3.4 THE CONDITIONAL & BICONDITIONAL

Definition. Any statement that can be put in the form “If 풑, then 풒”, where 푝 and 푞 are basic statements, is called a conditional statement and is written symbolically as 풑 → 풒. The component 푝 is called the premise (or antecedent) and the component 푞 is called the conclusion (or consequent). A conditional statement is also called an implication and can be rewritten in the form “풑 implies 풒.” Note how the arrow follows the logical direction of the implication expressed by the statement.

Example. Consider the following (true) conditional statement:

“Numbers that are divisible by 2 are even.”

This is a conditional statement of the form 푝 → 푞 with premise 푝: “The number is divisible by 2” and conclusion 푞: “The number is even.”

The above statement could also be written in other logically equivalent ways, such as:

“If a number is divisible by 2, then it is even.” “The fact that the number is divisible by 2 implies that it is even.” “A number is divisible by 2 so / hence / therefore it is even.” “All numbers that are divisible by 2 are even.” “Every number that is divisible by 2 is even” “A number is even provided it is divisible by 2.” “A number is even if it is divisible by 2.” “A number is even whenever it divisible by 2.” “A number is odd (or not even), unless it is divisible by 2.”

Words like so, hence, therefore, and then are straightforward conditional connectives that are immediately followed by the conclusion of the conditional statement (푞). Other connectives, however, such as provided, whenever, and unless are logically trickier as they logically reverse the direction of the implication. In these cases, the premise (푝) and the conclusion (푞) are swapped within the sentence! The connective unless is particularly tricky as it also negates the conclusion 푞 of the conditional statement (check that the form “~푞 unless 푝” is equivalent to the conditional “If 푝, then 푞”). You must be careful when you read conditional statements to identify the premise and conclusion properly. The rule of thumb is to always revert back to the explicit conditional form “If p , then q .” Basic Truth Table for the Conditional

To understand the basic truth table for conditional statements, let’s first examine the truth value of the following simple conditional statement:

“All rental boats in Venice are gondolas.” or “If it’s a rental boat in Venice then it’s a gondola.”

Symbolically, we then have 푝 → 푞, where the premise 푝 is basic statement “It’s a rental boat in Venice” and the conclusion 푞 is the basic statement “It’s a gondola.”

All logical possibilities follow the usual four rows of a truth table for a compound statement with two components 푝 and 푞.

. Row 1 If both 풑 and 풒 are true (it’s a rental boat in Venice and it’s a gondola) then the overall statement is clearly true since the implication is not violated.

. Row 2 If 풑 is true and 풒 is false (it’s a rental boat in Venice and it’s NOT a gondola), then the overall statement is false since the implication has been violated!

. Rows 3 & 4 These two rows are a little counter-intuitive. However, in both rows the premise 풑 is assumed false (it’s NOT a rental boat in Venice), so the conclusion becomes automatically irrelevant as far as the logical implication is concerned. In other words, once the premise is false the implication is no longer applicable. This suggests that the truth value for the overall statement should be true in these cases, and not false.

Below is the basic truth table for the conditional statement “If 푝, then 푞.”

풑 풒 풑 → 풒 T T T T F F F T T F F T

Basic Truth Table for the Conditional Statement “If 푝 then 푞” Notes . The conditional 푝 → 푞 is only false when its premise 풑 is true and its conclusion 풒 is false. . The conditional 푝 → 푞 is always true when its conclusion 풒 is true. . The conditional 푝 → 푞 is always true when its premise 풑 is false. This means that absurd conditional statements containing premises that are obviously false are technically true. For example, statements such as “If 4 is odd, then 5 is even” or “All tricycles with four wheels are red ” are true according to the truth table above! Such sentences are said to be vacuously true. . The logical conditional connective “implies” is not to be confused with the word “causes”. While causal conditional statements are common (e.g. “When water reaches 100 degrees Celsius, it boils”), they are nevertheless not the same as the weaker material implication used in propositional logic that follows the truth table above. In other words, to say “푝 implies 푞” is not the same as saying “푞 because 푝”.

Principles from set theory can be put to use in a powerful way to justify the basic truth table for conditionals. Consider the conditional statement “If 푝, then 푞.” Each basic statement 푝 and 푞 can be, respectively, written in set notation as 푥 ∈ 푃 and 푥 ∈ 푄, where a subject 푥 belongs to set 푃 (the predicate of the premise) implies that the same subject 푥 belongs to set 푄 (the predicate of the conclusion). We exclude here any conditional that relates different subjects with different predicates (e.g. “If it rains today, it will be wet tomorrow.”)

The statement (푥 ∈ 푃) → (푥 ∈ 푄) is, by definition, stating that 푃 is a subset of 푄 (푃 ⊆ 푄). It then follows that if 푝 is true then 푞 must be true. This is consistent with the first two rows of the truth table. Row 1 is trivial. Row 2 can be justified since it can’t possibly be true that 푥 belongs to 푃 yet doesn’t belong to 푄 (see figure below).

As for rows 3 and 4 in the table, we need to assume that 푝 is false (so 푥 ∉ 푃). In that case, set 푃 is empty and it still follows that the statement (푥 ∈ 푃) → (푥 ∈ 푄) is true by the fact that the empty set is a subset of every set, so 푃 ⊆ 푄. These are, again, true only in a technical sense. Equivalent Disjunctive Form of the Conditional

Using basic truth tables, it turns out that the conditional “If 푝, then 푞” is equivalent to the disjunction “Not 푝 or 푞.” [Check this by constructing the truth table of ~푝 ∨ 푞].

Therefore, we have the following important equivalency:

풑 → 풒 ≡ ~풑 ∨ 풒

Example 1. The statement “All squares are rectangles” is equivalent to the statement “It’s not a square or it’s a rectangle.”

Example 2. The statement “I am not scared to hide in the dark room, unless it’s full of spiders” is equivalent to the statement “I’m scared to hide in the dark room or it’s not full of spiders.” [Check this.]

Based on this disjunctive form, we can draw the circuit of the conditional statement 푝 → 푞 as follows:

Circuit for the Conditional Statement “If 푝 then 푞”

Note that the light bulb in the circuit above can only be off if both switches ~푝 and 푞 are open. This is consistent with the fact that the conditional statement 푝 → 푞 is only false when ~푝 and 푞 are both false (so when 푝 is true and 푞 is false in row 2).

Negation of the Conditional

Applying De Morgan’s Laws to the disjunctive form of the conditional, we then have the following: ~(풑 → 풒) ≡ ~(~풑 ∨ 풒) ≡ 풑 ∧ ~풒

Example. The negation of the statement “I will go to the beach unless it rains” is given by the statement “It rains and I go to the beach.” [Check this.]

Related Forms of the Conditional

There are three important forms related to the conditional 푝 → 푞. These are given below.

The Converse: 풒 → 풑 This form switches the premise and conclusion of the conditional. In other words, the converse reverses the implication of the sentence (i.e. the arrow): 푞 → 푝 ≡ 푝 ← 푞.

The Inverse: ~풑 → ~풒 This form negates both the premise and the conclusion of the conditional. The implication of the sentence, however, remains unchanged.

The Contrapositive: ~풒 → ~풑 This form negates both the premise and the conclusion of the conditional; it also reverses the implication.

Based on the basic truth table for 푝 → 푞 and its related statements (the converse, the inverse and the contrapositive), we have the following two important equivalences:

풒 → 풑 ≡ ~풑 → ~풒 and 풑 → 풒 ≡ ~풒 → ~풑

As a result, a conditional statement is logically equivalent to its contrapositive and its converse is logically equivalent to its inverse.

Example 1. The statement “If the square of a natural is odd, then the natural is also odd” is equivalent to the much simpler statement “If a natural number is even, then its square is also even.” Can you see why? For natural numbers, the negation of odd is even and vice-versa. Therefore, here you are looking at a conditional (the last statement) and its contrapositive (the original statement) and so they must be equivalent.

Example 2. The statement “Every apple is green or red” is equivalent to “If it’s not green and it’s not red, then it is not an apple.” Can you see why? Here you must write the original statement as a conditional in the form “If a, then g or r,” where a is the statement “It’s an apple,” r is the statement “It’s red,” and g is the statement “It’s green,” and then write the contrapositive as “If (not g and not r), then not a” using De Morgan’s Law.

Basic Truth Table for the Biconditional

Definition. Any statement that has the form “풑 if and only if 풒”, where 푝 and 푞 are basic statements, is called a biconditional statement. It is defined as the conjunction of a conditional with its converse and is written symbolically as 푝 ↔ 푞:

(푝 → 푞) ∧ (푞 → 푝) ≡ (푝 → 푞) ∧ (푝 ← 푞) ≡ 푝 ↔ 푞

A biconditional statement is also called an equivalence and can be rewritten in the form “풑 is equivalent to 풒.” (Symbolically: 푝 ≡ 푞).

Below is the basic truth table for the biconditional statement “푝 if and only if 푞.”

풑 풒 풑 ↔ 풒 T T T T F F F T F F F T

Basic Truth Table for the Biconditional Statement “푝 if and only if 푞”

So the biconditional statement “푝 if and only if 푞” is only true when 풑, 풒 are both true or both false. Since 푝 ↔ 푞 ≡ (푝 → 푞) ∧ (푝 ← 푞) ≡ (~푝 ∨ 푞) ∧ (~푞 ∨ 푝), we can draw the circuit of the biconditional statement 푝 ↔ 푞 as follows:

Circuit for the Biconditional Statement “푝 if and only if 푞”

Example 1. The statement “1 + 1 = 2 if and only if 32 ≠ 9” is false while the statement “1 + 1 = 3 if and only if 32 ≠ 9” is true. [Check this.]

Example 2. The statement “푛2 = 9 if and only if 푛 = 3 or 푛 = −3” is a tautology. To prove this, check the truth value of the biconditional for all values of 푛:

 If 푛 = 3, then 푛2 = 32 = 9 and the disjunction 푛 = 3 or 푛 = −3 is true. Hence the biconditional is also true since 푇 ↔ 푇 ≡ 푇.  If 푛 = −3, then 푛2 = (−3)2 = 9 and the disjunction 푛 = 3 or 푛 = −3 is true. Hence the biconditional is also true since 푇 ↔ 푇 ≡ 푇.  If 푛 ≠ ±3, then 푛2 ≠ 9 and the disjunction 푛 = 3 or 푛 = −3 is false. Hence the biconditional is also true since 퐹 ↔ 퐹 ≡ 푇.

Example 3. Prove that the negation of the biconditional “푝 if and only if 푞” (~(푝 ↔ 푞)) is equivalent to the exclusive disjunctive form “Either 푝 or 푞, but not both” (푝 ⊕ 푞).

To show this, we first apply De Morgan’s Laws to 푝 ⊕ 푞 as follows:

푝 ⊕ 푞 ≡ (푝 ∨ 푞) ∧ ~(푝 ∧ 푞) ≡ (푝 ∨ 푞) ∧ (~푝 ∨ ~푞)

Next, we apply De Morgan’s Laws and the disjunctive form of the conditional to ~(푝 ↔ 푞) as follows:

~(푝 ↔ 푞) ≡ ~[(푝 → 푞) ∧ (푞 → 푝)] ≡ ~(푝 → 푞) ∨ ~(푞 → 푝) ≡ (푝 ∧ ~푞) ∨ (푞 ∧ ~푝)

Now, check that the circuit corresponding to (푝 ∨ 푞) ∧ (~푝 ∨ ~푞) is equivalent to the circuit corresponding to (푝 ∧ ~푞) ∨ (푞 ∧ ~푝).