So, here we are at the final chapter, and at this point you might be minded to ask ‘So what?’ Although some of you may have found this book to be so compelling that you have decided to become an epidemiologist, it is likely that most of you will looking for other ways for this epidemiology stuff to value add to your health science learning and ongoing professional or academic lives. In modern life, we are deluged with health information that is provided in multiple formats, including social media, news websites, online videos, televised news bulletins and chat shows, and even academic texts and other published literature. How are we to find something approaching the truth in this plethora of often contradictory information? In its focus on epidemiology, this book has aimed to provide you with the tools for evaluating scientific information using critical thinking – a way of identifying and evaluating evidence that has wide applicability to just about every area of human endeavour.

Epidemiology is the study of patterns and determinants of disease and other health states in populations. It primarily uses quantitative methods (those methods dealing with counting, measuring and comparing things) that definitely use statistics and include statistical methods, but in this book we will not be talking about performing any statistical acrobatics more complicated than completing a sudoku puzzle.

Arriving at evidence-based solutions requires strong evidence. Usually, this evidence will be derived from quality research, such as is often published in reputable scientific journals. But how do we know whether even these studies are good through and through? There is always the potential that pesky flaws, such as bias and confounding, might can beset even the most (otherwise) perfect of studies. This is why the methods taken to avoid bias and confounding are always well-described in all good published studies, as is the potential for remaining sources of error for which the design is (inevitably) unable to account, but which might still influence findings. There is always a bit of uncertainty about any evidence provided by studies and, to add to this, the very real possibility that we are not getting the full story at all times. In a phenomenon known as ‘publication bias’, even really high quality studies may not get published if they report non-significant results.

In this chapter, we are moving across to the experimental branch of the epidemiological research tree to focus specifically on randomised controlled trials (RCTs). These are often referred to as the ‘gold standard’ of health research designs, and are ranked pretty much at the top of the hierarchy of scientific evidence. Representations of the hierarchy of evidence in the form of pyramid are more or less ubiquitous on the internet and in textbooks.

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.