5.1.4. Examples

Since the order of the two components of a conditional matters, the chief problem in analyzing English conditionals lies in identifying the antecedent and consequent. The key to this is the rule of thumb that the arrow runs from the subordinate clause (the if-clause) to the main clause.

After providing symbolic analyses of the following examples, we will restate them with all arrows running rightwards. This avoids the problematic English notation for leftwards conditionals, and the logical principles that we will be going on to study will be stated directly only for rightwards conditionals.

Notice that these two sentences are not equivalent. If the first were stated in English with the if-clause to the left of main clause it modifies, we would have John drove, and, if it was raining, Sam rode along. So the first says definitely that John drove while the second does not.

Although it is easy to capture the content of the first sentence using an if clause that precedes the one it modifies, it is not easy to capture the content of the second sentence unambiguously with an if clause that follows the one it modifies. Indeed, that may be one reason that if-clauses are so often moved to the front. To get the same effect with an if-clause at the end we need to force a break before if by a long pause or analogous punctuation, such as John drove and Sam rode along—if it was raining.

In the next example, we tackle a conditional concerning the future. We will be forced to make a shift in tense when we state the subordinate clauses as independent components.

One of the uses of the simple present tense in English is to state the antecedents of indicative conditionals concerning the future. But once it is out of that grammatical context, a sentence in simple present tense does not speak of the future. In fact, some sentences in simple present tense have very few natural uses at all. For example, while If the meeting gets out early, I’ll call is unexceptional, the sentence The meeting gets out early would normally appear only either as part of certain style of narrative (e.g., The meeting gets out early. Sam calls. They go out to dinner.) or as a statement of a regularity (i.e., the sort of thing that might be stated more explicitly as The meeting always gets out early). The result of the analysis is the sort of double conditional mentioned in the discussion of Curry’s law in 5.1.2, and Curry’s law tells that the same thing could be said using If I’m in town and I get a chance, I’ll call.

The word if is, by far, the most common way of expressing a conditional in English but occasionally other expressions are used, the most common of which is provided (that). So the example above might have been expressed instead as If I’m in town, I’ll call provided I get a chance or If I’m in town, I’ll call provided that I get a chance.

Sometimes we wish to commit ourselves to different things when a condition is true and when it is false. One way of doing this is with the form (φ → ψ) ∧ (¬^± φ → χ), which we will refer to as a branching conditional (after the name of an analogous conditional command used in computer programming languages). A sentence of this form asserts one thing, ψ, if φ is true and something else, χ, if φ is false. In English, the term otherwise is often used to express the condition in the second conjunct, as in the following sentence:

In this use of the term, otherwise probably means something like if that is not the case and, in principle, the reference of that might be the consequent rather than the antecedent of the conditional that precedes it. That is, it might be possible to understand the example above to have the form (E → D) ∧ (¬ D → L). This alternative form is entailed by the form above (since E → D ⊨ ¬ D → ¬ E and ¬ D → ¬ E, ¬ E → L ⊨ ¬ D → L) but it is a slightly weaker claim since it does not rule out the possibility that E and L are false when D is true; that is, it does not rule out the possibility of going out to dinner instead of having a late supper even in a possible world where they do not arrive early.