Testing Logical Equivalence

by axellerate
Tags: equivalence, logical, testing
axellerate is offline
Jan30-13, 05:04 PM
P: 4
Hello hello, I'm not looking just for an answer per say, but am also wondering the thought process in solving problems such as the following:

Hopefully this doesn't take up too much of someones time.

Determine whether the following equations are logically equivalent:

1) (∃x)( P(x) → Q(x) )

2) (∀x)P(x) → (∃x)Q(x)
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micromass is offline
Jan30-13, 09:06 PM
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P: 16,561
Try to write the implication in terms of other connectives.
nomadreid is offline
Jan31-13, 01:09 AM
P: 497
After you have followed micromass's suggestion, the next step can make more sense to you if you contemplate an analogy (I stress that this is a way of thinking: it would not work as a formal proof)
all quantifier like a large "and",
existence quantifier like a large "or"
"and" like "intersection"
"or" like "union"
deMorgan Laws.
Formally, if you are not an intuitionist, you can try playing around with the equivalence between "[itex]\forall[/itex]x P" and "~[itex]\exists[/itex]x ~P", or between "[itex]\exists[/itex]x Q" and "~[itex]\forall[/itex]x ~Q"

(by the way, it's "per se")

MLP is offline
Jan31-13, 02:10 PM
P: 31

Testing Logical Equivalence

If you are familiar with how to determine a formula is logically true, then you can use the fact that formulas are logically equivalent just in case their biconditional is logically true. If there is an interpretation that makes the biconditional of (1) and (2) false, then they are not logically equivalent. If there is no such interpretation, then they are logically equivalent.

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