Notation thoughts on logic

  • Thread starter jostpuur
  • Start date
  • #1
2,111
18
With finite amount of sets unions and intersections can be written as

[tex]
A_1\cup A_2\cup\cdots\cup A_n
[/tex]

and

[tex]
A_1\cap A_2\cap\cdots \cap A_n.
[/tex]

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

[tex]
\bigcup_{i\in I} A_i
[/tex]

and

[tex]
\bigcap_{i\in I} A_i.
[/tex]

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

[tex]
A_1 \lor A_2\lor\cdots \lor A_n
[/tex]

and

[tex]
A_1 \land A_2\land\cdots \land A_n.
[/tex]

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:

[tex]
\bigvee_{i\in I} A_i
[/tex]

and

[tex]
\bigwedge_{i\in I} A_i?
[/tex]
 

Answers and Replies

  • #2
1,425
1
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.
 
  • #3
402
1
See "Infinitary Logic" Here:

http://plato.stanford.edu/entries/logic-infinitary/" [Broken]
 
Last edited by a moderator:

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