It is easy to prove that an apple is an apple: The equation x=x is a tautology, for all x. So if x=apple, we can substitute it in the equation x=x and get apple=apple, so an apple is an apple.

But how can one prove (using standard deductive logic) that an apple is not an orange, since x!=y is not a tautology for all x,y?
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It's not a *logical truth* that an apple is not an orange, but it's nevertheless *analytic* with respect to the meaning postulate: if predicate P holds uniformly of object a, then for all predicates Q contrary to P, ¬Q(a). For example, if something is uniformly black, then it's not of any color that is contrary to black (say red). Recall that predicates A, B are *contraries* iff they cannot simultaneously be true of an object:

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> **Definition.** Predicates P and Q are *contraries* =def for all x, either ¬P(x) or ¬Q(x).
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Here's how a proof could proceed. We have a meaning postulate, a convention that states that:

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> **P1.** For all P, x: if P(x), then for all Q: if P and Q are contraries, then ¬Q(x).
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Now we begin with the hypothesis that some object a is an A, and that A and O are contraries:

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> **P2.** A(a)  
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> **P3.** Predicates A and O are contraries
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Then we instantiate (P1) with a for x, obtaining:

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> **P4.** For all P: if P(a), then for all Q: if P and Q are contraries, then ¬Q(a)
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We then instantiate (P4) with A for P and O for Q, obtaining:

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> **P5.** if A(a), then if A and O are contraries, then ¬O(a)
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The rest is two applications of modus ponens starting with (P5) and using (P2) and then (P3):

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> **P6.** if A and O are contraries, then ¬O(a)  
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> **P7.** ¬O(a)
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the logic used here is mathematical logic . but a mathematical logic is only accepted if there is no way to disprove the given logic i.e to say 2+3 =5 and 3+2 also=5 assures the logic that counting 5 pieces from left or right will always remain same because we cant find any two no's which dont obey this logic .
so if we assume apple as x and orange as y such that we know always x=x and y=y is true(because any experiment cant disprove this .an apple remains an apple forever)
now if somehow apple could change into orange after a few years then x=x would have been wrong and then x=y would hold good
also falsness of only x=x directly assures the validity x=y.

but validity of x=x doesnt directly assures the falsness of x=y
note:apple=apple means apple will never change to anything else (including orange)
and orange=orange means orange will never change to anything else (including apple)
so both the chances are finished and there is no other way by which we can say that apple = orange.if one of the two statements x=x and y=y becomes false then x=y can be true means both x=x and y=y should be true