D (I) T, & L L
October ,
Tautologies, contradictions and contingencies
Consider the truth table of the following formula:
p (p ∨ p) ()
If you look at the final column, you will notice that the truth value of the whole formula depends on the way a truth value is assigned to p: the whole formula is true if p is true and false if p is false. Contrast the truth table of (p ∨ p) in () with the truth table of (p ∨ ¬p) below:
p ¬p (p ∨ ¬p) ()
If you look at the final column, you will notice that the truth value of the whole formula does not depend on the way a truth value is assigned to p. The formula is always true because of the meaning of the connectives. Finally, consider the truth table table of (p ∧ ¬p):
p ¬p (p ∧ ¬p) ()
This time the formula is always false no matter what truth value p has. Tautology A statement is called a tautology if the final column in its truth table contains only ’s. Contradiction A statement is called a contradiction if the final column in its truth table contains only ’s. Contingency A statement is called a contingency or contingent if the final column in its truth table contains both ’s and ’s. Let’s consider some examples from the book. Can you figure out which of the following sentences are tautologies, which are contradictions and which contingencies? Hint: the answer is the same for all the formulas with a single row.
() a. (p ∨ ¬p), (p → p), (p → (q → p)), ¬(p ∧ ¬p) b. ¬(p ∨ ¬p), (p ∧ ¬p) c. p, ((p → q) → p)
Substitutions salva veritate The truth value of any tautology or contradiction does not depend on the truth value of its atomic statements. It follows that we can freely substitute for the atomic statements of any tautology or contradiction without affecting the truth value of the whole. If we substitute (q → r) for p in (p ∨ ¬p), we still have a tautology:
q r (q → r) ¬(q → r) ((q → r) ∨ ¬(q → r)) ()
A sometimes quick way to tell whether a given statement is a tautology: try to construct a line in a truth table whose final value is . This is an application of a general reasoning strategy called reductio ad absurdum (Latin for “re- duced to an absurdity”). A proof by reductio consists on deriving a contradictory result from a given assumption and concluding that, since the assumption led to such a result, it has to be wrong. The method works quite well with complicated statements. Consider, for instance the following one:
() ((p → (q → p))
We can prove it is a tautology. Here’s how. . Assume ((p → (q → p)) is not a tautology.
. Then some rows in its truth-table must end in s. . Given the truth table of “→”, that means that in such a row, p has to be true and (q → p) false. . Given the truth table of the implication, that means that in such a row q is true and p is false. . But then we have concluded that p is true and false in some row! . Since p can’t be both true and false in the same row, we conclude that the assumption that ((p → (q → p)) is not a tautology must be wrong.
. ∴ It must be a tautology.
Logical equivalence, logical consequence and laws
. Logical equivalence
Two statements are logically equivalent if they have the same truth-values for all possible values for their atomic statements. Here’s an example. ¬(p ∨ q) is logically equivalent to (¬p ∧ ¬q), as you can see in the following truth-tables:
p q ¬p ¬q ¬(p ∨ q) ()
p q ¬p ¬q (¬p ∧ ¬q) ()
If we connect two logically equivalent statements by means of a biconditional, we end up with a tautology:
p q ¬p ¬q ¬(p ∨ q) (¬p ∧ ¬q) ¬(p ∨ q) ↔ (¬p ∧ ¬q) ()
Logically equivalent statements can freely replace one another in any statement without affecting its truth value. It is customary to write “P ⇔ Q” to denote logical equivalence between any two statements P, Q.
. Logical consequence
A statement Q is a logical consequence of a statement P if whenever the final row of P’s truth-table has a , the final row of Q’s truth table also has a . Here’s an example:q is a logical consequence of ((p ∨ q) ∧ ¬p)
p ¬p q (p ∨ q) ((p ∨ q) ∧ ¬p) ()
If Q is a logical consequence of P, then “P → Q” is a tautology:
p ¬p q (p ∨ q) ((p ∨ q) ∧ ¬p) (((p ∨ q) ∧ ¬p)) → q ()
It is customary to write “P ⇒ Q” to say that Q is a logical consequence of P. Logical equivalence preserves truth and falsity. Logical consequence preserves truth but not falsity. If the antecedent if a tautologous conditional is true, then the consequent must also be true, but if the antecedent is false, the consequent may be either true or false. We have said that if P is logically equivalent to Q, then they can be freely substituted in any statement without affecting its truth value. But if Q is a logical consequence of P, we cannot in general substitute Q for P in a larger formula without affecting its truth value. We can verify that (p ∧ ¬p) ⇒ p, but let’s convince ourselves that we cannot substitute p for (p ∧ ¬p) in ((p ∧ ¬p) → q) without affecting its truth value.
. Laws of statement logic
It is handy to have a small number of logical equivalences from which all others can be derived. The book (page ) lists some. It is possible to verify them. Consider, for instance the following equivalence:
() (p → q) ⇔ (¬p ∨ q)
If the statements (p → q) and (¬p∨q) are logically equivalent, the statement (p → q) ↔ (¬p∨q) should be a tautology. Let’s check that out. Since logically equivalent statements can be substituted for each other salva veritate, we can use the laws of statement logic to transform a complex statement into a simpler one. Here’s an example from the book:
() . (p → (¬q ∨ p)) . (¬p ∨ (¬q ∨ p)) (Conditional Law) . ((¬q ∨ p) ∨ ¬p))(Commutative Law) . (¬q ∨ (p ∨ ¬p)) (Associative Law) . ¬q ∨ T (Complement Law) . T (Identity Law)
Substitution of logical equivalents statements can be carried out without affecting the truth value of the formula containing them. The principle is sometimes called the Rule of Substitution.