Tenses contra logic

The two terms are not contradictions but who has not thought as much at one time or another? Often it is not the question of tense that makes things difficult but the elusive aspect of English verbs: ‘I went by bus’ and ‘I always went by bus’. There is nothing in the verb ‘go’ that indicates regularity in the second sentence and a single event in the first.

Logic and tenses get in one another’s way on other levels too. The common sentence:

Either the payment has not been made or it has not been processed yet.

has the pattern of a disjunction: not P or not Q. If we reason with a disjunctive syllogism in mind, that is, by adding the premise: ‘The payment has been made,’ or P for short, then the conclusion is: therefore, it has not been processed yet. All’s fine. But if we add: ‘The payment has been processed,’ or Q for short, then we are forced to conclude that the payment has not been made. But how could the payment have been processed if it hadn’t been made?

With disjunction and conjunction, order does not matter in our logic and mathematics (there are logics where it does). ‘I took the money and went to the shop’ is the same as ‘I went to the shop and took the money’. Yet, just as frequency, order is a critical part of the English tense system.

Another oddity involves sentences such as:

My apple tree will produce a crop only if I spray it.

Here, ‘I spray’ is the necessary condition. In material conditionals, necessary conditions come in the main clause, so we can paraphrase the sentence as:

If my apple tree produces a crop, then I will spray it.

Immediately, we sense it is wrong. But it is not the logic that is wrong; rather the tenses got out of kilter in the shift. If we insist on this kind of paraphrase, then we should adjust the sentence to something like:

If my apple tree produces a crop, then I will have sprayed it.

Besides frequency and order, tenses convey causation – to which logic is largely indifferent. No one would think it odd to say:

If you drop it you’ll break it.

where clearly ‘dropping’ precedes ‘breaking’, and something of this order is captured in the tense sequence (present followed by future). But speculation as to the causes of the breaking might well turn up a sequence such as:

If you broke it, then you had dropped it. It is as simple as that.

The verbs have been reversed (and so the necessary condition is in the right place) but the order of events hasn’t.

Consider also: ‘You touch it, you bought it.’ Here the grammar misleads us, but the logic doesn’t. ‘Touching’ is a sufficient condition for ‘buying’, so it goes in the antecedent:

If you have touched, you have bought it.