1.4.4. Conditional Statements

Now let us define another logical operator, which is the conditional statement. This is also called as if then statement and we use this notation p → q and the truth table for p implies q is defined as this. So you can see that p implies q is true for three possible combinations and it is defined to be false only when p is true, but q is false.

A conditional statement, represented as p → q, is a fundamental concept in logical reasoning. It states that if 'p' (the hypothesis) is true, then 'q' (the conclusion) must also be true. The only situation where this is not true is when 'p' is true and 'q' is false. Essentially, the conditional statement holds true in all scenarios except this one because it establishes that 'q' is reliant on 'p'.

For example, if we say 'If it rains, then the ground will be wet', this statement is only false if it actually rains (p is true), but the ground is not wet (q is false). In all other scenarios, the statement is considered true, even if both p and q are false or if p is false and q is true.

Imagine a vending machine: 'If I press the button for soda (p), then I will get a soda (q).' This relationship holds true as long as I press the button and the machine is working. However, if I press the button and the machine is out of order (which means q is false), then the statement is false. Otherwise, in all other circumstances, the statement is considered true.

Now, the question is why the truth table of p → q is defined like this? That means why p → q is true, even if both p and q are false or why p → q is defined to be true even if p is false, but q is true. So to understand that let me give an example, a very simple example. Suppose we make this logical statement. Suppose there is a PM candidate called John and he makes the election promise that if he becomes the prime minister then good days will come for the country.

The rationale behind the truth table for p → q hinges on how we interpret the implications of a conditional statement. The example regarding a political promise illustrates that we only deem a conditional statement false when the assumption (the hypothesis) is met but the promise (the conclusion) does not materialize. In other words, John’s claim can only be counted as false if he becomes prime minister, but good days do not follow. In cases where 'p' is false (John does not become prime minister), we'll state that the promise cannot be broken since it was never fulfilled, thus maintaining the statement's truth.

Think of a promise made by a parent: "If you clean your room, then you can have dessert." This promise is only broken if the child cleans their room but does not receive dessert. If the child does not clean their room, whether or not they have dessert is irrelevant to the promise, which remains unbroken and thus true.

So let us try to understand that why p implies q or why if p then q can also be interpreted as q is necessary for p and why it can be interpreted as p only if q. So to understand that let me first claim here that the statement p → q and the statement ¬q →¬p are both logically equivalent.

The relationship between p → q and ¬q → ¬p hinges on their logical equivalence. This means they yield the same truth values under all possible interpretations. If p implies q is true, it logically follows that if q is false, then p must also be false. Therefore, having q as necessary for p reflects that p can occur only when q is also true, which signifies a dependency in that p requires q.

This interpretation often causes confusion, since phrases like 'only if' or 'necessary condition' suggest that one cannot assume p without ensuring q is met.

Consider a student saying, 'I will graduate only if I pass my exams.' Here, passing exams (q) is a necessary condition for graduating (p). If the student doesn’t pass the exams, they certainly can’t graduate. Thus, their promise illustrates the logical dependency where graduation is contingent upon passing.

So, whenever we write a conditional statement of form, if then, p → q, then the p statement or whatever statements you have for the propositional variable p they are called as hypothesis or antecedent or premise and q denotes the conclusion or the consequence.

In a conditional statement, the first part, known as the antecedent (or hypothesis), is the condition, while the second part (the conclusion) is the outcome that follows if the condition is true. Understanding these roles allows better grasping of how arguments and logical structures function in reasoning. Importantly, this means logically we can have statements that have no real-world connection but still maintain logical integrity. That’s the hallmark of mathematical logic.

A classic analogy would be the statement: "If you sleep early (p), you will feel refreshed in the morning (q)." Here, sleeping early is the antecedent, and feeling refreshed is the consequence. No matter the truth of the outcome, as long as the structure follows, the logical form holds, independent of real-life occurrences.

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