Question on the concept of " Identity "

[Mathelogician]

Hi all;
Look at the attached part from Van Dalen's Logic and structure.
What is he doing exactly?
In axiomatizing 'Identity' as he does, what is gained rather than what we had before (i.e., looking at 'Identity' as a binary predicate)?!
Even in the axioms, he is again using a symbol in the language for identity as a binary predicate (i.e., = ) and then he proves the axioms (or says they are provable) in the language. [note that he also proves I3 and I4 that i haven't shown.]
Thanks.

 

[solakis1]

Hi all;
Look at the attached part from Van Dalen's Logic and structure.
What is he doing exactly?
In axiomatizing 'Identity' as he does, what is gained rather than what we had before (i.e., looking at 'Identity' as a binary predicate)?!
Even in the axioms, he is again using a symbol in the language for identity as a binary predicate (i.e., = ) and then he proves the axioms (or says they are provable) in the language. [note that he also proves I3 and I4 that i haven't shown.]
Thanks.

The axiom I4 is rather an axiom sceme ,because for each "t" and for each "φ" we a corresponding axiom.

Axioms I2 and I3 can be proved using axioms I1 and I4

 

[Mathelogician]

Of course they are axiom schemes; but I'm afraid my question is something else!
Thanks.

 

[Evgeny.Makarov]

When we are talking about semantics (i.e., $\models$), this section may be considered just as an observation that identity satisfies these axioms. The importance of the axioms comes when we consider the derivability relation $\vdash$ (Definition 1.4.2). Then we have to use special axioms or inference rules to say that $=$ is not just an arbitrary predicate symbol.

Axioms I2 and I3 can be proved using axioms I1 and I4

This is true. It is given as an exercise later in the text. The term version of $I_4$ can also be proved from the formula version.

 

[blue_raver22]

It appears that Van Dalen is attempting to formalize the concept of "Identity" by axiomatizing it in a logical structure. This means that he is creating a set of rules or principles that define the properties and relationships of identity. By doing this, he is providing a framework for understanding and reasoning about identity in a precise and consistent manner. This can be seen as a gain compared to simply considering identity as a binary predicate, as it allows for a more rigorous and systematic analysis of the concept.

In his axioms, Van Dalen is indeed using a symbol for identity as a binary predicate, but this is necessary in order to formalize the concept and make it amenable to logical analysis. The purpose of the axioms is to provide a set of statements that are true by definition and can be used as a basis for further logical reasoning. By proving these axioms, Van Dalen is demonstrating that they are logically consistent and can serve as a foundation for understanding and reasoning about identity.

Overall, Van Dalen's approach to axiomatizing "Identity" can be seen as a valuable contribution to the field of logic and philosophy. It allows for a more precise and rigorous understanding of this fundamental concept and enables us to reason about it in a systematic manner.