Difference between revisions of "Truth table proofs"
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==Disproof of a logical fallacy== | ==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. | + | Truth tables can also be used to disprove a logical fallacy. For example, [https://en.wikipedia.org/wiki/Affirming_the_consequent 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. |
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| + | 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. | Consider this truth table. | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| − | ! A !! B !! A → B ( | + | ! A !! B !! A → B !! ((A → B) ∧ B) → A |
|- | |- | ||
| − | | false || false || true | + | | false || false || true || true |
|- | |- | ||
| − | | false || true || true | + | | false || true || true || false |
|- | |- | ||
| − | | true || false || false | + | | true || false || false || true |
| − | |- | + | |- |
| − | | true || true || true | + | | true || true || true || true |
|} | |} | ||
| − | In the | + | 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. |
| − | |||
=Assignment= | =Assignment= | ||
Revision as of 15:20, 29 August 2022
Truth tables are used to show all possible values that a given logical expression might take.
Contents
Examples
Definition of AND
The following gives the definition of the logical AND operation.
| A | B | A ∧ B |
|---|---|---|
| false | false | false |
| false | true | false |
| true | false | false |
| true | true | true |
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) |
|---|---|---|---|---|
| false | false | false | true | true |
| false | true | false | true | true |
| true | false | false | true | true |
| true | true | true | false | false |
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 |
|---|---|---|---|
| false | false | true | true |
| false | true | true | false |
| true | false | false | true |
| true | true | true | true |
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.
