Chapter 14: If Wishes Were Horses: The Lojban Connective System
Now we are ready to express ✥1.1 in Lojban! The kind of logical connective which is placed between two Lojban bridi to connect them logically is an ijek:
✥4.1 la djan. nanmu .ija la djeimyz. ninmu John is-a-man or James is-a-woman.
Here we have two separate Lojban bridi, “la djan. nanmu” and “la djeimyz. ninmu”. These bridi are connected by “.ija”, the ijek for the truth function A. The “.i” portion of the ijek tells us that we are dealing with separate sentences here. Similarly, we can now say:
✥4.2 la djan. nanmu .ije la djeimyz. ninmu John is-a-man and James is-a-woman. ✥4.3 la djan. nanmu .ijo la djeimyz. ninmu John is-a-man if-and-only-if James is-a-woman. ✥4.4 la djan. nanmu .iju la djeimyz. ninmu John is-a-man whether-or-not James is-a-woman.
To obtain the other truth tables listed in c14-§2, we need to know how to negate the two bridi which represent the component sentences. We could negate them directly by inserting “na” before the selbri, but Lojban also allows us to place the negation within the connective itself.
To negate the first or left-hand bridi, prefix “na” to the JA cmavo but after the “.i”. To negate the second or right-hand bridi, suffix “-nai” to the JA cmavo. In either case, the negating word is placed on the side of the connective that is closest to the bridi being negated.
So to express the truth table FTTF, which requires O with either of the two bridi negated (not both), we can say either:
✥4.5 la djan. nanmu .inajo la djeimyz. ninmu John is-not-a-man if-and-only-if James is-a-woman. ✥4.6 la djan. nanmu .ijonai la djeimyz. ninmu John is a man if-and-only-if James is-not-a-woman
✥4.7 John is a man or James is a woman, but not both.
Here is another example:
✥4.8 la djan. nanmu .ijanai la djeimyz. ninmu John is-a-man or James is-not-a-woman. John is a man if James is a woman.
How's that again? Are those two English sentences in ✥4.8 really equivalent? In English, no. The Lojban TTFT truth function can be glossed “A if B”, but the “if” does not quite have its English sense. ✥4.8 is true so long as John is a man, even if James is not a woman; likewise, it is true just because James is not a woman, regardless of John's gender. This kind of “if-then” is technically known as a “material conditional”.
Since James is not a woman (by our assertions in c14-§1), the English sentence “John is a man if James is a woman” seems to be neither true nor false, since it assumes something which is not true. It turns out to be most convenient to treat this “if” as TTFT, which on investigation means that ✥4.8 is true. ✥4.9, however, is equally true:
✥4.9 la djan. ninmu .ijanai la djeimyz. ninmu John is a woman if James is a woman.
This can be thought of as a principle of consistency, and may be paraphrased as follows: “If a false statement is true, any statement follows from it.” All uses of English “if” must be considered very carefully when translating into Lojban to see if they really fit this Lojban mold.
✥4.10, which uses the TFTT truth function, is subject to the same rules: the stated gloss of TFTT as “only if” works naturally only when the left-hand bridi is true; if it is false, the right-hand bridi may be either true or false. The last gloss of ✥4.10 illustrates the use of “if … then” as a more natural substitute for “only if”.
✥4.10 la djan. nanmu .inaja la djeimyz. ninmu John is-not-a-man or James is-a-woman. John is a man only if James is a woman. If John is a man, then James is a woman.
The following example illustrates the use of “se” to, in effect, exchange the two sentences. The normal use of “se” is to (in effect) tranpose places of a bridi, as explained in Chapter 5.
✥4.11 la djan. nanmu .iseju la djeimyz. ninmu Whether or not John is a man, James is a woman.
If both “na” and “se” are present, which is legal but never necessary, “na” would come before “se”.
The full syntax of ijeks, therefore, is:
- .i [na] [se] JA [nai]