Is it Appropriate to Use Big Vee and Big Wedge for Infinitary Logic?

[jostpuur]

With finite amount of sets unions and intersections can be written as

$$A_1\cup A_2\cup\cdots\cup A_n$$

and

$$A_1\cap A_2\cap\cdots \cap A_n.$$

If we have an arbitrary collection of sets, $$(A_i)_{i\in I}$$, then we can still write unions and intersections as

$$\bigcup_{i\in I} A_i$$

and

$$\bigcap_{i\in I} A_i.$$

If we have a finite amount of logical statements, then logical "or" and "and" of them can be written as

$$A_1 \lor A_2\lor\cdots \lor A_n$$

and

$$A_1 \land A_2\land\cdots \land A_n.$$

I don't think I've ever seen anything being done with arbitrary collections of logical statements. Have you? Is it okey to write something like this:

$$\bigvee_{i\in I} A_i$$

and

$$\bigwedge_{i\in I} A_i?$$

 

[Werg22]

If I is finite, this is legal. If I is infinite however, I don't know of any logic where this is legal, albeit meaning is clear, i.e. at least one proposition in I is true, all propositions in I are true.

 

[JSuarez]

See "Infinitary Logic" Here:

http://plato.stanford.edu/entries/logic-infinitary/"