TITLE: Infinite Disjunctions and Conjunctions
QUESTION [7 upvotes]: While doing work on propositional logic (namely, proving the Generalized De Morgan's Laws), I found myself wondering why precisely an infinite conjunction or disjunction are not permitted, due to the fact that in predicate logic (also a finitary logic), quantification over domains of discourse with infinite size are allowed.  Using the definition of the existential and universal quantifiers as:
$ \exists( x \in X) P(x) \Leftrightarrow \bigvee_{x_i \in X}{P(x_i)} $ and $\forall (x \in X)P(x) \Leftrightarrow \bigwedge_{x_i \in X}{P(x_i)} $, along with the duality of the quantifiers (namely $ \neg\exists x P \Leftrightarrow \forall x \neg P $ and $ \neg\forall x P \Leftrightarrow \exists x \neg P $), does this not imply that 1) infinite conjunctions and disjunctions must be allowed in order to quantify over infinite domains and 2), at least in predicate logic, De Morgan's Laws hold for the infinite case?
Thanks

REPLY [4 votes]: This is not really an answer, but I thought I would talk a little bit about one class of logics that are related to the question.
One class of logics which allow infinite conjunctions/disjunctions are Infinitary Logics.  These logics specify the number of conjunctions/disjunctions allowed, and also the number of quantifiers allowed.  In short, given infinite cardinals $\kappa, \lambda$ the logic $L_{\kappa\lambda}$ allows for the following infinitary constructions:

If $\delta < \kappa$ and $\{ \phi_\xi : \xi < \delta \}$ is a family of formulas, then $\bigvee_{\xi < \delta} \phi_\xi$ and $\bigwedge_{\xi < \delta} \phi_\xi$ are formulas;
If $\epsilon < \lambda$ and $\{ x_\xi : \xi < \epsilon \}$ is a family of individual variables, then given any formula $\phi$ the expressions $( \forall x_0  \ldots x_\xi \ldots ) \phi$ and $( \exists x_0  \ldots x_\xi \ldots ) \phi$ are formulas.

(In this formulation, $L_{\omega\omega}$ is the standard (finitary) first-order logic.)  (There are also "unbounded versions" where $\kappa$ or $\lambda$ may be $\infty$, meaning unrestricted lengths of conjunctions/disjunctions or quantifiers, respectively.)
The infinitary de Morgan's Laws should be theorems in any formal axiomatization, and are likely to be taken as axioms themselves.  However, these logics are often unsuitable from a "practical" standpoint.

Only $L_{\omega\omega}$ and $L_{\omega_1\omega}$ satisfy the following Completeness property: if a sentence $\phi$ is true in every model, then $\phi$ is a theorem.
Given $\kappa > \omega$, natural extensions of the Compactness Theorem for $L_{\kappa\kappa}$ fails except when $\kappa$ is a large cardinal.