A conditional statement or conditional proposition (sometimes referred to as if-then statement) is a compound statement that is connected by the words “If…then” or just “then.” Most logicians used the sign horseshoe (⊃) to mean “if…then”. Let us consider the example below.
If the airship Albatros has a powerful weapon, then it could destroy objects on the ground. (S, T)
If we let S stand for “The airship Albatros has a powerful weapon” and T for “It could destroy objects on the ground,” then the statement above is symbolized as follows:
S ⊃ T
It must be noted that the statement that comes before connective horseshoe (⊃) is called the “antecedent” and the statement that comes after it is called “consequent.”
It must be noted as well that there are instances wherein the words “if…then” are not mentioned in the statement, yet the statement remains a conditional one. Let us analyze the statement below:
Passage of the law means morality is corrupted. (S, T)
If we analyze the statement above, it is obvious that it is a conditional statement because it implies a “cause and effect” relationship. Thus, the statement can be restated in the following manner:
If the law is passed, then morality will be corrupted.
If we let S stand for “The law is passed” and T for “Morality will be corrupted,” then the proposition is symbolized as follows:
S ⊃ T
It is also important to note that sometimes the antecedent is stated after the consequent. If this occurs, then we have to symbolize the statement accordingly. Let us consider the statement below.
The forest will be destroyed should the logging law is passed. (S, T)
If we analyze the statement, it is obvious that the antecedent is “The logging law is passed” and the consequent is “The forest will be destroyed.” Hence, the statement “The forest will be destroyed should the logging law is passed” is symbolized as follows:
T ⊃ S
As we can notice, the variables provided after the statement represent the component statements in the entire statement respectively. Thus, in the statement
The forest will be destroyed should the logging law is passed. (S, T)
The variable S stands for “The forest will be destroyed” and T stands for “The logging law is passed.” Again, since T is our antecedent and S is our consequent, and since in symbolizing a conditional statement we need to write the antecedent first and then the consequent, so the statement “The forest will be destroyed should the logging law is passed” is symbolized as follows:
T ⊃ S
Rules in a Conditional Statement
- A conditional statement is false if the antecedent is true and the consequent false.
- Thus, other than this form, the conditional 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 observe, the rules in a conditional statement say that the only instance wherein the conditional statement becomes false is when the antecedent is true and the consequent false. Let us take this statement:
If the airship Albatros has a powerful weapon, then it could destroy objects on the ground. (S, T)
Now, the first row in the truth table above states that p is true and q is true. So, obviously, p ⊃ q is true. This is because, if it is true that “The airship Albatros has a powerful weapon,” then it must also be true that “It could destroy objects on the ground.”
The second row states that p is true and q is false. So, p ⊃ q must be false. This is because if it is true that “The airship Albatros has a powerful weapon” then it should necessarily follow that “It could destroy objects on the ground.” However, it is stated that q is false, that is, the “It could not destroy objects on the ground”; therefore, the conditional statement is false. For sure, it is not sound to conclude that the airship Albatros does not have the capability to destroy objects on the ground given that it has a powerful weapon. Hence, again, the conditional statement is false.
The third row states that p is false and q is true. If this is the case, then p ⊃ q is true. This is because if it is not true that “The airship Albatros has a powerful weapon”, then it does not necessarily follow that it could not destroy objects on the ground. In fact, even if the airship Albatros does not have a powerful weapon, it is still possible for the airship Albatros to destroy objects on the ground.
Finally, the last row in the truth table above states that 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 states that “The airship Albatros does not have a powerful weapon” and that “it could not destroy objects on the ground.” Hence, obviously, the conditional statement is true.