Material Conditional Not Logically Equivalent?

Question

Hi all this is my first time using this site so I hope I am presenting this properly, apologies if not. It is said that “a → b” and “¬(a and ¬ b)” are logically equivalent but I do not understand why/agree. However I agree that “a → b” and “¬a or b” are logically equivalent due to their truth tables being the same.

Here is why I do not understand (please point out where I made a mistake – I know I must have): The truth table for a → b is like this:

a   b   a → b

T   T   T
T   F   F
F   T   T
F   F   T       

I can understand this and am happy with it. However this is now where I do not agree with what is accepted. The truth table for ¬(a and ¬b):

a   b   ¬(a and ¬ b)

T   T   T
T   F   F
F   T  (F)
F   F   T       

Now I disagree on the third row and now I will show my working for that: ¬(a and ¬ b) in words is:

not(a and not b)        insert truth values for a and b
not(F and not T)        which simplifies to 
not(F and F)            which simplifies to 
not(T)                  which simplifies to 
F

This is why I think that it should be false on the third row which would disagree with the third row for a → b. I know I must have made a mistake/logical error but at the moment I cannot see it. I am new to studying logic so if anyone could point out what I have done wrong that would be great.

Answer 1

(F and F) is FALSE. An AND statement is true only in the case that both sides are true. You are confusing it with an equivalence statement, which is true when both sides have the same value.

Answer 2

Chris captured the mistake exactly: from not(F and F) you've concluded not(T), but (F and F) or (0 ∧ 0) is simply 0. A simple way, without using tables, to verify the truth of the equivalence is to notice that:

Fact. (¬φ ∨ ψ) ↔ ¬(φ ∧ ¬ψ) is true by De Morgan's Law.

And since → is defined in terms of ∨, (φ → ψ) is equivalent by that Fact to ¬(φ ∧ ¬ψ).