A contextual proposition is a statement that means something to a given relevant observer. Such a proposition relevant to one observer need not mean anything to any other observer.

In science, observers aim to assign truth values to contextual propositions on the basis of experimental outcomes. A truth assignment is the association, by a relevant observer, to a contextual proposition of one element of a set of truth values. Truth value sets are in general binary, that is, they have two elements denoted 0 and 1 respectively. Element 0 will often represent the truth value false whilst element 1 represents the truth value true, but the opposite convention could be used when required. Truth value sets containing three or more elements can occur: in Scottish law, the set of potential verdicts in a trial is {guilty, innocent, not proven}. Mathematicians have extended the classical Aristotelian theory of binary logic to multi-valued logics.

According to this line of thinking, contextual truth values are not intrinsic properties of propositions per se but assignments by relevant observers. In other words, this form of truth is contextual.

In this book, we define contextuality as the recognition of the circumstances or context under which a truth value is assigned. A contextual statement consists of a proposition or statement, and an associated context. In propositional calculus, contextual statements are known as material conditionals: the contexts are known as antecedents and the propositions are called consequents. A material conditional is usually written in the form C ^ A, interpreted as ‘if context C is true then assertion A is true’.

There are several points to keep in mind about contextual statements, because they impinge on our discussion of time in this book:

1. Contextual statements are not automatically true: truth values have to be determined by relevant observers. In science, truth values are validated by experiments.

2. A falsified contextual statement is not metaphysical: the knowledge that a physical theory gives bad predictions can be of great importance to physicists. Metaphysical statements simply have no truth values, so are of no use to physicists.

3. The truth value of a proposition relative to one context carries no implication as to its truth value relative to any other context, even if the difference between contexts appears to be small, negligible, or insignificant. This principle was recognized early in the history of quantum mechanics, a famous example being the double slit experiment. If any attempt is made to detect the slit ‘through which a particle had gone’, then the quantum interference pattern hitherto observed is replaced by a classically predicted pattern that shows no interference.

4. Contextual statements should not be interpreted automatically in terms of cause and effect.

Empirical truths

The laws of physics are examples of empirical truths, which are contextual truths that have been validated under such a broad range of contexts that they may be regarded FAPP (for all practical purposes [Penrose, 1990]) as absolute. Examples are the laws of conservation of energy and electric charge. The difference between empirical truths and absolute truths is that physicists can test the validity of the former. For instance, Einstein’s principle of equivalence, which is based on the assertion that inertial mass and gravitational mass are strictly proportional, has been validated to an extraordinarily fine precision [v. Eotvos, 1890], but in principle could be proved false.

For two thousand years, the proposition ‘space is Euclidean’ was regarded as an absolute truth. Following the work of Gauss, Lobachewsky, and Bolyai, it is now regarded as an empirical truth valid only when gravitation can be neglected.