5.1.1. Conditions

The use of or is not the only way of hedging what we say. Instead of hedging a claim by offering an alternative, we can limit what we rule out to a certain range of possibilities. For example, instead of saying It will rain tomorrow, a forecaster might say It will rain tomorrow if the front moves through. The subordinate clause if the front moves through limits the forecaster’s commitment to rain tomorrow to cases where the front does move through. If it does not move through, the forecaster’s prediction cannot be faulted even if it does not rain.

We will refer to the connective marked by if as the (if-)conditional and to sentences of the form ψ if φ as (if-)conditionals. The qualification if- is used here to distinguish this connective from connectives associated with only if and unless that we will consider in 5.2. The three connectives are closely related, we will refer to all three as conditionals. However, the if-conditional is the most important of the three we will consider, and a reference to the conditional without qualification will be to it. Outside of contexts where we are discussing several sorts of conditional sentence, a reference to conditionals will be to the various compounds formed using it rather than to the three sorts of connective. In fact, we will analyze the other two connectives in a way that makes the if-conditional the main component of the result, so compounds formed using the other two connectives will count as special sorts of if-conditionals.

Although we take the word if, like the words and and or, to mark a two-place connective, it raises somewhat different grammatical issues. Since it is used mainly to join full clauses, there is less often a need to fill out the expressions it joins to get full sentences (though, of course, pronominal reference from one component to another must still be removed). And there are special problems associated with it. The conditional is an asymmetric connective: it makes a difference which component is having its content trimmed and which expresses the condition used to trim the content. For example, there is a considerable difference between the following sentences:

The first is a truism about contests and merely rules out cases of Mike winning the prize without entering the content. On the other hand, the second suggests confidence in Mike’s success and rules out cases where he entered the contest without winning.

Still, no fixed order between the two clauses of a conditional is imposed by English syntax. Like other subordinate clauses, if-clauses can be moved to the beginning of the sentence. Thus the two sentences above could be rephrased, respectively, as the following:

Sometimes the word then will precede the main clause when conditionals are stated in this order; but, as the examples above show, this is not necessary.

We will use the asymmetric notation → (the rightwards arrow) or ← (the leftwards arrow) for the conditional. The subordinate if-clause will contribute the component at the tail of the arrow, and the main clause of a conditional sentence will contribute the component at the head. We will refer to these two components, respectively, as the antecedent (i.e., what comes before, in the direction of the arrow) and the consequent (what comes after, again in the direction of the arrow).

Since the difference between the conditioned claim and what it is conditional on is marked by the difference between the two ends of the arrow, the order in which we write these components makes no difference provided that the arrow points from the antecedent to the consequent. For example, Adam opened the package if it had his name on it could be written as either of the following:

This means that the reordering of clauses in English can be matched by our symbolic notation, with φ → ψ corresponding to If φ then ψ and ψ ← φ corresponding to ψ if φ. When we are not attempting to match the word order of English sentence, the rightwards arrow will be the preferred notation, and generalizations about conditionals will usually be stated only for the form φ → ψ.

We will use if φ then ψ as English notation for φ → ψ. Here the word if plays the role of a left parenthesis (as both and either do). We will not often use English notation for the leftwards arrow, but it can help in understanding the relation of the two to have some available. If we are to have anything corresponding to the form ψ ← φ, we will put if between the two components, so we need another word to the role of left parenthesis. English usage provides no natural choices, so we will have to be a bit arbitrary. The interjection yes does not disturb the grammar of the surrounding sentence, so it can be easily placed where we want it. So we will write yes ψ if φ as our English notation for the form ψ ← φ. This way of tying the words yes and if is not backed up by an intuitive understanding of English, so the yes in the form yes ψ if φ does not help in understanding the symbolic form. On the other hand, it does not interfere with the help that if provides; and, as an interjection, yes can help to mark breaks in a sentence in much the way punctuation does.

On the other hand, the leftwards arrow ← is the easier of the two to accommodate if we look for a simple English substitute to use along with parentheses, for ← corresponds directly to if. We will not often need to use English notation with parentheses in the case of conditionals, so finding something for the rightwards arrow → is not a pressing practical problem. However, the way this problem is typically solved emphasizes an important point about the conditional

Of course, we cannot use if also for the rightwards arrow. And, even if we were not using if for the leftwards arrow, it would not work for → since if in English must precede rather than follow the subordinate clause. And then will not do either since it is if that bears the meaning of the connective in English. The usual approach is to look further afield and employ the word implies. Lacking a better alternative, we will follow this practice and use the word implies (in this typeface) as an English version of → to use with parentheses.

There is some danger of confusion in doing this, for we have used implies as a synonym for entails in the case of a single premises, and the signs → and ⊨ have quite different meanings. In particular, the notation φ → ψ refers to a sentence that speaks only of the actual world while, in saying that φ ⊨ ψ, we make a claim about all possible worlds. One way to avoid the confusion is to say that φ → ψ expresses material implication while, when saying that φ ⊨ ψ, we express logical implication. We will discuss this distinction further in 5.3.1; but, for now, we can note that this terminology is intended to capture a distinction between a claim about what is a matter of fact on the one hand and a claim about logical necessity on the other. And, however we describe the difference, this is a case where the typeface definitely matters, for

is the use of an English word to provide an alternative notation for φ → ψ while

is our way of saying in ordinary English what is expressed in notation as φ ⊨ ψ.

To give an example of some of this notation in action, let us return to the idea that a conditional serves to trim the content of its consequent. This can be expressed in symbolic notation as the entailment

which says that the argument ψ / φ → ψ is a valid one. If we use English notation for the conditional, we might express the same entailment as either

or

and we express the relation in English, using implies to express entailment, by saying that ψ implies φ → ψ, that ψ implies if φ then ψ, or that ψ implies φ implies ψ. Of course, because we have all these options, we have many ways of avoiding potentially confusing expressions; but trying to discern the meaning of a potentially confusing but really unambiguous expression is a good exercise in sorting out the range of concepts we are working with.