Last time, I wrote a brief review of Steven Pinker’s latest book on rationality. I received no comments and will not add anything further to this discussion, other than this: I read two other books on critical thinking and rationality at about the same time and it was interesting to see the contrast, especially with different backgrounds of authors.

*Possibles*

We’ve talked a lot about possibility and necessity in this column over the years. Here’s a way into some work that I’ve come across recently in the journals. Consider the following list.

- If hydrogen is sparked in oxygen, it burns.
- If xenon is sparked in oxygen, it burns.
- If Oswald did not kill Kennedy, someone else did.
- If xenon uranide is synthesized, apples would be edible by humans.
- If Oswald had not killed Kennedy, someone else would have.
- If the sky were green, Aristotle would have been a photographer.
- If the sky were blue, Aristotle would have been a philosopher.
- If 2+2=4, then 4+4=8.
- If 2+2=5, then 4+4=8.
- If 2+2=5, then 4+4=9.
- If 13 were composite, then it would have more than 2 factors.
- If 13 were composite, then it would have 2 and 6 as factors.
- If 13 were composite, there would be no cicadas of 13-year period.
- If modus ponens were invalid, reasoning would be harder.
- If 2+2=4, then for all
*p*, either*p*or*not**p*. - If 2+2=5, then it is not the case that for all
*p*, either*p*or*not**p*. - If intuitionistic logic is correct, then it is not the case that either
*p*or*not**p*. - If a particular paraconsistent logic is correct, then it is the case that for some
*p*, both*p*and*not p*. - If ZFC then GCH.
- If ZFC then not GCH.

You may have to look up some of these to understand them. I will analyze all of them in turn and provide some take on what is going on. Your job will be to analyze my analysis and see what, if anything, you find funny about it.

- This seems to be straightforward; a straightforward conditional with the two “parts” related to each other. (I am not calling them “antecedent” and “consequent” for reasons we will go into in the followup, next month.) It also seems to be true, based on grade 10 physical science, anyway.
- This one is very similar to the previous, except that it is false: Xenon does not react with oxygen so easily.
- This one requires a bit more thought, but is true assuming that Kennedy was in fact killed. (Rather than dying of old age or the like.)
- This one is harder, since it involves what might be called a “counterlegal.” Uranium does not likely form such a molecule, and there are good reasons for this supposition. On the other hand, the second part is true, if read as a proposition in its own right. This makes the overall conditional true, according to the analysis of the condition as a so-called material conditional. Some people adopt the view that this is wrong, however. Why? The lack of connection between what I called the “parts.” This connection idea is, however, vague. One approach is to “share a variable,” in the so-called relevant logics. I am sympathetic, but it seems that this approach is syntactic; it might need semantic connections.
- This is our first example of what some call a counterfactual conditional. It is in the subjunctive mood. It, however, strikes many people as intuitively false, somehow. On the other hand, it is unclear what grounds this judgment. You decide!
- This is not only a counterfactual, but one with seemingly no connection involved. People are less likely to judge this as true (or false, even).
- Some people think this one is true, despite its similarity to the previous. It isn’t obvious why this is so. Some think this is because the first “part” and second “part” are in fact true. I would like to put some cold water on that idea, though: “… the sky were blue” and “Aristotle would have been a photographer” are
*not*true. So, it is not a case of a conditional with true parts and hence being true as if it were a material conditional. (Note the problem with stating this: It is self applicable!) - This seems true to many people, but it is also unclear why. But note that this is a statement within mathematics, too, so the connection between the propositions is not obvious. Perhaps the domain of mathematics (or arithmetic?) is enough.
- This is like the previous but involves a false first “part” and a true second “part.” Traditionally, this is regarded as true as a whole, treating it as a material conditional.
- For some reason, many people regard this as more implausibly true, despite the fact that it is also arguably a material conditional as well.
- This is a subjunctive conditional again, but this time people balk at it. The number 13 cannot (for some “value” of “cannot”) be composite, so it seems weird to think of a situation where it would be.
- This one is utterly baffling. A counterfactual in mathematics sounds bizarre to most people, especially not specific instances.
- This one is hard to understand, since it appears to appeal to mathematics in a “factual” or just “biological” context. Cicadas appear to have been selected for periods which are prime numbers of years.
- This is a “counterlogical.” People find these incomprehensible a lot of the time. Yet there are many alternative logics in the literature. Some of these assert that modus ponens is invalid — i.e., not truth preserving. So how do we evaluate claims of which logic to use? These do not seem to be even claims of truth.
- This one “quantifies” over propositions — the “p” in the text. Many people do think that “
*p*or*not p*” is a logical truth, so the given proposition may well be true. - This one has the complexities of the previous two.
- This is very much like the previous, except that it states the logics involved rather than alluding to them by a defining principle.
- This is exactly like the previous, except it uses a quantifier over logics, and also uses a more specifically “weird” logic (i.e., contentious, hard to understand, etc.).
- This is elliptical. ZFC is a family of theories within a set theory, and the GCH is a specific statement within it. However, it is a statement which the theory can state but cannot prove or disprove (provided ZFC is consistent). Note that the view of “mathematical platonism” (or mathematical realism, of some kind) holds that GCH is true or false independently of our knowledge of the specific theory in which it might be stated (ZFC).
- Exactly like the previous except that the second “part” is negated. Some views of negation come into play here; some people might think that GCH is not true but also not false. Another view holds that it is true in one theory, false in others — a different view of truth than in factual science.

Re your statement #3: Since 1963, I have been obsessed with the idea that the current events and historical reporting on the Kennedy assassination (OK, that’s an assumption) is just a cover-up for the real facts. I have been doing my own research on this for decades, culminating with a recent visit to Dallas’ 6th Floor Museum, the site of the JFK killing (if we may use that term). I urge all serious critical thinkers to visit (Nay, scrutinize) said Dallas locale. They may find that your statement #3 is not just true, but Screamingly True!

I’m having some fun by recasting the conditional as a disjunction (i.e. “P=>Q” = “not-P or Q”). Under that interpretation, we get that #5 is true (since P is false), which seems wrong because, had Oswald failed, it doesn’t follow that some other assassin would have succeeded. Same with #6, I think. Overall, it seems like the disjunctive form fails to capture what we mean by implication.

Unclear what you’re getting at in #7. The sky is blue, and A was a philosopher (is “photographer” a typo?) — does the subjunctive mood make it not a truth-apt proposition?

For #11-13, I find it helpful to universalize them, which removes the intuitive absurdity of imagining a prime number as composite. Then we get:

11. Forall X, if X is composite, then X has more than two factors. (True under both the implicative and disjunctive forms across all values of X; 13 is an instance where the antecedent is false).

12. Forall X, if X is composite, then X has 2 and 6 as factors. (False, because there are instances where the antecedent is true but the consequent false; plugging in 13 for X doesn’t create a novel situation).

13. Forall X, if X is composite, then there would be no cicadas of X-year period. (True, assuming that all cicada species are strongly selected for prime-number periods. Or does the “would be” mean something different than just “are” here?)