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Notation thoughts on logic

  1. Jan 22, 2010 #1
    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]
     
  2. jcsd
  3. Jan 22, 2010 #2
    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.
     
  4. Jan 22, 2010 #3
    See "Infinitary Logic" Here:

    http://plato.stanford.edu/entries/logic-infinitary/" [Broken]
     
    Last edited by a moderator: May 4, 2017
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