The conditional sentence (or implication), is a compound sentence of the form
“if P then Q”
From a purely logical point of view, conditional sentences do not necessarily imply a cause and effect between P and Q, although generally there is a definite cause and effect. For example the conditional sentence
If 1 + 1 = 3, then pigs fly.
is a true conditional sentence, although the reader would have to think long and hard to find a cause and effect relation between 1 + 1 = 3 and flying pigs. A more common implication in mathematics would be
If a positive integer n is composite, then n has a prime divisor less than or equal to .
which provides an important cause and effect between P and Q. No doubt the reader has seen conditional sentences in Euclidean geometry, where the subject is explained through cause and effect implications of this type. The sentence, “If a polygon has three sides, then it is a triangle,” is a conditional sentence relating two important concepts in geometry.
The conditional statement P ⇒ Q can be visualized by the Euler (or Venn) diagram as drawn in Figure 1.4.
For example all polygons are triangles that we illustrate by the diagram in Figure 1.5.
The conditional sentence “if P, then Q” is best understood as a promise, where if the promise is kept, the conditional sentence is true, otherwise the sentence is false. As an illustration suppose your professor makes you the promise:
If pigs fly, then you will receive an A for the course.
The proposition is true since your professor has only promised an A if pigs fly, but since they do not, all bets are off. However, if you see a flying pig outside your classroom and your professor gives you a C, then you have reason to complain to your professor since the promise was broken, hence the proposition false.
The conditional sentence P ⇒ Q is often called an inference, and we say P implies Q. Another way of stating P ⇒ Q is to say P is a sufficient condition for Q, which means the truth of P is sufficient for the truth of Q. We also say that Q is a necessary condition for P, meaning the truth of Q necessarily follows from the truth of P.
The implication P ⇒ Q gives rise to three related implications shown in Table 1.22, one equivalent to the implication, the others not.
Table 1.22 Converse, inverse, contrapositive.
Implication | Converse | Inverse | Contrapositive |
P ⇒ Q | Q ⇒ P | ∼P ⇒ ∼ Q | ∼Q ⇒ ∼ P |
It is easy to show by truth tables that
A fundamental principle of logic, called the law of the syllogism, states:
“if P implies Q, and Q implies R, then P implies R”
which is equivalent to the compound conditional sentence
This sentence is a tautology since Table 1.23 shows all T's in column (5).
Table 1.23 Truth table verification of the syllogism.
(5) | |||||||
P | Q | R | |||||
T | T | T | T | T | T | T | T |
T | T | F | T | F | F | F | T |
T | F | T | F | T | T | F | T |
T | F | F | F | T | F | F | T |
F | T | T | T | T | T | T | T |
F | T | F | T | F | T | F | T |
F | F | T | T | T | T | T | T |
F | F | F | T | T | T | T | T |
The implication P ⇒ Q is true if either P is false or Q is true. Hence, we have the useful logical equivalence
which we verify by means of the truth table in Table 1.24.
Table 1.24 Equivalence of P ⇒ Q ≡ ∼ P ∨ Q.
P | Q | P ⇒ Q | ∼P | ∼P ∨ Q |
T | T | T | F | T |
T | F | F | F | F |
F | T | T | T | T |
F | F | T | T | T |
Also the negation of the implication P ⇒ Q is another useful equivalence that we obtain by one of De Morgan's laws:
In other words, an implication is (only) false when the premise is true and the conclusion false.
Compound sentences of the form
“P if and only if Q”
are fundamental in mathematics, which leads to the following definition.
Identify the assumption and conclusion in the following conditional sentences and tell if the implication is true or false.
Write the contrapositive of the conditional sentences in Problem 1.
Let P be the sentence “4 > 6,” Q the sentence “1 + 1 = 2,” and R the sentence “1 + 1 = 3.” What is the truth value of the following sentences?
Let P be the sentence “Jerry is richer than Mary,” Q is the sentence “Jerry is taller than Mary,” and R is the sentence “Mary is taller than Jerry.” For the following sentences, what can you conclude about Jerry and Mary if the given sentence is true?
Construct truth tables to verify the following logical equivalences.
Translate the given English language sentences to the form P ⇒ Q.
Without making a truth table, say why the following implications are true.
For P, Q, and R verify the distributive laws
One of the following sentences is logically equivalent to the implication P ⇒ Q. Which one is it?
For the two sentences not equivalent to P ⇒ Q, find examples illustrating this fact.
Is the following statement a tautology, a contradiction, or neither?
Show that the following five implications are all logically equivalent.
Is the statement
true for all truth values of P and Q, or is it false for all values, or is it sometimes true and sometimes false?
Is the statement
true for all truth values of P and Q, or is it false for all values, or is it sometimes true and sometimes false?
Find the negation of the following sentences.
“If N is an integer, then 2N is an even integer.”
write the converse, contrapositive, and inverse sentences.
Rewrite the sentence
in an equivalent form in which the symbol “⇒” does not occur.
The statement
can be read “If P is true, then P follows from any Q.” Is this a tautology, contradiction, or does its truth value depend on the truth or falsity of P and Q?
The statement
can be read “For any two sentences P and Q, it is always true that P implies Q or Q implies P.”
Is this a tautology, contradiction, or does its true value depend on the truth or falsity of P and Q?
Two‐valued (T and F) truth tables were basic in logic until 1921 when the Polish logician Jan Lukasiewicz (1878–1956) and American logician Emil Post (1897–1954) introduced n‐valued logical systems, where n is any integer greater than one. For example, sentences in a three‐valued logic might have values True, False, and Unknown. Three‐value logic is useful in computer science in database work. The truth tables for the AND, OR, and NOT connectives are given in Table 1.28.
Table 1.28 Three‐valued logic.
P | Q | P OR Q | P AND Q | NOT P |
True | True | True | True | False |
True | Unknown | True | Unknown | False |
True | False | True | False | False |
Unknown | True | True | Unknown | Unknown |
Unknown | Unknown | Unknown | Unknown | Unknown |
Unknown | False | Unknown | False | Unknown |
False | True | True | False | True |
False | Unknown | Unknown | False | True |
False | False | False | False | True |
From these connectives, derive the conditional P ⇒ Q and biconditional P ⇔ Q by drawing a truth table.
Write Modus Ponens and Modus Tollens as compound sentences, and show they are both tautologies.
Are the following two statements equivalent?
Figure 1.6 below shows the totality of 16 relations between 2 logical variables. One expression can be proven from another if it lies on an upward path from the first. For example
Verify a few of these implications using truth tables. The compound sentence PΔQ refers to the exclusive OR, which means either P or Q true but not both.
There is a wealth of information related to topics introduced in this section just waiting for curious minds. Try aiming your favorite search engine toward conditional connectives, biconditional, truth tables, and necessary and sufficient conditions.