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Semantic structure of FOL is defined as follows: $\mathcal{M}$ is a structure of $\mathcal{L}$ if and only if $\mathcal{M} = \left(M, m_f, m_p\right)$ for some $M, m_f, m_p$ such that (1) $M \neq \emptyset$; (2) for each $n$-ary function symbol $f$ in $\mathcal{L}$, $m_f\left(f\right)$ is some map $F: M^{n} \to M$; (3) for each $n$-ary predicate symbol $p$, $m_p\left(p\right)$ is some set $S \subseteq M^{n}$. Specifically, $m_{p}\left(=\right) = \left\{\left(s, s\right) \vert s \in M\right\}$.

Let $P_{0}$ be an atomic predicate that takes zero argument. What is $m_{p}\left(P_{0}\right)$? Further, how to interpret $\mathcal{M} \vDash P_{0} \left[\sigma\right]$?

Ziqi Fan
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  • If $P_0$is an atomic predicate with zero arguments, it is a propositional variable and thus its interpretation will be a truth value. – Mauro ALLEGRANZA Apr 12 '23 at 12:41
  • @MauroALLEGRANZA Is this understanding forced to true or false? After all, we cannot tell from the formal relation $\in$. – Ziqi Fan Apr 12 '23 at 12:42
  • This depends on the semantics... if it is bi-valent, then Yes. – Mauro ALLEGRANZA Apr 12 '23 at 12:49
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    See this post: "In certain cases, it can be convenient to introduce 0-ary relations. A nullary relation symbol is therefore interpreted in a structure either by the true or by the false. These nullary relations are not very interesting in and of themselves, but occur naturally as a calculation tool: If a formula in n free variables represents an n-ary relation (the set of n-tuples that satisfy it), a statement represents a 0-ary relation, either the true or the false." – Mauro ALLEGRANZA Apr 13 '23 at 09:32

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