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2.1.F Implications in English

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  • The proposition → is called an implication or a conditional. It is usually read as “pimplies q”. In such an implication and also get special names of their own. is called the hypothesis or antecedent and is called the conclusion or consequent.

    Furthermore we say that if the implication → holds, then is sufficient for q. That is if is true that is sufficient to also make true. Conversely we say that is necessary for p. Without being true, it is impossible for to be true. That is if is false, then also has to be false.

    In English, → is often expressed as ‘if then q’. For example, if represents the proposition “Karel Luyben is Rector Magnificus of TU Delft” and represents “Prometheus is blessed by the gods”, then → 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, is false and is also false. Checking the definition of → q, we see that p → 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.

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    Image: 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 → where is the proposition “Feyenoord is a great team” and is the proposition “I’m the king of the Netherlands”. Now, demonstrably I am not the king of the Netherlands, so is false. Since is false, the only way for → to be true is for to be false as well. (Check the definition of → in the table, if you are not convinced!) So, by asserting → 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 → is true, where represents “The party is on Tuesday” and represents “I will be at the party”. Suppose that 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 is true and → is true, 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 is false. When is false and → is true, the definition of → allows 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.