12. Inverse probability: you can’t be indifferent about it!

People often confuse probabilities with their inverses. Many inductive arguments require us to reason about inverse probabilities. ‘Inverse probability: you can’t be indifferent about it!’ looks at the relationship between inverse probabilities, illustrating it with the Argument to Design, which asks: does not the fact that the physical cosmos is ordered in the way that it is give us reason to believe in the existence of a god of a certain kind? Logicians use the term Principle of Indifference to describe an important part of intuitive reasoning about probability: given a number of possibilities, with no relevant difference between them, they all have the same probability.

7. Conditionals: what’s in an if ?

A conditional is a sentence in the form ‘if a then c’. Logicians call a the antecedent of the conditional, and c the consequent. ‘Conditionals: what’s in an if’ looks at the importance of the conditional, which is fundamental to much of our reasoning. Yet, it is deeply puzzling. Conditionals have been studied in logic ever since its earliest times. It is important to note that in the kind of situation where you learn that a premiss is true by being informed of it, other factors are usually operating. We often make such inferences without thinking about them.

6. Necessity and possibility: what will be must be?

We often say that something must be so (e.g. ‘It must be going to rain’). We also have ways of saying that, though something may not be the case, it could be (e.g. ‘It could rain tomorrow’). ‘Necessity and possibility: what will be must be?’ considers the notion of things that may happen, things that could happen, and modal operators (it is possible for something to be the case—so it is not necessarily the case that it is false). The argument for fatalism illustrates modal operators. Fatalism is the view that whatever happens must happen. We suppose that every situation comes furnished with a bunch of possibilities, or possible worlds.

8. The future and the past: is time real?

We are familiar with the concept of time: past, present, and future. We make many inferences about time. It might seem simple, but if we start thinking about time, we can get very confused. ‘The future and the past: is time real?’ examines time in relation to logic. One particularly puzzling thing is: time seems to flow. How can time change? Time measures the rate at which everything else changes. This problem is central to several interesting time conundrums. Every situation comes with an associated collection of earlier and later situations. Where does space come into thoughts about time? Space does not flow in the same way that time does.

10. Vagueness: how do you stop sliding down a slippery slope?

The idea of identity is beset with problems. Everything wears out in time. Parts get replaced. Do such changes affect the identity of the object? How about if we eventually change all the parts of the original object? ‘Vagueness: how do you stop sliding down a slippery slope?’ considers identity and time. Another good example is of a child, who is still a child a second later, and again another second later. Eventually he or she isn’t a child anymore. How can logic explain that? Such sorites paradoxes provide some of the most annoying paradoxes in logic. The issue here is one of vagueness, where nothing is straightforward.

9. Identity and change: is anything ever the same?

We all think that objects can survive change, but the opposite can be argued. If a person changes their hairstyle, they are both the same person and a different person. ‘Identity and change: is anything ever the same?’ examines problems arising when things change and the identity of objects that change through time. Any change means that the old object no longer exists and is replaced by a different object. Logicians point to the difference between the object and its properties. Is this a language issue? In English, the verb ‘to be’ and its various grammatical forms can be used to express both the object and its properties.

2. Truth functions—or not?

We are strongly intuitive about the validity of various inferences. This may or may not be hard-wired into us. Intuition gets us into trouble sometimes. ‘Truth functions—or not?’ plays around with the idea of invalid and valid inferences and, using phrases such as ‘or’ or ‘it is not the case that’, shows that chaining valid inferences together doesn’t necessarily result in an invalid inference. People’s intuition can be misleading. Sentences can be true or false. Gottlob Frege called these truth values. If we assume that every sentence is either true or false, but not both, we can predict the conditions in what logicians call a truth table.

1. Validity: what follows from what?

To tell someone that they are not being logical is seen as a criticism. Logic is the study of what counts as a good reason for what, and why. ‘Validity: what follows from what?’ explains how logic works in terms of premisses, inferences, conclusions, and validity of premisses. A valid inference is one where the conclusion follows from the premiss or premisses. A deductively valid inference is one for which there is no situation in which all the premisses are true, but the conclusion is not. How do we recognize inferences as valid or invalid? Is there some special insight contained within us or is it hard-wired into us through evolution?

15. Maybe it is true—but you can’t prove it!

‘Maybe it is true—but you can’t prove it!’ first considers David Hilbert’s Program in the Foundations of Mathematics, which was to prove that mathematics was consistent. Hilbert’s ambitious proposal required all of mathematics to be axiomatized. The existence of such an axiom-system was disproved by the Austrian mathematician Kurt Gödel (1906–78). What Gödel showed was that such an axiom system cannot be provided even for the fragment of mathematics that concerns natural numbers, let alone the rest of it. Gödel’s result has been held to have many other philosophical consequences, concerning the nature of numbers, our knowledge of them, and even the nature of the human mind.

14. Halt! What goes there?

‘Halt! What goes there?’ looks at some results about what formal reasoning can and can’t do, and some of the philosophical implications of these facts. It considers the workings of computers and explains computations—or algorithms. It asks if there is an algorithm we can apply to a program and inputs to determine whether or not a computation with that program and those inputs terminates, discussing Georg Cantor’s Halting Theorem along with diagonalization. The answer is no and this is what Alan Turing proved. The Church–Turing Thesis, which claims that one can write a computer program for every algorithm, is also considered.