The proposition p → q is called an implication or a conditional. It is usually read as “pimplies q”. In such an implication p and q also get special names of their own. p is called the hypothesis or antecedent and q is called the conclusion or consequent.
Furthermore we say that if the implication p → q holds, then p is sufficient for q. That is if p is true that is sufficient to also make q true. Conversely we say that q is necessary for p. Without q being true, it is impossible for p to be true. That is if q is false, then p also has to be false.
In English, p → q is often expressed as ‘if p then q’. For example, if p represents the proposition “Karel Luyben is Rector Magnificus of TU Delft” and q represents “Prometheus is blessed by the gods”, then p → q could be expressed in English as “If Karel Luyben is Rector Magnificus of TU Delft, then Prometheus is blessed by the gods.” In this example, p is false and q is also false. Checking the definition of p → q, we see that p → q is a true statement. Most people would agree with this, even though it is not immediately obvious.
The letter ‘T’ in the TUDelft logo bears a stylized flame on top, referring to the flame that Prometheus brought from Mount Olympus to the people, against the will of Zeus. Be- cause of this, Prometheus is sometimes considered as the first engineer, and he is an important symbol for the univer- sity. His bronze statue stands in the Mekelpark at the centre of campus.
Image: weblog.library.tudelft.nl/2016/ 01/04/english-prometheus-is-back/.
It is worth looking at a similar example in more detail. Suppose that I assert that “If Feyenoord is a great team, then I’m the King of the Netherlands”. This statement has the form m → k where m is the proposition “Feyenoord is a great team” and k is the proposition “I’m the king of the Netherlands”. Now, demonstrably I am not the king of the Netherlands, so k is false. Since k is false, the only way for m → k to be true is for m to be false as well. (Check the definition of → in the table, if you are not convinced!) So, by asserting m → k, I am really asserting that the Feyenoord is not a great team.
Or consider the statement, “If the party is on Tuesday, then I’ll be there.” What am I trying to say if I assert this statement? I am asserting that p → q is true, where p represents “The party is on Tuesday” and q represents “I will be at the party”. Suppose that p is true, that is, the party does in fact take place on Tuesday. Checking the definition of →, we see that in the only case where p is true and p → q is true, q is also true. So from the truth of “If the party is on Tuesday, then I will be at the party” and “The party is in fact on Tuesday”, you can deduce that “I will be at the party” is also true. But suppose, on the other hand, that the party is actually on Wednesday. Then p is false. When p is false and p → q is true, the definition of p → q allows q to be either true or false. So, in this case, you can’t make any deduction about whether or not I will be at the party. The statement “If the party is on Tuesday, then I’ll be there” doesn’t assert anything about what will happen if the party is on some other day than Tuesday.