1.3.6. Indexicality

We will give less direct attention indexicality than to implicature or presupposition in later chapters but it would be hard to ignore the phenomenon. There are certainly many sentences without such indexical words as I, that, here, and yesterday, but there are other features of a sentence, most notably its tense, that can make the proposition it expresses vary with context in which it is asserted.

If the propositions expressed by sentences vary with the context, it seems that the logical properties and relations of these sentences (which we trace to the propositions they express) may vary as well. Let’s look at one example. The proposition expressed by the sentence I am here will depend on the speaker, the speaker’s location, and the time of utterance. And this sentence may express the same proposition as the sentence You are there when the latter is used by a second speaker in an appropriately related context. However, there are many contexts in which these sentences might be asserted where they would not express the same proposition. But sentences are supposed to be logically equivalent when they express the same proposition, so it seems these sentences would be equivalent when used in some contexts and not equivalent when used in other. And this sort of concern affects deductive properties as well as relations; a sentence that is a tautology when used in one context might not be a tautology when used in another.

More broadly it seems that we really should not speak of sentences as having deductive properties and standing in deductive relations. If a sentence expresses no fixed proposition independent of the context in which it is asserted, we can really only talk about the deductive properties and relations of sentences-in-context, of sentences each taken together with a context of use. The term statement has sometimes been used to speak of a particular use of a sentence. If we use this terminology, we can say that certain statements made using the sentences I am here and You are there are equivalent and that it statements rather than sentences have deductive properties and stand in deductive relations. Something like this approach would be required if we really were to study the phenomenon of indexicality. However, that is beyond the scope of our study of deductive logic since the logical forms on which we will focus do not include indexical elements.

We can set aside issues of context and speak of deductive properties and relations as holding of sentences in virtue of the specific ways the propositions they express vary with the context of use—i.e., in virtue of the characters of these sentences. For example, we can say that sentences are equivalent if their characters lead them to express the same proposition in any context of use, and we can say that a sentence is a tautology if its character leads it to express a tautologous proposition in every context of use. This means that, even though I am here and You are there may be used to make statements that are equivalent, we will not count these sentences as equivalent because it is not the case that, in each context, the propositions expressed by these sentences are the same. (Indeed, it is not easy to imagine even one context with respect to which the two would express the same proposition.)

We will not need to consider the characters of sentences and generalizations about context explicitly. We will simply take it for granted that sentences are being compared with respect to some context (but only one at a time), and we will speak freely of the propositions they express. We must be sure that what we assume about this context will hold for all contexts of use, but our consideration of deductive properties and relations will be based on very general principles and hold for any single context of use. There is an analogy here to a typical use of variables in algebra. When numerical laws are used to manipulated algebraic formulas, it is assumed that variables appearing in those formulas have been assigned numerical values. But there is often no need to consider what those values are since the laws being used apply to all numbers.

Still there are things we will miss by ignoring character and context. Shifts of context in the courses of conversation is one of these. The assertion I am here followed by the confirmation Yes. You are there is a simple example of this. Another phenomenon we will miss is the exploiting of context dependence to convery information about the context. If I assert Today is Tuesday, the proposition expressed may be no more informative than is Tuesday is Tuesday, but my assertion can still be helpful because someone who tries to accommodate it will need to take it to have been asserted on Tuesday, and will thus know what day it is. In short, even if the proposition expressed by Today is Tuesday is a given context is a tautology and conveys no information, the assumption that this sentence expresses a tautology (rather than an absurdity) in that context yields information about the context. And this way of deriving information can support a form of non-deductive inference.

When assessing the deductive validity of an argument we consider all contexts and compare the propositions expressed by the premises and conclusion in each one of them. And, when comparing these propositions, we look at all possible worlds. So, in effect, we ask whether the conclusion of the inference is true for every context and possible world for which the premises are all true. The context serves us only to fix the contextually variable meanings of indexical terms. Any relation between such terms that holds for any context will be reflected in judgments of validity and other deductive properties and relations. For example, we can expect that the terms today and tomorrow are related in such a way that Tomorrow is the day after today is a tautology and Today is Tuesday implies Tomorrow is Wednesday (here asssuming that Wednesday is the day after Tuesday is also a tautology).

Inferences can be evaluated in a way that incorporates accommodation to indexicals by considering each context and comparing the truth value of the premises and conclusion in the actual world of that context. If we assume that any possible world is the actual world of some context, this will yield the same result as the assessment of entailment for terms whose meaning does not vary contextually and also in the case of relations of meaning, like that of now and today that are established contextually but hold for all possible worlds in any context. But there are relations between the meanings of indexical terms that are established in any context, but only for the actual world of that context. For example, whoever is the speaker will actually be in the location of the utterance at the time of utterance, so the premise Today is Tuesday would justify the conclusion I am speaking on Tuesday in spite of the fact that this conclusion would not be true in every possible world in which the premise is true—even given the contextually assigned meanings of the terms. (To see why this is not a case of entailment, it may help to look at another example: although I am here now is true in the actual world of any context, the fact that it is also false in other possible worlds can be important for the meaning of sentences, like I am here now but I almost didn’t make it, that speak of other possibilities.)