# The law of identity: two interpretations

Some philosophers state the law of identity as "if p, then p" or "p iff p" (i.e. in propositional logic). On the other hand, others state it in a completely different language, for example: "A is A".

What are the origins of these two styles?

• Do these ways of articulating identity strike you as different on a deeper level than wording? (I'm trying to better understand your concern about "the origin of these two styles") – virmaior Apr 19 '15 at 16:03
• At least, "p iff p" contains two propositions, but "A is A" contains only one... ? – Barış Akalın Apr 19 '15 at 16:08
• Hello. Could you provide a specific reference to a philosopher that regards "if p, then p" as the law of identity? – Ram Tobolski Apr 19 '15 at 17:43
• You do realise though, that "if p then p" is not equivalent with "p iff p"? The first is implication, the second equivalence. – user2953 Apr 19 '15 at 19:49

A is A is the "traditional" form of the Law of identity dating from the medieval commentators to Aristotle.

Aristotle' logic is a logic of terms; terms are (roughly speaking) "names" for objects : individuals, etc.

Thus, the identity is a relation between names or objects, and thus it is expressed with "is" (in formal logic : "=", and the usual law : x=x).

Propositional logic is formulated in terms of sentential variables, standing for sentences: thus, the relations between them are expressed with the connectives : "and, or, not, if__ then___".

"If p, then p" is one of the most elementary law of propositional calculus (in many textbook it is the first one derived).

In Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (2nd ed - 1927), page 99, we have the law :

*2.08. ⊢p ⊃ p

i.e. any proposition implies itself. This is called the "principle of identity". It is not the same as the "law of identity" ("x is identical with x") [...].

I presume that this is the origin of the modern usage in logic.

• But why we have call it a law/principle? I mean couldn't we derive it from the law of excluded middle and law of non-contradiction? – ado sar Jul 15 at 19:33
• @adosar - the form x=x is not a propositional formula. This it cannot be derived. The propositional form p ⊃ p is derived (as in Principia above) from propositional axioms, as well as LEM and LNC. – Mauro ALLEGRANZA Jul 16 at 11:36