Truth table proofs
Truth tables are used to show all possible values that a given logical expression might take.
Definition of AND
The following gives the definition of the logical AND operation.
|A||B||A ∧ B|
Proof of one of De Morgan's laws
A truth table can also be used to prove a logical identity. The following proves De Morgan's law that ¬ (A ∧ B) is equivalent to (¬ A) ∨ (¬ B). Notice that the truth values for these (the last two columns in the table) are always the same.
|A||B||A ∧ B||¬ (A ∧ B)||(¬ A) ∨ (¬ B)|
Disproof of a logical fallacy
Truth tables can also be used to disprove a logical fallacy. For example, one fallacy is to assume the converse of an implication holds. If you know that (A → B) is true, and you know that B is true, the fallacy would be to then conclude that A must also be true. Note that the statement - "if an animal is a mammal then it is also a vertebrate" - is true. If we have an animal that is a vertebrate (for example, a dog), it would be a fallacy to now conclude that the animal must also be a mammal.
We will show that this is a logical fallacy - regardless of the statements A and B, just because we know A → B is true, we should not in general assume the converse is also true. First we remind you that (A → B) is equivalent to ¬ A ∨ B. What we want to look at is the claim: ((A → B) ∧ B) → A. If this expression is always true, then we could rely on the converse to always be used. Let us first simplify the expression. It is equivalent to ((¬ A ∨ B) ∧ B) → A. This is equivalent to ¬ ((¬ A ∨ B) ∧ B) ∨ A. We can keep this formula in mind as we evaluate it's truth value for each possible value of A and B.
Consider this truth table.
|A||B||A → B||((A → B) ∧ B) → A|
In the second row, we see that the last column is false. This is where we should expect to see a problem - the place where B is true, but we should not be able to conclude that A must also be true. We have shown that we cannot always apply the converse.
You will be assigned a logical identity to prove and a logical fallacy to disprove. For each, you are permitted and encouraged to simplify the expressions to make it easier to evaluate the expressions. You then write a truth table for each part - proving the logical identity, and disproving the logical fallacy. Include a similar amount of explanation as above.
Unless otherwise indicated by your instructor, you can submit your solutions in whatever format you like - word document, pictures of it worked out on paper, physical paper, OneNote notebook link. If you send a link to an electronic document, make sure it is set so it is shared with your instructor.
Pass rating check Each part must be correct to earn a 1/1 on that part. The total for the problems would be 2 points.