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I am doing some research into logic, I have gotten far into Propositional Calculus and its generalizations, my issue comes when I go into Predicate Calculus.

I am trying to look into it in a more generalized sense and not the specific one we are using most commonly. So for quantifiers I consider things beyond $\forall$ (and by extension $\exists$)

I understand we can subdivide the "alphabet" of GFOL (Generalized First Order Logic) accordingly

Variables $V_i$, Function $F_i^k$, Predicates $P_i^k$, Operators/Connectors, $O_i^k$ and Quantifiers, this is where I get uncertain, does it have an arity? Imma assume here no but please correct me! $Q_j$.

I know that terms are inductively defiend as this

  • $V_i$ are Terms
  • If $T_i$ is a term, then $F_i^k(T_1,\ldots,T_k)$ is a term.

and for WFF they are also defined inductively. Let $T_i$ be terms as before and $W_i$ be a WFF.

  • $P^k_i(T_1,\ldots,T_k)$ is a WFF.
  • $O_i^k(W_1,\ldots,W_k)$ is a WFF
  • $Q_i V_j W_k$ is a formula

I feel secure with the Term one, but WFF one I do not as I am not certain how to generalize quantifiers, if they are all unary it would be done but are they always unary in FOL?

I next consider substitution for these things, that is $A[B/x]$. I understand the general idea of it but looking at This it only speaks about existential quantifier, but how would I generalize it to a general quantifier $Q_i$? Are there any good sources for this?

Thank you in advance.

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    $\begingroup$ I've never seen a treatment in which terms are considered atomic formulae. The point of formulae is that they are the sorts of things which might be evaluated to have truth values. Quantifiers are always unary in FOL, and nothing about substitution cares whether the quantifier is universal or existential, only whether the variable is bound or free. $\endgroup$ – Malice Vidrine Nov 17 '17 at 20:30
  • $\begingroup$ Thank you, I made a mistake there, it was meant to be a predicate on it and operator on the second. But the question is does it care wether quantifiers are beyond those 2? $\endgroup$ – Zelos Malum Nov 18 '17 at 0:28
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    $\begingroup$ I think the "Generalized Quantifier" entry from SEP might interest you. $\endgroup$ – Poypoyan Nov 18 '17 at 2:18
  • $\begingroup$ @Poypoyan and do i did! $\endgroup$ – Zelos Malum Nov 19 '17 at 1:08

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