Logic - Monday May 5
We worked through some problems from the current homework and the quiz that was given on Friday. (I "downgraded" the quiz to an "in-class exercise".)
No new reading was assigned.
Reminder: participation logs due Friday.
In class I was temporarily stumped by two questions about possible ways of symbolizing "Every dog likes some cat". The best symbolization (using what I hope is obvious notation) is
(1) (x)(Dx => (Ey)(Cy . Lxy))
This is the symbolization I used in class. Joe B. asked if it could be symbolized as
(2) (x)(Ey)(Dx => (Cy . Lxy))
It turns out that (2) is logically equivalent to (1). It is an inferior translation, however. (Note that any given sentence has infinitely many logical equivalents.)
Proving that (1) and (2) are equivalent is quite difficult. (Try it if you don't believe me!)
Matt M. asked if the sentence could be symbolized as
(3) (x)(Ey)((Dx . Cy) => Lxy)
The answer is "no." Consider an interpretation with exactly one object in its domain -- something that is a dog and not a cat. This interpretation makes (3) true but (1) and (2) false.