The equality relationship is symmetric, "For every a and b, if a = b, then b = a" [1]. This holds for the negation as well.
Or, "Properties of Equality... The [equality] relation must also be symmetric. If two terms refer to the same thing, it does not matter which one we write first in an equation. ∀x.∀y.(x=y ⇒ y=x)" [2]
The 'is' relationship in logic is understood to be equality. The 'is a' relationship in logic is subset. Colloquially, the word "is" can be either one though.
I notice in your counter example you swapped the "is" relationship with "is a". Keeping the "is" relationship: "A rectangle is not square" (true generally, but false for specific cases for rectangles). That is really the distinction, the sometimes true vs always true.
Let's stick to the precise definition of "is" to mean a logical equality from here on please, and be precise when we use "is" vs "is a" and never infer "is" to actually mean "is a".
So, with "Math is a language" vs "Math is language". Which do you mean?
Or, "Properties of Equality... The [equality] relation must also be symmetric. If two terms refer to the same thing, it does not matter which one we write first in an equation. ∀x.∀y.(x=y ⇒ y=x)" [2]
The 'is' relationship in logic is understood to be equality. The 'is a' relationship in logic is subset. Colloquially, the word "is" can be either one though.
I notice in your counter example you swapped the "is" relationship with "is a". Keeping the "is" relationship: "A rectangle is not square" (true generally, but false for specific cases for rectangles). That is really the distinction, the sometimes true vs always true.
Let's stick to the precise definition of "is" to mean a logical equality from here on please, and be precise when we use "is" vs "is a" and never infer "is" to actually mean "is a".
So, with "Math is a language" vs "Math is language". Which do you mean?
[1] https://en.wikipedia.org/wiki/Equality_(mathematics)
[2] http://logic.stanford.edu/intrologic/extras/equality.html