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A set $\Gamma$ of (first-order) formulas is said to contain instances when, for every existentially quantified formula $\exists x A$ of the language, the following holds:

There is some term $t$ in the language s.t. $\Gamma \models \exists x A \rightarrow [A]\frac{t}{x}$.

Now, let $\frak{I}, \sigma$ be such that for every $d \in D$ there is a $t \in \mathcal{T}_\mathcal{S}$, s.t. $\varphi^\sigma_\frak{I}(t) = d$. Define $$\Gamma := \{A \in \mathcal{F}_\mathcal{S}~|~\frak{I}, \sigma \models A\}$$

My question is: How do you use the information on $\frak{I}, \sigma$ above to prove that $\Gamma$ contains instances?

Note on the notation:

  • $D$ stands for a domain, and $d$ for an object of the domain.

  • $\frak{I}$ stands for an interpretation (structure);

  • $\sigma$ stands for a variable assignment;

  • $\mathcal{F}_\mathcal{S}$ stands for the set of formulas in the language $\mathcal{S}$.

paul bernays
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  • Do you know what the symbols mean? – Mauro ALLEGRANZA Mar 08 '21 at 13:52
  • What is $\varphi$? The def IMO means that for every object $d$ in the domain $D$ of the interpretation there is a term of the language that is a "name" $t$ for it. – Mauro ALLEGRANZA Mar 08 '21 at 13:57
  • If so, the result is obvious: if formula $\exists x A \to A[t/x]$ is true, when the antecedent is true, also the consequent is. And thus, there must be a suitable term $t$. – Mauro ALLEGRANZA Mar 08 '21 at 13:59
  • I just added a remark on the notation. – paul bernays Mar 08 '21 at 14:03
  • @MauroALLEGRANZA You might be right in saying it's obvious, but then it's not clear to me exactly at which point of the proof we require the information that for every object in the domain there is a term referring to it... – paul bernays Mar 08 '21 at 14:10
  • Because you have to prove that the formula $\exists x A \to A [t/x]$ holds. If in the language there are no "names" for object, there is no way to produce $A[t/x]$. – Mauro ALLEGRANZA Mar 08 '21 at 14:11
  • @MauroALLEGRANZA I assumed it was the other way round. To produce $A[t/x]$ for an arbitrary $t$, every term would need to refer to an object. Said otherwise: Let's say $t$ refers to an object $d$, but some other object $d'$ has no name. Why would we then fail to produce $A[t/x]$? – paul bernays Mar 08 '21 at 14:17
  • The issue is: assume that $\exists x A$ holds in $I$. We can image a situation where there is no name for some object: consider e.g. a language that has only the names for the natural number. We can express the formula $\exists z (z \times z= 2)$, but what term $t$ we can use to write $(z \times z = 2)[t/z]$? – Mauro ALLEGRANZA Mar 08 '21 at 15:55

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