They were great for settling questions of logical truth, validity, equivalence, and so on, but became unwieldy in an exponential hurry as the number of relevant atomic sentences increased. They also foundered on the rocks of $\mathcal{Q}\mathcal{L}$’s non-truth-functional constructions.

In this chapter, a number of important notions surrounding logical truth, logical equivalence, contradiction, and logical consequence will be explored and clarified. A shocking fact about classical logic will be encountered and examined: every argument with contradictory premises is deductively valid. Contradictions entail everything. This is not a feature of every formal logical system …

If the purported counterexamples to modus ponens (that appear in §7.7 of the previous chapter) are genuine, then modus ponens isn’t the only argument form that is in trouble – modus tollens and hypothetical syllogism look like they’re on no better footing. But are the counterexamples genuine? It would be easier to answer this question if we had a better grip on the semantics of indicative conditionals.

Heaven forbid we have eight. If we have eight relevant atomic sentences, we’re going to need 256 rows for our truth table. If we double that to sixteen atomic sentences, we are all of a sudden at 65,536 rows. The problem is that the number of rows we need grows exponentially with every added atomic sentence. I was ready to tap out at the 256 rows for eight atomic sentences. Preparing 65,536 rows for sixteen atomic sentences is not going to happen. An argument with thirty-two distinct atomic sentence would require 4,294,967,296 rows. Even if I could write out about 120 characters per minute (which I can’t), it would take me almost sixty-eight years of solid writing to fill out such a truth table. Factor in a bit of time for sleep, and that’s more than one whole lifetime just to fill out a truth table for an argument with thirty-two atomic sentences. To say the least, the truth table method doesn’t scale up very well.

This chapter centers on the major descriptive findings of L2 research, focusing on ordered and systematic development. We review and discuss such things as morpheme orders, developmental stages/sequences, unmarked before marked, and U-shaped development, among others. We also review the evidence for L1 influence on ordered development. We touch on the nature of internal (e.g., Universal Grammar, general learning mechanisms) and external constraints (e.g., quantity and quality of input and interaction with that input, frequency) as underlying factors in ordered development. We also briefly touch upon variability during staged development.

It would be a scandal of philosophy and of human reasoning in general if we were unable to cast helpful light on the logic of conditionals. Conditionals loom large in both everyday and theoretical reasoning. They figure in the tight, rigorous proofs of mathematics, the subtle theoretical reasoning of quantum physics, the strategies of financial planners and generals, and even the loose contingency planning of vacationers and of educators trying to cobble together a plan to teach during a pandemic.

Carl Linnaeus dubbed his own species Homo sapiens, meaning something like “wise (or knowledgeable) man.” This is a bit overly self-congratulatory, but it does focus attention on a feature that sets humans apart. Humans inquire about the world and about themselves, and – sometimes, anyway – thereby acquire knowledge, wisdom, and understanding that surpasses that of even the most clever ostriches, squirrels, and mushrooms. Humans engage in inquiry about everything under the sun, and a good many things above it as well. Humans will even engage in inquiry about things that have no spatiotemporal relationship to the sun at all – things like the number 7, the orthocenter of a triangle, and the intricacies of the fictional world imagined in Frank Herbert’s Dune. At some critical stage in evolutionary history, humans even began to turn their inquiring gaze back on inquiry itself.

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.