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The law of non-contradiction (LNC) and law of excluded middle (LEM) are both formulated in terms of statements. That is "A and not A" is always false and "A or not A" is always true. According to Wikipedia the law states that:

"Each thing is identical with itself".

This law isn't formulated in terms of statements. What is a "thing"? Can someone explain what this law "tries" to state? I mean even if a thing is equal to itself (a=a) then it maybe true that a=b. The law doesn't state that "each thing is identical only to itself".

Also if we replace "thing" with statement the law becomes: "Each statement is identical with itself" or equivalent "A=A". But again in that case why we need such a law? The meaning of a sentence is completely determine by the sentence itself. So every statement is unique. Why we need to state that?

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    It is a modern view that truth and logic apply only to statements. But Aristotle was a realist, to him, the laws of thought "in the strictest sense" were the most basic laws of being that apply to things themselves, see Crivelli, Aristotle on Truth. That every thing is always itself and never its opposite is the metaphysical basis for identity and non-contradiction in logic. It may seem odd in our functional times that something so non-functional needs to be asserted, but back then there were those, like Heraclitus, who were willing to deny it.
    – Conifold
    Commented Jul 17, 2020 at 7:54
  • If some of the answers below satisfies you, please accept it.
    – Mauro ALLEGRANZA
    Commented Sep 7, 2020 at 14:12

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As per Wiki's entry regarding the Law of identity, expressing it in a form like e.g. "Every A is A", it is not a propositional one (compare to LNC and LEM).

It seems that there is no explicit statement of it in Aristotle; he used it only once, in Pr.An., Book B, Ch.22, 68a:

... and B is predicated both of itself and of C...

In modern reading, terms used by Aristotle are "class" names and an universal proposition like "B is predicated of all A" is translated with: "every A is B".

If so, due to the fact that a reading in terms of self-predication is highly implausible (the predicate Man is not itself a man), we can try to translate the statement "B is predicated of itself" as:

"every B is B",

that in symbols amounts to:

∀x(Bx ≡ Bx).

A more modern formulation can be found in Leibniz, Nouv.Ess. IV,2:

Les vérités primitives, qu'on sait par intuition, sont de deux sortes comme les dérivatives. Elles sont du nombre des vérités de raison,ou des vérités de fait. Les vérités de raison sont nécessaires et celles de fait sont contingentes. Les vérités primitives de raison sont celles que j'appelle d'un nom général identiques, parce qu'il semble qu'elles ne font que répéter la même chose, sans nous rien apprendre. Elles sont affirmatives ou négatives; les affirmatives sont comme les suivantes.

Chaque chose est ce qu'elle est. ["Every thing is what it is."] Et dans autant d'exemples qu'on voudra A est A, B est B.

The term "thing" is the most general one. In modern logic it is expressed with an individual variable of First order logic, from which the usual equality axiom:

∀x(x = x).

In High-order Logic, where we can quantify over predicates, we can define identity between individuals:

x = y ↔ (∀F)(Fx ↔ Fy).

From this defintion, using the tautology Fx ↔ Fx (that is basically Aristotle principle above), we can derive:

x=x.

See also Ivo Thomas, On a passage of Aristotle, NDJFL (1974).

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Ɐx (x=x) is a statment in predicate logic, which extends propositional logic .

A proposition is a truth valued variable. The symbol A in the law A v ~A represents a proposition.

Where as the symbol a in the law Ɐa.(a=a), represents a variable (sometimes called an object, or entity), where as the equality symbol = is a relational operator, and the predicate a=b is a function that is valued as true if "a equals b".

A predicate is a function which maps its variable(s) to a truth value. These variables are not themselves truth-valued; they reference objects in the domain of discorse -- whatever it is that is being discussed, which might often be, for example, integers, rational numbers, real numbers, people, quokas, et cetera.

We usually write predicates as capital letters followed by a bracketed list of lower letters (the variables), such as P(x), Q(y), R(x,y) or such, but some bivariate predicates (relations) are written with the predicate operator (often a capital letter or special symbol) between two variables: a R b, a = b, a > b or such.

The law doesn't state that "each thing is identical only to itself".

Indeed, it does not. The law is not that restrictive, by itself. Ɐa.(a=a) merely states that "anything is equal to itself".

I mean even if a thing is equal to itself (a=a) then it maybe true that a=b.

It may indeed, and we have other laws that allow us to make inferences when that happens.

Why we need to state that?

It is a seemingly self-evident statement and in many proofs it is surprisingly useful to be able to state that something is equal to itself. We may as well take it as an axiom.

[In some systems we do this by a rule of inference, 'equality introduction'. Effectively the same thing.]

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  • Ok I got the difference between a and A. Still having hard time to understand what Aristotle tried to state. That in every statement if we know a=b we can interchange a for b and vice versa like we do in maths?
    – Antonios Sarikas
    Commented Jul 16, 2020 at 16:05
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I'm not sure but we can make it statement by stating that "A and A" is always true

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