Logical truths, contradictions and disjunctive normal form

Truth tables and logical truths Let’s write out the truth table for (A ∨ ~A) ∧ (B ∨ ~B) on the board. Notice anything interesting about it?

Truth tables and logical truths No matter what line of the truth table we look at, this sentence always comes out as true. Which is to say that for every possible assignment of truth values to the atomic sentence letters, this sentence is true. A sentence that has this property is called a logical truth.

Logical truths As a precise definition: a sentence of sentence logic is called a logical truth if and only if it is true in every possible case, that is, for all possible assignments of truth values. (Sometimes people call logical truths tautologies.)

Contradictions Contradictions are the converse of logical truths. Logical truths are always true, no matter the assignment of truth values. Contradictions are always false, no matter the assignment of truth values. So the truth table for a contradiction has an ‘f’ in every line of the sentence’s column.

Contradictions More precisely: a sentence of sentence logic is called a contradiction if and only if it is false in every possible case, that is, for all possible assignments of truth values.

Checking for logical truth or contradiction To check whether or not a particular sentence of sentence logic is a logical truth or a contradiction (or neither), simply draw up a truth table for it, and check the outcome. If it’s always true, it’s a logical truth, if always false it’s a contradiction, and if neither it’s neither.

Checking for logical truth and contradiction Using partial truth tables is a good way to cut corners when doing this. For example: let’s check together whether or not the following sentence is a contradiction: ~((A ∨ ~B) ∧ (~C ∨ ~~~(D ∨ E))) ∧ ((A ∨ ~B) ∧ (~C ∨ ~~~(D ∨ E)))

Some more laws With these concepts in hand we can now give a couple more handy laws. The law of logically true conjunct (LTC): If X is any sentence and Y is a logical truth, then X ∧ Y is logically equivalent to X. And the law of contradictory disjunct (CD): If X is any sentence and Y is a contradiction, then X ∨ Y is logically equivalent to X.

Disjunctive normal form Now we’ve got a good grasp on logical equivalence, we can introduce another concept: disjunctive normal form. Let’s start with an example – let’s draw out the truth table for C ∨ ~~D.

Disjunctive normal form The table tells us that C ∨ ~~D is true in cases 1, 2 and 3, and false in case 4. I.e. it is true if C is true and D is true, or if C is true and D is false, or if C is false and D is true. Disjunctive normal form is just a way of expressing this information in sentence logic.

Disjunctive normal form And it’s very easy to do. Because it is easy to write ‘it is true if C is true and D is true, or if C is true and D is false, or if C is false and D is true’ in sentence logic. It’s just ‘(C ∧ D) ∨ (C ∧ ~D) ∨ (~C ∧ D)’. This is logically equivalent to our original sentence ‘C ∨ ~~D’. Whenever that sentence is true, ‘(C ∧ D) ∨ (C ∧ ~D) ∨ (~C ∧ D)’ will be true too.

Disjunctive normal form So our method is fairly easy. If we are asked to put a sentence into disjunctive normal form, we just write out the truth table, and then express the information contained in the truth table in the manner just specified. Since we can produce a truth table for any sentence, we can say that every sentence has a disjunctive normal form (with the special exception of contradictions).

Disjunctive normal form We can specify all this more precisely with the following. A sentence is in disjunctive normal form if it is a disjunction, the disjuncts of which are themselves conjunctions of sentence letters and negated sentence letters. (In this characterization we allow as a special case that a disjunction may have only one disjunct and a conjunction may have only one conjunct.)

Disjunctive normal form For any sentence, X, of sentence logic, the disjunctive normal form of X is given by a sentence Y if Y is in disjunctive normal form and is logically equivalent to X. Except for contradictions, the disjunctive normal form of a sentence is the sentence's truth table expressed in sentence logic.

Example Put ‘P ∨ Q’ into disjunctive normal form. Put ‘(C ∧ D) ∨ ~E’ into disjunctive normal form.