/sci/ - Science & Math

>>15768301
So, all men are mortals

Have you solved that one?

If not you are still bounded by aristotelian logic

>>15768301
That which is indiscernible is the same.

>All this seems to state is that ~(P(x)^P(x))

Typo. I meant to write ~(P(x)^~P(x))

>>15768301
>What exactly is the point of the law of identity?
It's an axiom that defines the notion of equality.
https://en.wikipedia.org/wiki/Schrödinger_logic

>>15768301
can you prove that x=x?

>>15768355
Sure tell me the value of x

What's the point? You start with obvious things and you put them together to prove more complicated things. Let's try a simple example.

x = y [assumption]
x = x [law of identity]
y = x [substitution]
If x = y, then y = x. [discharging assumption]
name of theorem: symmetric property

a+b = a+b [law of identity]
(a+b) + 0 = a+b [definition of +]
a+b = (a+b) + 0 [symmetric property]
b+0 = b [definition of +]
b = b+0 [symmetric property]
(a+b)+0 = a+b [substitution]
(a+b)+0 = a+(b+0) [substitution]

(a+b)+c = a+(b+c) [assumption]
S((a+b)+c) = S((a+b)+c) [law of identity]
S((a+b)+c) = S(a+(b+c)) [substitution]
(a+b)+S(c) = S((a+b)+c) [definition of +]
S((a+b)+c) = (a+b)+S(c) [symmetric property]
a+(b+S(c)) = a+S(b+c) [definition of +]
a+S(b+c) = S(a+(b+c)) [definition of +]
a+(b+S(c)) = S(a+(b+c)) [substitution]
S(a+(b+c)) = a+(b+S(c)) [symmetric property]
(a+b)+S(c) = S(a+(b+c)) [substitution]
(a+b)+S(c) = a+(b+S(c)) [substitution]
If (a+b)+c = a+(b+c), then (a+b)+S(c) = a+(b+S(c)). [discharging assumption]

(a+b)+0 = a+(b+0) [see proof above]
If (a+b)+c = a+(b+c), then (a+b)+S(c) = a+(b+S(c)). [see proof above]
(a+b)+c = a+(b+c) [principle of induction]

Normally you wouldn't write proofs out in such a nitty-gritty way, but when you do, you see the law of identity is used all over the place.

>>15768301
X must be equivalent to itself. Otherwise, X would not equal X which is simply False and also a contradiction.

>>15768358
why? you have to prove that x always equals x

>>15768574
Is easier if you give me a value for x

After all is question about the identity of x

>>15768301
> A and non-A do not exclude each other
> philosophy of Hegel and of Marx

So just useless trash, for sure commies must be exterminated to the last parasite.

In type theory, one way to define propositions is by listing out the things that count as a proof of that proposition. Equality is that which can be proven by the principle x=x.

>>15768580
>Is easier if you give me a value for x
if I give you a value for x you are going to prove it for that value alone, but that's not a proof for all values of x which is what I asked

>>15768301
>What exactly is the point of the law of identity?
The point is to specify the meaning of '='.

>>15768355
Sure with the law of noncontradiction. Assume that x is something other than x. In this case, x = not x, which violates noncontradiction and is false.

>>15768799
Are you stupid? What does x = not x contradict, if not x=x?

>>15768599
X is a value defined by instance specification in a variable such that P(x) is a statement regarding X like "X is a dog". If p(dog) Then we assert the statement is true because a dog is a dog. All values of X depends on the domain upon which a statement is invoked.

To gain specificity, you keep asking questions about the instance being invoked. A dog may be a dog if it has dog dna. Dna from a dog composes a certain set of information. The set of information is related to other sets which produce specific relations and functions.

>>15768799
Noncontradiction

>not contradicting

>>15768301
It has less to do about stating a property of "x" and more to do about stating a property of "=".