4.6: Footnotes

1. Kant, I. 1997. Critique of Pure Reason. Guyer, P. and Wood, A. (tr.). Cambridge: Cambridge University Press. p. 106. ↵
2. This form is often called the “Disjunctive Syllogism”. Notice that the word ‘syllogism’ is used there. By the Middle Ages, Stoic Logic hadn’t disappeared entirely. Rather, bits of it were simply added on the Aristotelian system. So, it was traditional (and still is in many logic textbooks), when discussing Aristotelian Logic, to present this form, along with some others, as additional valid forms (supplementing Barbara, Datisi, and the rest). But this conflation of the two traditions obscures the fundamental difference between a class-centered approach to logic and one focused on propositions. These should be kept distinct. ↵
3. That’s actually a controversial claim about the role of semantics. Your humble author, for example, is one of the weirdos who thinks it not true (of natural language, at least). But let’s leave those deviant linguists and philosophers (and their abstruse arguments) to one side and just say: semantics gives you truth-conditions. That’s certainly true of our artificial language SL. ↵
4. You might think ‘Beyoncé is’ is a part of the sentence that qualifies as a sentence itself—a sentence claiming that she exists, maybe. But that won’t do. The word ‘is’ in the original sentence is the “‘is’ of predication”—a mere linking verb; ‘Beyoncé is’ only counts as a sentence if you change the meaning of ‘is’ to the “‘is’ of existence”. Anyway, stop causing trouble. This is why we didn’t give a rigorous definition of ‘component part’; we’d get bogged down in these sorts of arcane distinctions. ↵
5. Play along. ↵
6. You may have learned an “order of operations” in grade school, according to which multiplication takes precedence over addition, so that there would be no ambiguity in this expression. But the order of operations is just a (mostly arbitrary) way of removing ambiguity that would be there without it. The point is, absent some sort of disambiguating convention—whether it’s parentheses or an order of operations—the meanings of expressions like this are indeterminate. ↵
7. Pausing briefly to note, once again, that this talk of sentences, rather than the propositions that they express, having truth-values is a bit fast and loose. Reaffirming our earlier stance on this: not a big deal. ↵
8. As was the case when we had to make a choice about the word ‘some’ in Aristotelian logic, the argument makes the case that the inclusive sense is the core meaning of ‘or’, and the exclusive sense is a meaning that’s often, but not always, conveyed when we use ‘or’ in particular circumstances—an implicature. This line of reasoning has both adherents and detractors. ↵
9. The triple-bar is a logical equals-sign; it indicates that the components have the same truth-conditions (meaning). ↵
10. They’re often referred to as “DeMorgan’s Laws,” after the nineteenth century English logician Augustus DeMorgan, who was apparently the first to formulate them in the terms of the modern formal system developed by his fellow countryman and contemporary, George Boole. DeMorgan didn’t discover these equivalences, however. They have been known to logicians since the ancient Greeks. ↵
11. In fact, it’s possible to get by with only one symbol: if we defined a new two-place operator that’s true when both components are false, and false otherwise, that would do the trick. The symbol typically used for this truth-function is ‘|’, called the “Sheffer stroke” after the logician (Henry Sheffer) who first published this result. ↵