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Can we do without equality in first order logic? I looked at some cases in which equality is essential and found that it seems enough to have inequality implicit in the variables. Let $\phi(x,y)$ be a formula not containing equality.

Instead of writing $(\forall x)(\exists y) x \neq y \wedge \phi(x,y)$ one can write $(\forall x)(\exists y) \phi(x,y)$, supposing that different variables denote different objects. Correspondingly:

$(\exists x)(\exists y) \phi(x,y)$ instead of $(\exists x)(\exists y) x \neq y \wedge \phi(x,y)$

$(\forall x)(\forall y) \phi(x,y)$ instead of $(\forall x)(\forall y) x = y \vee \phi(x,y)$

$(\exists x)(\forall y) \phi(x,y)$ instead of $(\exists x)(\forall y) x = y \vee \phi(x,y)$

The other way around, if we want to express that $(\forall x)(\exists y) \phi(x,y)$ allowing the case $x=y$, we have to write $(\forall x)(\exists y) \phi(x,x) \vee \phi(x,y)$, and correspondingly

$(\exists x)(\exists y) \phi(x,x) \vee \phi(x,y)$ instead of $(\exists x)(\exists y) \phi(x,y)$

$(\forall x)(\forall y) \phi(x,x) \wedge \phi(x,y)$ instead of $(\forall x)(\forall y) \phi(x,y)$

$(\exists x)(\forall y) \phi(x,x) \wedge \phi(x,y)$ instead of $(\exists x)(\forall y) \phi(x,y)$

Another case is "there is exactly one object $x$ with $\phi(x)$":

$(\exists x)(\forall y) \phi(x) \wedge \big(\phi(y) \rightarrow y = x\big)$ becomes $(\exists x) \phi(x) \wedge \neg (\exists x)(\exists y)\phi(x) \wedge \phi(y)$

I wonder if every sentence of first order logic including equality can be equivalently written without equality when interpreting different variables to denote different objects. For the time being I'd like to restrict the question to languages without function symbols and individual constants, i.e. comprising only relation symbols and variables.

Tara B
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Hans-Peter Stricker
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  • There is no reason for different variables to denote different objects. Different variables are different objects but they can be interpreted as the same object if they are free in a formula. It depends of the model you choose. – leo Apr 16 '13 at 07:59
  • If you allow relation symbols, a theory can always just add a new binary relation symbol EQUALS and the axioms ensuring it is an equivalence relation.... –  Apr 16 '13 at 08:36
  • I don't want to add a new binary relation, but to get rid of one. – Hans-Peter Stricker Apr 16 '13 at 08:43
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    If you want to be able to talk about functional relations, you will need equality. – Dan Christensen Apr 18 '13 at 04:25
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    That's why I restricted my question to languages without function symbols (to be on the safe side). On the other hand I tried to show how to circumscribe "there is exactly one x..." – Hans-Peter Stricker Apr 18 '13 at 09:12
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    @Hurkyl, equivalence is necessary but not sufficient for equality. Two objects can be isomorphic but distinct, and isomorphism is also an equivalence. – alancalvitti Jul 26 '19 at 14:29
  • @alancalvitti: ... but you can't tell the difference between equivalence and equality if you only use the equivalence predicate. Also, isomorphism is a more complicated issue: we usually don't care merely about the proposition "are these objects isomorphic?", but instead to talk about specific (and often implicit) maps between them being isomorphisms. –  Jul 26 '19 at 19:43
  • @Hurkyl, OP wrote 'comprising only relation symbols and variables' - he didn't restrict it to equivalence relations. Re isomorphism, why does placing the emphasis on the map vs objects change anything? The categorical view (eg Mazur's "When is one thing...") is that equality can be replaced by "up to unique isomorphism" - the focus being on the objects 'the things'. Isomorphism, as well as congruence, are equivalence relations. How do you tell them apart from equality? – alancalvitti Jul 26 '19 at 20:12

2 Answers2

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The question is this:

Can we do without equality in first order logic, and get something equally expressive using a language with the semantics for quantifiers tweaked so that different variables get assigned different values?

The proposal here in fact goes back to Wittgenstein's Tractatus 5.53, where he writes, ‘Identity of the object I express by identity of the sign and not by means of a sign of identity. Difference of the objects by difference of the signs.’ Can this proposal, not really developed out by Wittgenstein, be made to work?

The answer is it that it can, as shown by Hintikka in 1956 ('Identity, Variables, and Impredicative Definitions', Journal of Symbolic Logic). Hintikka distinguishes the usual 'inclusive' reading of the variables (i.e. we are allowed to assign the same object to distinct variables) from the 'exclusive' reading, and then proves the key theorem (summarized on p. 235):

[E]verything expressible in terms of the inclusive quantifiers and identity may also be expressed by means of the weakly exclusive quantifiers without using a special symbol for identity.

So yes, Hans Striker's conjecture is right. For a more recent revisiting of Hintikka's result, in the context of interpreting the Tractatus see e.g. Kai F. Wehmeier's 'How to Live without Identity - And Why', Australasian Journal of Philosophy 2012, downloadable at http://www.academia.edu/949632/How_to_live_without_identity_--_and_why

Peter Smith
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For finite $\mathcal{A}$, identity is first order definable in $\mathcal{A}$ if and only if $\mathcal{A}$ is asymmetric (has no non-trivial automorphisms).

Samuel Schlesinger
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