Universals, Identity and Logical Language

Biruni

This is a thought that came to me when I formulated the proposition of the form "If you can have A without B, A is not B". I don't think anyone could be in any doubt about this if the operator IS is taken to represent logical identity.
In that case, A → B and B → A, so to reject it you end up saying A is not A. As a result, you always end up with a true answer when the A is a noun and B is a noun and their definitions are constant. (at least as far as I can tell)

However, it's different when you think in terms of adjectives (or universals if you will). Then you can end up saying "If you have roses without red, roses are not red". It still holds up that roses are not linked to red by identity. They are not the same thing as one another. However, this is still seen as an acceptable use of the word IS and is the form used in that syllogism everyone knows:

Socrates is a man
Men are mortal
Socrates is mortal

The relationship here is clearly not identity. The thing about the word mortal, however, it is almost always used in the adjective sense. The proposition "If you can have socrates without mortal" is just nonsensical. You have to change the word to mortality for it to make sense.

So, when I say "if you can have roses without red, roses are not red", can we say the definition used for red1 and red 2 are the same, or are the definitions not consistent? I think this might be the case, since it is clearly an adjective in the second use, but it appears to be a noun in the first.

But what I really want to know is this. Is the proposition "If you can have A without B, A is not B" true for all interpretations of "IS" as a logical operator where the definitions of A and B are constant, which they have to be for it to be valid anyway.

Regardless of what your answers are, I think I have gained a much better understanding of why philosophers prefer to express logic in mathematical notation than put up with the confusing grammatical uses of certain words.

Morbert

Biruni wrote: »

This is a thought that came to me when I formulated the proposition of the form "If you can have A without B, A is not B". I don't think anyone could be in any doubt about this if the operator IS is taken to represent logical identity.
In that case, A → B and B → A, so to reject it you end up saying A is not A. As a result, you always end up with a true answer when the A is a noun and B is a noun and their definitions are constant. (at least as far as I can tell)

However, it's different when you think in terms of adjectives (or universals if you will). Then you can end up saying "If you have roses without red, roses are not red". It still holds up that roses are not linked to red by identity. They are not the same thing as one another. However, this is still seen as an acceptable use of the word IS and is the form used in that syllogism everyone knows:

Socrates is a man
Men are mortal
Socrates is mortal

The relationship here is clearly not identity. The thing about the word mortal, however, it is almost always used in the adjective sense. The proposition "If you can have socrates without mortal" is just nonsensical. You have to change the word to mortality for it to make sense.

So, when I say "if you can have roses without red, roses are not red", can we say the definition used for red1 and red 2 are the same, or are the definitions not consistent? I think this might be the case, since it is clearly an adjective in the second use, but it appears to be a noun in the first.

But what I really want to know is this. Is the proposition "If you can have A without B, A is not B" true for all interpretations of "IS" as a logical operator where the definitions of A and B are constant, which they have to be for it to be valid anyway.

Regardless of what your answers are, I think I have gained a much better understanding of why philosophers prefer to express logic in mathematical notation than put up with the confusing grammatical uses of certain words.

In logic, an equivalence relation between A and B would be

A ⇔ B
Or "iff A then B"

Or (as you say)

(A → B ) ∧ (B → A)