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I recently had a test which asked various things about propositional logic: natural deduction, truth table's, analytic tableau's, ...

The very first question asked to create a truth table of φ and then give the amount of "elements" in Σ.

Given: Σ = {φ, p , q} and φ = ((p → (q ∨ r)) ↔ ¬(p ∧ ¬q))

Asked: create the truth table of Σ and give the amount of elements in Σ as a number.

I managed to create a truth table of Σ but I don't understand what they mean with "elements". I looked through my notes, slides and an additional book but could not find what I was suppose to do.

Could somebody give me an example? This is probably a basic question but I'm at a loss here and can't find anything online.

Note: for clarification, I assume the answer might be 3. But I am unsure if this is the case given that φ is a formula.

user3152069
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  • I'm guessing the goal is to identify whether $\phi$ is the same as p or q, in which case the set has 2 elements. – TomKern Nov 05 '22 at 23:32
  • Hello @TomKern thanks for the reply, could you elaborate a bit on how you arrived to that conclusion? If I understand correctly does this mean I need to count the amount of unique atomic proposition letters? I have a hard time understanding how this relates to a truth table given the question was grouped in with it. – user3152069 Nov 06 '22 at 00:48
  • Check to see if $\phi$ is true exactly when p is true. if so, they are the same. – TomKern Nov 06 '22 at 02:11
  • @TomKern I remember from the truth table that was not case, so in that case I guess the answer is 3? I still feel a bit bewilderd, it feels like an obvious answer so it feels wrong that the reasoning would be that simple but perhaps it is. – user3152069 Nov 06 '22 at 15:06

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