Friday, November 28, 2014
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logic, a spice rack, and some other things
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3 comments:
Thanks very nice blog!
I noticed that there’s a possible bug in your software. For suppose (P→Q) is true. (P→Q) is equivalent to (¬P∨Q∨¬P∧Q) which is also equivalent to (P∧¬Q→¬P∧Q). Suppose we use Modus Ponen on (P∧¬Q→¬P∧Q). Then it seems we get a contradiction unless both P∧¬Q and ¬P∧Q are false.
The modus ponens argument
P&~Q->~P&Q
P&~Q
,',
~P&Q
is valid, but only because the two premises are inconsistent; they can't both be true. The software notes this if you use the "therefore" operator (",',") in the argument. The premises lead to a contradiction, and anything follows from a contradiction. In other words, all arguments with contradictory premises are valid. This seems like a problem, but you have to remember that not all valid arguments are sound; an argument doesn't prove anything unless its premises are true, and contradictions are always false, so an argument with contradictory premises will never be sound.
It is not surprising that the two premises are inconsistent because, as you argue, (P∧¬Q→¬P∧Q) is equivalent to (P→Q). (P→Q) is equivalent to ¬P∨Q, which is equivalent to ¬(P∧¬Q), which is the negation of P∧¬Q.
Is there anything that the software appears to be doing wrong? From your comment, I can't tell if the problem is with the software or with the logical argument itself.
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