I have a doubt. I need help.

Can the basic axioms of equality be presented as "open axioms"?

I) (reflexivity)
$\qquad x = x$

II) (Substitutivity)
$\qquad (x = y) \to \big(F (x, x) \to F (x, y)\big)$

  • 5
    $\begingroup$ The convention is that free variable in "axioms" are slapped with a universal quantifier. $\endgroup$ – Asaf Karagila Sep 10 '17 at 0:08
  • 1
    $\begingroup$ Well, you should also include "symmetry", if x= y then y= x, and "transitivity", if x= y and y= z then x= z. $\endgroup$ – user247327 Sep 10 '17 at 0:09
  • 2
    $\begingroup$ @user247327 Symmetry and transitivity can be deduced using reflexivity and suitable instances of substitutivity, provided $F$ is allowed to contain additional variables beyond those exhibited. For symmetry, if you take $F(a,b)$ in substitutivity to be $b=a$, then substitutivity reads $(x=y)\to((x=x)\to(y=x)$, which, in view of reflexivity, simplifies to $(x=y)\to(y=x)$. For transitivity, take $F(a,b)$ to be $z=b$. Then substitutivity says $(x=y)\to((z=x)\to(z=y))$. $\endgroup$ – Andreas Blass Sep 10 '17 at 2:36
  • $\begingroup$ In Mendelson's textbook, the substitutivity axiom is as you have written (II). It uses the Hilbert proof system. Notice that $x$ and $y$ are metavariables denoting object variables. I believe that this presentation is acceptable because we can generalize variables in these axioms. However, we can't move such axioms to premises of implication by the Deduction Theorem (maybe you need this). $\endgroup$ – beroal Sep 10 '17 at 7:51
  • $\begingroup$ @beroal - not clear; they are axioms: thus, you have no need to use as premises. You van review the def of deduction (of a formula $\mathcal C$) from $\Gamma$ [page 28]: the (logical) axioms are usable directly as lines in the proof, without listing them in $\Gamma$. In any case, using Gen we can imemdiately retrieve the closed versions of I) and II) [see Mendelson's proof, page 93]. $\endgroup$ – Mauro ALLEGRANZA Sep 12 '17 at 8:27

Yes, we can.

Consider the Natural Deduction proof system.

We can derive $(x=x)$ from the "closed" version: $\forall x \ (x=x)$ simply using $\forall$-elim rule.

Conversly, from the open axiom $\vdash (x=x)$, we can derive $\vdash \forall x \ (x=x)$ using $\forall$-intro.

There are no assumptions (i.e. $\Gamma = \emptyset$) and thus the proviso of the rule: "$x$ not free in $\Gamma$", is satisfied.

The open equality axiom $(x=x)$ is obviously valid.

See e.g.:

The definition of true in a structure $\mathfrak A$ is restricted to sentences, i.e. "closed" formulas.

For open ones, it adopts the convention that:

$\mathfrak A \vDash \varphi \ $ iff $ \ \mathfrak A \vDash \text {Cl} (\varphi)$,

where $\text {Cl} (\varphi)$ is the universal closure of $\varphi$.

A different approach is adopted by:

In this case, the meaning and the truth-value of an open formula with respect to an interpretation $\mathfrak A$ is defined for specific "instances" of the formula, obtained through the variable assignment device.

The basic notion is that of satisfaction of a formula $\varphi$ by an assignment $s$ in a interpretation $\mathfrak A$ (in symbols: $\mathfrak A, s \vDash \varphi$).

A formula $\varphi$ is true in $\mathfrak A$ when it is satisfied by every assigment.

Consequently, $(x=x)$ is true in every interpretation.


I agree with Mauro ALLEGRANZA.

The use of open instead of closed formulas as axioms comes from Hilbert proof systems which have a rule of inference Universal Generalization statting that $\forall x F(x)$ can be derived from $F(x)$. The rule $\forall$ introduction of Gentzen's Natural Deduction is similar to Universal Generalization.

David Hilbert and Wilhelm Ackermann: Grundzüge der Theoretischen Logik, Springer, Berlin, Germany, 1928 (An English translation was published in 1950, see below)

David Hilbert and Wilhelm Ackermann: Principles of Mathematical Logic, AMS Chelsea Publishing, Providence, Rhode Island, USA, 1950


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