An if-then statement or conditional statement is a type of compound statement that is connected by the words “if…then”. Logicians usually used horseshoe (⊃) as the symbol for “if…then”. In some cases, logicians used the mathematical symbol “greater-than” (>) instead of a horseshoe.
Let us consider the example below:
If the company closes down, then obviously many workers will suffer. (p, q)
If we let p stand for the statement “The company closes down” and q for the statement “Obviously many workers will suffer”, then the conditional statement is symbolized as follows:
p ⊃ q
If we use the greater-than symbol, then the statement above is symbolized as follows:
p > q
It is important to note that the statement that precedes the connective horseshoe (⊃) is called the “antecedent” and the proposition that comes after it is called “consequent.” Hence, in the example above, the antecedent is “The company closes down”, while the consequent is “Obviously many workers will suffer”.
It is also important to note that there are cases wherein the words “if…then” is not mentioned in the statement, yet it remains a conditional one. Let us consider the following example:
Provided that the catalyst is present, the reaction will occur. (p, q)
If we analyze the statement, it is very clear that it is conditional because it suggests a “cause and effect” relation. Thus, the statement can be stated as follows:
If the catalyst is present, then the reaction will occur. (p, q)
If we let p stand for the statement “Provided that the catalyst is present” and q for “The reaction will occur”, then the statement is symbolized as follows:
p ⊃ q
It is equally important to note that sometimes the antecedent is stated after the consequent. If this happens, then we have to symbolize the statement accordingly. Let us take the example below.
The painting must be very expensive if it was painted by Michelangelo. (p, q)
If we analyze the statement, it is clear that the antecedent is “It was painted by Michelangelo” and the consequent is “The painting must be very expensive”.
Now, if we let p stand for “The painting must be very expensive” and q for “It was painted by Michelangelo”, then statement “The painting must be very expensive if it was painted by Michelangelo” is symbolized as follows:
q ⊃ p
Please note that we symbolized the statement “The painting must be very expensive if it was painted by Michelangelo” as q ⊃ p because in symbolizing if-then or conditional statements, we always write the antecedent first and then the consequent. By the way, please note that the variables provided after the statement represent the statements in the entire statement respectively. Thus, in the statement
The painting must be very expensive if it was painted by Michelangelo. (p, q)
the variable p stands for the statement “The painting must be very expensive” and q stands for the statement “It was painted by Michelangelo”. Again, since q is our antecedent and p is our consequent, and since in symbolizing if-then statement we need to write the antecedent first and then the consequent, so the statement “The painting must be very expensive if it was painted by Michelangelo” is symbolized as follows:
q ⊃ p
Rules in If-then Statements
- An If-then statement is false if the antecedent is true and the consequent false.
- Thus, other than this form, the If-then statement is true.
The truth table below illustrates this point.
The truth table above says:
- If p is true and q is true, then p ⊃ q is true.
- If p is true and q is false, then p ⊃ q is false.
- If p is false and q is true, then p ⊃ q is true.
- If p is false and q is false, then p ⊃ q is true.
As we can see, the rules in If-then statements or conditional statements say that the only instance wherein the conditional statement becomes false is when the antecedent is true and the consequent false. Let us consider the example below.
If it rains today, then the road is wet.
Now, the first row in the truth table above says that p is true and q is true. So, obviously, p ⊃ q is true. This is because, if it is true that “it rains today,” then it must also be true that “the road is wet.”
The second row says that p is true and q is false. So, p ⊃ q must be false. This is because if it is true that “it rains today” then it must necessarily follow that “the road is wet.” However, it is said that q is false, that is, the road is not wet; hence, the conditional statement is false. Again, it is impossible for the road not to get wet if it rains.
The third row says p is false and q is true. If this is the case, then p ⊃ q is true. This is because if it is false that it rains today (in other words, it does not rain today), it does not necessarily follow that the road is dry. Even if it does not rain, the road may still be wet because, for example, a fire truck passes by and spills water on the road.Lastly, the fourth row in the truth table above says p is false and q is false. If this is the case, then p ⊃ q is true. This is because, based on the example above, it says “it does not rain today” and the “road is not wet.” So, obviously, the conditional statement is true.
