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On p126 in §3. Extensions by Definitions in VIII Syntactic Interpretations and Normal Forms In Ebbinghaus' Mathematical Logic: $S$ is a (non-logical) symbol set

3.1 Definition. Let $\Phi$ be a set of $S$-sentences.

(a) Suppose $P \notin S$ is an $n$-ary relation symbol and $\phi_P(v_0, ... , v_{n-1})$ an $S$-formula. Then we say that $$ \forall v_0, .... \forall v_{n-1} \quad (P v_0 ... n_{n-1} \leftrightarrow \phi_P(v_0, ... , v_{n-1})) $$ is an $S$-definition of $P$ in $\Phi$.

  • How is $ \forall v_0, .... \forall v_{n-1} \quad (P v_0 ... n_{n-1} \leftrightarrow \phi_P(v_0, ... , v_{n-1})) $ a $S$-sentence or even a $S$-formula?

  • $P v_0 ... n_{n-1}$ is on the left hand side of $\leftrightarrow$. Does that assume $P v_0 ... n_{n-1}$ to be a $S$-formula? But $P \notin S$, so how can $P v_0 ... n_{n-1}$ be a $S$-formula?

Thanks.

Tim
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1 Answers1

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To save some writing, let's let $\sigma$ stand for $\forall v_0, \ldots, \forall v_{n-1} (Pv_0, \ldots, v_{n-1} \leftrightarrow \phi_P(v_0, \ldots, v_{n-1}))$.

You're correct that $\sigma$ is not an $S$-formula, because $\sigma$ involves the symbol $P$, which is not in $S$. On the other hand, $\sigma$ is an $(S \cup \{P\})$-sentence. That's kind of the point here: $\sigma$ is telling you that the symbol $P$, which is not in $S$, is equivalent to an $S$-formula. The terminology "$S$-definition" refers to the fact that $\sigma$ defines $P$ in terms of $S$, it does not mean that $\sigma$ itself is an $S$-sentence.

Chris Eagle
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    Thanks. What do you mean by "symbol , which is not in , is equivalent to an -formula"? Which " -formula"? – Tim Aug 19 '20 at 16:23
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    The $S$-formula $\phi_P$, which appears on the right side of the equivalence in $\sigma$. – Chris Eagle Aug 19 '20 at 16:25
  • "an -definition of in Φ" sounds to say that it is in Φ and therefore is an S-sentence. – Tim Aug 19 '20 at 16:31
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    I agree the naming is a bit unfortunate. The "in $\Phi$" part appears in the next two parts of the definition (for $S$-definitions of functions and constants), and refers to $\Phi$ implying that the $S$-formulas $\phi_f$ and $\phi_c$ have the requisite properties (namely defining a function or having a unique witness, respectively). – Chris Eagle Aug 19 '20 at 16:35
  • Is it more accurate to say that (0,...,−1)) is an -definition of in Φ, than to say that is? – Tim Aug 19 '20 at 19:35
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    I've heard both. While I personally prefer to call $\phi_P$ the $S$-definition, it looks like the book you're referring to is calling $\sigma$ an $S$-definition. – Chris Eagle Aug 19 '20 at 19:37
  • Do you see the concept of Syntactic Interpretation in some other mathematical logic books? – Tim Aug 19 '20 at 19:39
  • Is an -definition of in Φ an interpretation of symbol P as an S'-sentence? (as part of a syntactic interpretation of S' in S' itself? c.f. p120) – Tim Aug 19 '20 at 21:13