[j®y] (material implication)

[j®y] has the following truth-table:

One can read it most simply as "j arrow y".

Notice that the whole sentence will be true just when the first constituent is false or the second constituent is true. So [j®y] is equivalent to [¬jÚy]. Notice too that the whole sentence will be false just when the first constituent is true and the second is false. So [j®y] is equivalent to ¬ [jÙ¬y].

Commonly [j®y] is used to translate some conditional sentences. So one might translate "If John is Welsh, he likes rugby" as "[John is Welsh ® John likes rugby]". (Notice that we have used "John likes rugby" rather than "he likes rugby", to avoid cross reference.)

Similarly one might translate "Liverpool will win the premiership if they beat Manchester United" as "[Liverpool will beat Manchester United ® Liverpool will win the premiership].

How good are these translations? Well certainly if John is Welsh but does not like rugby, "If John is Welsh, he likes rugby" is false. So if it is true, the suggested translation must also be true too, because it will be false only if John is Welsh but does not like rugby. But it is not so clear that the converse holds. If John is not Welsh, that is enough to make the translation true; but is it enough to make the original true? And, if John likes rugby, that is enough to make the translation true, but is it enough to make the original true? Suppose, for instance, that all Welshmen, except possibly John, loath rugby.

Some people have argued that such translations are perfectly good and that in each case the original and the translation would be true in exactly the same circumstances. The apparent differences, some have claimed, lie in the fact that the original implicates something additional. So, if we are tempted to think that the original is false in certain circumstance while the translation is true, we are mistaking the falsity of what is implicated (which does not make the original false) for the falsity of what is logically implied (which would).

For a good discussion of these issues, see Mark Sainsbury, Logical Forms: an Introduction to Philosophical Logic, chapter 2, sections 4-8.

Meanwhile consider the following.

Nonetheless, it is often safe for the purposes at hand to translate "if j, y" as "[j®y]", because of the fact that whenever the former is true, the latter is too.

There are, however, certain translations which would certainly be a mistake. It would be wrong to translate "If Derby County had beaten Southampton in their first match of the season, they would have won the premiership" as "[Derby County beat Southampton in their first match this season ® Derby County win the premiership this season]".

Conditionals of this sort are called counter-factuals. They are used talk about what would have happened if circumstances had been different: i.e. if their antecedents had been true. They would, of course, not be worth using if they were automatically made true by the fact that their antecedents were false: i.e. by the fact that circumstances were not different.

Again, one needs to beware of conditionals which say something general. For example, it would be a mistake to translate "If a letter comes, I will answer it" as "[a letter will come ® I will answer it]". What is does "it" refer to? Given that in "[a letter will come ® I will answer it]" we have treat "I will answer it" as meaning what it would if used on its own, the "it" must refer to some specific letter. But there is no specific letter which the "it" in the original refers to. The trouble is that in the original "I will answer it" involves cross-reference, and we cannot get rid of it without changing the meaning. 

Just as we can sometimes, with some possible loss of accuracy, translate "if j, y" as "[j®y]", so we can sometimes translate "j only if y" as "[j®y]". So while "If John is Welsh, he likes rugby" becomes "[John is Welsh ® John likes rugby]", "John likes rugby only if is he is Welsh" becomes "[John likes rugby ® John is Welsh]". We could equally well translate it as "[¬John is Welsh ® ¬John likes rugby]". It comes to the same thing, of course, since it is false just if John isn't Welsh but does like rugby.

Note: you should, OF COURSE, not confuse "only if" with "if and only if". You will win the jackpot in the lottery only if you buy a ticket. It is not, however, true that you will win the jackpot if and only if you buy a ticket. If you are ever tempted to treat "only if" as "if and only if", remember the lottery. Temptation is indeed possible. If a father tells his son, "You can have your pocket money on Saturday only if you tidy your room", and the son does tidy the room, but the father refuses to give him his pocket money, the son may feel he has been unfairly treated; he may even think that his father has misled him, especially if his behaviour has been impeccable throughout the week. But, if he has been misled, it will presumably be because he should have been warned about other conditions that had to be met. At worst we will have a case where what is implicated is false (that there were no conditions except the obvious ones).

We noticed, when discussing "[jÚy]", that one can sometimes translate "j unless y"as "[jÚy]". Some people find it more natural to translate it as "j if not-y", and so as "[¬y®j]". Notice that the two translations come to the same thing. "[¬y®j]" is true just when "¬y" is false or "j" is true; that is, just when "y" is true or "j" is true; i.e. just when "[jÚy]" is true.