Logical Uncertainty

1crowJan’s working on character development, and I’m not feeling very poetic, so here’s some logic.

In formal logic the conditional statement “if A, then B” specifically means: if A is true, then B is true; and if A is false, then B can be either true or false.  Thing is, spoken language isn’t formal, and spoken conditional statements don’t necessarily translate to logical conditional statements.  Consider the following sentences, and their interpretations:

  • “If I hold the dog’s leash, then he won’t run away.”  Ignoring the possibility that the dog could slip his collar, and run away, this means that holding the leash will guarantee the dog’s presence, but not holding the leash won’t ensure anything; the dog might stay, or not.  Logically this form of “if A, then B” breaks down like this:
    • A = false, and B = false is possible
    • A = true, and B = false is not possible
    • A = false, and B = true is possible
    • A = true, and B = true is possible
  • “If dad comes home early, then we can get ice cream.”  As used by my children, this means that ice cream will happen if I come home early, but ice cream won’t happen if I don’t come home early.  This is actually a logical biconditional.
    • A = false, and B = false is possible
    • A = true, and B = false is not possible
    • A = false, and B = true is not possible
    • A = true, and B = true is possible
  • “If I go to bed, then the sun will rise tomorrow.”  As any frat boy will tell you, the sun will rise whether you go to bed or not.  The first clause, “I go to bed”, is unnecessary information.
    • A = false, and B = false is not possible
    • A = true, and B = false is not possible
    • A = false, and B = true is possible
    • A = true, and B = true is possible
  • “If trees are made of wood, then the Pope is Catholic.”  Trees are made of wood, and the Pope is Catholic; these are both true by definition.  Logically this is just a conjunction of two truisms; neither can be false.
    • A = false, and B = false is not possible
    • A = true, and B = false is not possible
    • A = false, and B = true is not possible
    • A = true, and B = true is possible

Each of these linguistic conditional statements have two conditions in common.  The first clause being true when the second clause is false is not possible, and both clauses being true is possible.  The other two conditions can vary.  In fact, every conceivable arrangement possibility, of the remaining clause conditions, is represented.  In the same way that the logical conditional was uncertain about the truth of B when A was false, the linguistic conditional is uncertain about the possibility of A and B being false simultaneously, and A being false when B is true.  Essentially, the nature of the uncertainty is uncertain.

For the sake of continuity, let’s define the uncertainty of linguistic conditionals in logical terms.  First we’ll name the truth relationship between A and B; we’ll call it x, and write it (A,B)x.  Then, we’ll define some variables that represent the possibility of the four truth conditions of x.

  • x0 represents the possibility of A and B being false at the same time.
  • x1 represents the possibility of A being true when B is false.
  • x2 represents the possibility of A being false when B is true.
  • x3 represents the possibility of A and B being true at the same time.

With these we can simply say x is possible if x1 is false and x3 is true.  Or, in logical terms:

The possibility of (A,B)x is (NOT x1) AND x3

~Nick

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