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Disclaimer: None of the answers I searched for on Math.SE seemed to match my specific problem, e.g. the answer to this question is too informal for my goals from my POV. I am new to the subject.

There seem to be roughly two ways in the literature to build up the syntax of logic:

  1. Use variables, connectives, and functions(, ...) in the structure
  2. Use variables, connectives, functions(, ...) and constants in the structure

In other words: One with constants, one without. As one can see constants as 0-ary functions, the results are the same. How to make this sentence mathematically rigorous? Is there a theorem considering a structure isomorphism, category equivalence or something similar "unnecessarily formal abstract nonsense" to make the statement "constants give the same results as 0-ary functions" strict?

Mauro ALLEGRANZA
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user7427029
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    If $A$ denotes the domain then a $0$-ary function $f$ can be looked at as a function $A^0\to A$ where $0:=\varnothing$ and $A^0$ is the set of functions $\varnothing\to A$. There is exactly one such function wich is the empty function. So $A^0$ is a singleton. Then $c:=f(\varnothing)\in A$ is the constant corresponding with $f$. Also in the category of sets an element of a set $A$ corresponds with an arrow ${}\to A$ where ${}$ denotes a singleton. – drhab Oct 02 '22 at 12:51
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    The issue is simply that a 0-ary function is a "function" whose output value does not depend on the input value (there are none). This means that it behaves as a constant. – Mauro ALLEGRANZA Oct 02 '22 at 14:27

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There are sometimes mathematical objects which profit from defining them in a different way. Let me elaborate with an example. Polynomials can be seen as finite sequences of the factors, or can be seen as functions. Different areas of mathematics define them as such and such.

In your case, concerning constants, this is not the case. To differentiate between constants and functions is done only for pedagogical purposes. They are 0-ary functions, in universes where functions of higher arity exist, or just constants.

Jason
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To use a special syntactic cathegory for constant is wrong for pedagogical reasons.

It would be like teaching induction by first dealing with 0 as an exceptional case.

@drhab explains it well. As $A^0=\{\varnothing\}$, $\varnothing$ is the only 0-ary tuple. Hence constants are funtions that depends on 0 variables.

Note that the same happens with formulas: formulas that depends on a 0-tuple of variables are sentences.

Nothing deep, but it heps keeping math clean.

Primo Petri
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  • As is sometimes the case, we hide details from the students. Constants not seen as functions is such a case. – Jason Oct 02 '22 at 21:14