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[COLLEGE LOGIC] Why does the addition of constants to a first-order theory not result in more closed wffs?

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A proper axiom is a non-logical axiom that taken in addition to first-order logic forms a first order theory. Proper axioms take the form of closed wffs, but not all wffs are proper axioms. For instance, there are seven proper axioms of Robinson Arithmetic, such as Axiom 6: **For all natural numbers x, x·0=0**.

First-order logic can be axiomatized a number of ways, but one schema for logical axioms might be for any wffs, A and B, **A->(B->A)**.

Now say we are working in a first order theory like Robinson Arithmetic and we add a constant, "cat", to it. This new constant allows us to express new wffs, like "5=cat". This in turn gives us new logical axioms, like **5=cat -> (1+1=2 -> 5=cat)**. However, this does not affect the seven proper axioms of Robinson Arithmetic. Being able to express the term 'cat' does not, for instance, change the axiom **For all natural numbers x, x·0=0**.

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I hope this helps. If you have any questions please ask.

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