.
First, you've mentioned before in another context the notion of "the domain of discourse"
ReplyDeleteYes, that's normally explicit in these kinds of logical formulations. I avoided it here in order to show something about how you could pass over all things universally in order to say something about crows.
Here's an article on universal derivation, referencing yet another of Plato's dialogues:
https://milnepublishing.geneseo.edu/concise-introduction-to-logic/chapter/14-universal-derivation/