What is correct form to write A,E,I,O syllogistic relations in predicative form?

Question

I'm trying to draw a simple and comprehensive chart about syllogisms for my collegues. I need to represent relations like "All cows are mammals" in predicative form and draw Euler's diagram (not a problem) for it. I think about something that looks like (C - cows, M - mammals): $$ \forall a\in C:P(a \in M) $$ But I'm not really sure about it. It would be really cool to have some examples. Thanks!

Answer 1

See Aristotle's Logic:

$Aab$ is a shorthand for "$a$ belongs to all $b$" and we read it as "Every $b$ is $a$".

Thus:

$Aab$ will be $\forall x (b(x) \to a(x))$.

$Iab$ is "$a$ belongs to some $b$" and we read it as "Some $b$ is $a$".

Thus:

$Iab$ will be $\exists x (b(x) \land a(x))$.

And similar for those involving negation.

You can also use restricted quantifiers: $(\forall x : b)a(x)$ that, using a "mixed" symbolization using $\in$ will be : $(\forall x \in b) a(x)$.

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Switching from predicates to their extensions (i.e. sets or classes), we have that:

$Aab$ is $b \subseteq a$

while:

$Iab$ is $b \cap a \ne \emptyset$.