In the last chapter, we saw that no truth-function could capture the logic of natural language indicative conditional reasoning better than the material conditional. But this leaves open the question: can ordinary indicative conditional reasoning be properly captured with any truth function? There are good reasons to think that the logic of material conditionals departs in important ways from the logic of natural language indicatives, whatever their logical similarities might be. This is a topic which we will begin to explore in this chapter, and continue to explore in the next.

Semantics is the study of meaning. But ‘meaning’ in what sense? We use this term in many different contexts and it’s not clear exactly what – if anything – unites them. We might ask, for instance, about the the meaning of life. But we also might ask about the meaning of ‘life’, and that is a very different kettle of fish. Getting a full and confident grip on the meaning of ‘life’ might, sadly, leave one quite in the dark as to the meaning of life.

Yes, the developers of contemporary classical logic had a utopian vision. Instead of trying to rehabilitate the festering logical mess that is natural language, they’d develop a new, logically pure, language – one that was free from all of the defects that make it so hard to track or model right reasoning using natural languages. If they succeeded, they’d have a language that would give them some hope of systematizing logic and clarifying our reasoning (instead of one that constantly bewitched them into philosophical confusions).

Descriptions like ‘the man’ and ‘a turkey’ seem so simple and foundational to the way we talk that it’s shocking that one can muster more than a few short paragraphs to elucidate their semantics. The more one thinks about how we use these structures in language, however, the more puzzling and intractable they become. Over a century ago, Bertrand Russell tried to set out a simple, elegant theory of descriptions. That should have been the end of it. But it was just the beginning.

The move from $\mathcal{P}\mathcal{L}$ to $\mathcal{Q}\mathcal{L}$ can seem like a significant ramping up in terms of the complexity and difficulty of proof-making. In this chapter we’ll pause for a bit and work through some more proofs.

In chapter 10, we saw that contemporary classical logic departs from its Aristotelian roots in its tolerance for empty predicates like ‘– is a unicorn’, ‘– is a leprechaun’, and ‘– is a tasty kale recipe’. From the standpoint of formal semantics, such predicates can simply be assigned the null set as their extensions. This makes for some awkwardness, to be sure. It means that claims like ‘All leprechauns are Canadian’ should be counted as true (albeit vacuously so).

In addition to our stock of generic predicates – ready to be interpreted as one needs for whatever context one is using them – we have also introduced one (and only one) special predicate that has the same interpretation across all $\mathcal{Q}\mathcal{L}$ models: the identity predicate, ‘=’. Given the meaning that we have given to this predicate, it is possible to give Intro and Elim rules for it.

In its heyday, “classical” Newtonian physics was the simplest and most comprehensive model of basic physical phenomena that had come along in the history of science. It’s still taught today, not because it’s accepted as true by the physics community, but because (i) as models go, it’s pretty darn good at explaining and predicting the behavior of a large number of physical systems with reasonable accuracy, (ii) it’s simple, and (iii) it’s elegant. This makes it the perfect entering wedge for learning physics.

As we began to explore at the end of the previous chapter, in addition to making simple claims about particular individuals in the domain of discourse, we will sometimes want to make more general claims about all individuals in the domain of discourse.

The simplest sentences of natural languages like English, on the other hand, do have internal structure, and the logical properties of those sentences (and the logical relations between them) are, in part, a function of these internal structures. In this chapter, we will begin to explore and model the fine-grained structure of propositions (and the sentences that express them), drawing inspiration from Aristotle’s categorical logic, which is the historical inspiration for classical logic (and the reason we call it classical logic).

The metatheoretic formal semantics that we gave in Chapter 12 was rife with talk of functions, like the semantic valuation function for assigning extensions to formulae of $\mathcal{Q}\mathcal{L}$. Functions again took center stage in Chapter 19, which discussed intensions as functions from possible worlds onto extensions. Functions, functions, functions.

$\mathcal{P}\mathcal{L}$ can be thought of as the logic of truth functions. The connectives of $\mathcal{P}\mathcal{L}$ are introduced with characteristic truth tables because they all express truth functions – each is used to produce complex sentences the truth values of which are completely determined by the truth values of the component subsentence(s).

As has hopefully become increasingly clear throughout this text, the task of logical theory construction is a difficult one – one that’s full of choices. These choices have ramifications that aren’t always easily foreseen. Frege thought that he was on his way to reducing all of mathematics to classical logic when Russell pointed out that seeds of contradiction had been built into the foundations of Frege’s system.