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I understand that a truth table can prove the Distributive Law as a Logical Equivalence:

p V (q ^ r) <=> (p V q) ^ (p V r)

However, this makes no intuitive sense to me. Here is the contradiction I see: if p and q are both true, then wouldn't that result in p ^ q? that can work with the expression on the right, but that doesn't seem to work with the expression on the left. As I see it (and there must be something wrong with how I see it), either only p is true, or only q and r are true, according to the left expression.

Is anyone able to explain to me how this makes sense?

Let me know if I need to clarify anything.

John Bollinger
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Tyabetus
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2 Answers2

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The left hand equation is saying that either p is true or q and r are true. It does not say either p and only p is true, or q and r are only true.

For your example, p^q=> p (it also implies q, and pvq), which makes both sides true.

For example, in English the first equation says that at least one of the following is true

  • Pablo can swim OR
  • Quincy and Reginald can swim

If all three of them are true the statement is also true.

The one on the right says both of the following are true

  • Pablo or Quincy can swim AND
  • Pablo or Reginald can swim

If we have Pablo and Quincy can swim (your example), then we see that both statements hold. Pablo can swim so the first expression works because of its first clause. For the second expression since Pablo can swim both of its parts are true so it also holds.

Adam Martin
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I suspect you are using a colloquial meaning of "or", in the sense of "one or the other, but not both." E.g., "Choose the red pen or the blue pen." The meaning of "or" in formal logic is "at least one is true." In your hypothetical, certainly p^q, but the the values of q & r are irrelevant when p.

Molhan Treeney
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