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Suppose, a FO language has two binary relation symbols $U_1, U_2$ and two binary function symbols $g_1, g_2.$ In each structure of this language, the function $g_1$ shall be defined only on pairs $(x_1, x_2)$ where $U_1(x_1, x_2)$, and the function $g_2$ is defined only on $(x_1, x_2)$ where $U_2(x_1, x_2).$ Can there be structures with partial functions? How to describe them?

Marina
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    The usual semantics for first-order logic does not allow for partial functions. However, it's easy to either whip up a modified semantics which does (together with a sound-and-complete proof system for same), or simply replace each partial function with a corresponding relation (which is a bit hacky but does the job). – Noah Schweber Sep 16 '21 at 16:16
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    There was a recent similar question about your concerns of partial functions in this site, hopefully it helps. – cinch Sep 16 '21 at 16:24
  • @NoahSchweber Thanks. I may use ternary relations instead of functions: $G_1(x_1, x_2, x_3) \equiv (g_1(x_1, x_2) = x_3).$ If it is correct, it would solve it for me. – Marina Sep 16 '21 at 16:44
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    @Marina Yes, and you can actually add an axiom that the ternary relation $G_i$ is actually a partial function. – Berci Sep 16 '21 at 18:00
  • @Berci It may be like this: $$G(x, y, z) & (z^\prime \neq z) \Rightarrow \lnot(G(x, y, z^\prime))$$ – Marina Sep 16 '21 at 19:42

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Partial functions could be presented as relations in structures. In fact, since constants are functions and functions are relations, sometimes we only have relation (predicate) symbols in first order languages which are called predicate languages. But since constants and functions have their own properties different from relations and we want to study objects in mathematical practice involving constants, functions and relations, in mathematical logic we usually put constant symbols and function symbols into the languages.

M. Logic
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