4.5 Proposition properties

There are some key properties of these propositions which we also saw when discussing probabilities: mutual exclusivity and exhaustivity. Consider the second pair of competing propositions again:

• \(H_p^2\): Suspect 2 fired the gun,
• \(H_d^2\): Suspect 2 did not fire the gun.

These propositions are mutually exclusive since when one is true then the other is necessarily false (and vice versa); they cannot both be true. This occurs when two propositions have no logical overlap. Suspect 2 firing the gun and not firing the gun cannot both be true, they do not overlap. However, mutually exclusive propositions can both be false. If the defence’s proposition was instead that ‘Suspect 1 fired the gun’ (exonerating Suspect 2), then this excludes other possibly reasonable scenarios such as there being a third suspect who fired the gun and was not caught. ‘Suspect 2 did not fire the gun’ covers this alternative scenario.

These propositions are also exhaustive since they cover the entire set of all possibilities for the event in question. This means that one of the competing propositions must be true – there is no scenario in which both are false. We know that a gun was fired, so either Suspect 2 fired the gun or they did not. The proposition \(H_p^2\) covers one specific event and \(H_d^2\) covers everything else. When competing propositions are exhaustive and mutually exclusive then they cover all possible events and are logically separable, meaning that one of them must be true and the other is false. This either one or the other property makes competing propositions easier to evaluate.

These propositions were guaranteed to be mutually exclusive and exhaustive because of the way they were constructed. First we specified \(H_p\) then we negated \(H_p\) to get \(H_d\). The glass propositions provide another example:

• \(H_p^3\): The glass fragments originate from the smashed window,
• \(H_d^3\): The glass fragments originate from a source other than the smashed window.

If \(H_p^3\) is not true, then the glass fragments must originate from a source other than the smashed window, and so we get \(H_d^3\). This makes the propositions necessarily mutually exclusive and exhaustive.

The logic of negating propositions is in line with the burden of proof in UK criminal cases. It is for the prosecution to prove beyond reasonable doubt that their assertion of criminal liability is true. The defence, in proving reasonable doubt about the prosecution assertions need only show that the opposite of the prosecution story has reasonable credibility. Negating prosecution propositions is one logical way of doing this.

There are many situations in which negating the prosecution proposition is inappropriate however, e.g. when the defendant has a genuine alternative narrative for events. In this situation, the following assertion might be used:

• \(H_d^{3^*}\): the glass fragments originate from a glass barrier at a shooting range that Suspect 2 has recently been performing construction work on.

This proposition is still competing with \(H_p^3\), and \(H_p^3\) and \(H_d^{3^*}\) are still mutually exclusive. However, they are no longer exhaustive. This is because all possible alternatives to \(H_p^3\) are no longer considered by \(H_d^{3^*}\). It is now possible for both \(H_p^3\) and \(H_d^{3^*}\) to be false. This could happen if the suspect lies or is mistaken about their alternative version of events.