Logic for Understanding Probability

There is no substitute for a good logic course as a preface to the study of probability. But if you have had such a course and are a bit rusty, the following review should bring you back up to speed.

The language of elementary propositional logic

In propositional logic we use an artificial symbolism for simplicity and clarity. Atomic formulas (sometimes called “atomic sentences” of the language) are individual capital letters: A, B, C, .... If for some reason we were to need more than 26 of these, we would start over again with a hash mark to indicate a new atomic formula: A', B', C', .... We can add further hash marks as needed. This device secures an infinite supply of atomic formulas.

Intuitively, the atomic formulas stand for propositions, that is, the meanings of declarative sentences. Each atomic formula may be true or be false but not both.

We build up more complex formulas out of atomic formulas by the use of connectives and parentheses. The following table gives the connectives together with their names, their intuitive meanings, and some examples of their use.

Symbol	Name	Meaning	Example
~	Negation	not ___	~P
&	Conjunction	... and ___	(P & Q)
v	Disjunction	... or ___	(P v Q)
÷	Conditional	if ..., then ___	(P ÷ Q)
ø	Biconditional	... if and only if ___	(P ø Q)

It may seem a bit odd to think of the negation sign as a connective since it applies only to a single formula, e.g. ~P (“not P”). But it simplifies our construction of the language to have a single term available for referring to all of the symbols in the left column. We will refer to them all as connectives and to the last four as binary connectives – connectives that take two formulas, one to the left and one to the right, to fill them out.

Now we are in a position to give a general definition of a formula (or sentence) of the language of propositional logic. The definition is recursive: several of the clauses allow us to take formulas of the language, do something to them (often combining them in a certain way), and get additional formulas out. It runs like this, where “L” stands for the language of propositional logic:

1. Every atomic formula is a formula of L.

2. If n is a formula of L, then ~n is also a formula of L.

3. If n and R are both formulas of L, so are each of the following:

(n & R)

(n v R)

(n ÷ R)

(n ø R)

4. Nothing else is a formula of L.

All of the formulas constructed according to parts 2 and 3 of this definition are molecular formulas.

It may seem puzzling at first that we are using the little Greek letters n and R, which aren’t formulas of the language at all, in this definition. But we need variables that won’t be confused with the formulas of the language proper. It wouldn’t do to replace 2, for example, by saying this:

2*. The result of putting ‘~’ in front of any atomic formula is a formula.

The problem is not that 2* gives us things that aren’t formulas: it’s true as far as it goes. Rather, the problem is that it doesn’t go far enough. We need 2, not just 2*, to tell us that the following string of symbols counts as a formula:

~(P & Q)

This is the result of putting ‘~’ in front of ‘(P & Q)’ – but that latter string of symbols is not an atomic formula but rather a molecular one, so 2* wouldn’t allow us to say that ‘~(P&Q)’ is a formula of L. Similar considerations apply to the part 3 of the definition. This is why we need variables like n and R in our definition.

It is worth noticing a few consequences of our definitions. First, these rules will generate only finitely long sentences: no formula will be infinitely long. Second, the rules require “internal” parentheses for complex molecular formulas. Sometimes these are strictly necessary: the formulas ‘(P & (QvR))’ and ‘((P&Q) v R)’ do not mean the same thing (because they have different truth conditions), and the expression ‘(P & Q v R)’ – which is not a formula of L according to our definition since it lacks internal parentheses – is liable to be misread since it might be an attempt to express either of the first two.

Some authors (e.g. Lemmon) dispense with most of the parentheses by introducing a hierarchy of connectives by analogy with the hierarchy of arithmetic operations in mathematics. Others keep internal parentheses when there is any possibility of ambiguity but drop them when all of the connectives are ‘&’: ‘((P & Q) & R)’ has the same truth conditions as ‘(P & (Q&R))’ and may therefore be written as ‘(P & Q & R)’ since there is no ambiguity. Something similar applies when all of the connectives are ‘v’. But this simplification cannot be applied when the connectives are ‘ ÷’ or ‘ø’ since, e.g., ‘(P ÷ (Q ÷R))’ does not mean the same thing as ‘((P÷Q) ÷ R)’ does.

Propositional logic has limited expressive power. It cannot give us the fine structure of sentences involving quantifiers like “all” and “some,” and therefore we cannot display the validity of certain arguments in propositional logic. There is an extension of propositional logic, however, that greatly extends our expressive and derivational powers by adding predicates, names, variables and quantifiers to L, along with suitable formation and transformation rules to create quantificational (or “predicate”) logic. Further extensions are also possible, including the addition of modal operators and/or predicates or the generalization of quantification beyond what is allowed in ordinary quantificational logic. But for the purposes of elementary probability theory we do not need to develop these other systems.

Premises and conclusions

One of the major uses of logic is the evaluation of arguments. An argument, in the sense in which we use that term in logic, has a set of sentences called the premises and a sentence called the conclusion. Intuitively, we want to say that in a valid deductive argument the conclusion is a consequence of the premises – that the premises, if true, would ensure the truth of the conclusion. Here we are using the notion of truth, which is the central semantic notion, to explicate the notion of consequence. The formal counterpart to the notion of consequence is the notion of derivability: from the premises, we can derive the conclusion by rules specified in advance. In a well-behaved system of logic, all of the consequences are derivable, and vice versa.

Actually, matters are a bit more slippery than this. Our intuitive notion of consequence doesn’t tell us what happens when the set of premises is inconsistent, for example. We resolve this by specifying the notion of consequence a bit more tightly: the conclusion is a consequence of the premises if and only if it is impossible for all of the premises to be true and the conclusion to be false. When the premises are inconsistent, this requirement is automatically satisfied (since it is impossible for all of the premises to be true). So on the tightened definition, any formula is a consequence of an inconsistent set of premises.

This tightening also tells us what to do when the set of premises is empty. A conclusion will be a consequence of an empty set of premises just in case it is impossible for the conclusion to be false – that is, in case the conclusion is itself a logical truth.

Note that the terminology of premises and conclusion marks a distinction not between types of formulas but between roles formulas may play. The very same formula may appear as a premise and as a conclusion, even in the same argument.