Logical Consequence

Conceived as a field of inquiry, logic deals primarily with the concept of logical consequence. The aim is to determine what follows from what, and to justify why this is the case.

Part of logic's concern is to explain why such judgments are correct - when they are indeed correct - and to provide an account of other, often more controversial, cases in which it may not be clear whether the con­clusion does follow from the premises. As a result, it is no surprise that logical consequence becomes such a central concept. But logic is not any old study of logical consequence; it is a particular sort of study of this relation.

2.1 Formality and universality

From its early stages, logical studies has addressed the concept of logical consequence in a particular way. Since its inception in Aristotle’s work, and continuing through the mathematically developed work of Gottlob Frege (1967), logic has been conceived as a study of the relation of consequence having certain features. Central among them is the fact that logic is both universal and formal. It is formal in the sense that the validity of an argument is a matter of the form of the argument. It is universal in the sense that once a valid argument form is uncovered, it does not matter how that form is instantiated: every particular instance of the argument form will be valid as well.

But under what conditions is an argument valid - or, equivalently, under what conditions is the conclusion of an argument a logical con­sequence of its premises? The usual formulation, which is cashed out in modal terms, is: an argument is valid if, and only if, it is impossible for the premises to be true and the conclusion false.

(P1) Julia and Olivia are having dinner.

(C) Therefore, Julia is having dinner.

Clearly, if we have the conjunction of the premise (“Julia and Olivia are having dinner”) and the negation of the conclusion (“Julia is not having dinner”), we obtain a contradiction, namely: Julia is having dinner and Julia is not having dinner. Therefore, the argument is valid.

What is the logical form of the argument above? Note that the first premise of the argument, (P1), is equivalent to:

(PT) Julia is having dinner and Olivia is having dinner.

Thus, (PT) is a statement of the form: P and Q. The form of the argument is then:

(PT) P and Q

(C) Therefore, P

Now, we can clearly see why any argument with this form is bound to be valid. The conjunction of the premise (P and Q) and the negation of the conclusion (not-P) leads immediately to the contradiction: P and not-P.

This is, of course, an extremely simple example, but it already illustrates the two significant features of logic: universality and formality. The valid­ity of the argument ultimately depends on its logical form: the interplay between the logical form of the premise and the conclusion. And once the logical form of the argument is displayed, it is easy to see the universality of logic at work. It does not matter which sentences are replaced by the variables P and Q. Any instance will display the same feature: we obtain a contradiction if we conjoin the premise and the negation of the conclu­sion. The argument is indeed valid.

These two features of logic go hand in hand. By being formal, logic can be universal: the logical form of a valid argument ultimately grounds the fact that every instance of the argument will also be valid. Suppose, in turn, that all instances of a given argument form are valid.

2.2 The model-theoretic account: truth preservation and a priority

But there is another way of thinking about what grounds the validity of an argument. Why is it that in a valid argument it is impossible to conjoin the premises and the negation of the conclusion? On the truth­preservation account, this is because valid arguments are truth-preserving. If the premises are true, the conclusion must be true. Thus, if we were to make the conjunction of the premises of a valid argument with the nega­tion of the conclusion, we would obtain a contradiction. But even on the truth-preservation account, the formal aspect of logic is crucial. After all, it is ultimately due to its logical form that an argument is truth-preserving.

All of these features - formality, universality, and truth preservation - are clearly articulated in the most celebrated account of logical consequence in the literature. It is the proposal advanced by Alfred Tarski in terms of mathematical models (see Tarski 1936b). In order to formulate his account, Tarski first introduces what he takes to be a necessary condition for logical consequence, something that can be called the “substitution requirement.” This requirement connects a given class K of sentences and a particular sentence X as follows:

If, in the sentences of the class K and in the sentence X, the constants - apart from purely logical constants - are replaced by any other constants (like signs being everywhere replaced by like signs), and if we denote the class of sentences thus obtained from K by K', and the sentence obtained from X by X, then the sentence X must be true provided only that all sentences of the class K' are true. (Tarski 1936b, 415; emphasis omitted)

What is required here is a systematic replacement - or interpretation - of the expressions in the sentences of K and in X, so that the truth of the new sentence X is guaranteed by the truth of the new sentences of K.

We have in place here the core ideas of truth preservation, formality, and universality. The sort of replacement that Tarski envisions with the substitution requirement is one in which the logical form of the sentences in question is preserved, even though everything else changes. After all, the logical form of a sentence is ultimately given by the connections among the logical constants that occur in that sentence, and these constants are never replaced when the interpretation of the sentences of K and X is given. What emerges from this is truth preservation: the truth of the sentences in K' guarantees the truth of X'. And this result holds universally - as long as the conditions in the substitution requirement are met.

As an illustration of the substitution requirement, suppose that the class K contains two sentences:

(K₁) Either Olivia is sleeping or she is playing with her toys.

(K₂) Olivia is not sleeping.

And suppose that X is the sentence:

(X) Olivia is playing with her toys.

We can now form the class K' by replacing all the non-logical terms in the sentences of K; that is, all of the expressions in K’s sentences except for “either...

(Kj) Either Julia is dancing or she is resting.

(KJ) Julia is not dancing.

In this case, sentence X' will be:

(X ) Julia is resting.

Now, it is easy to see that if all the sentences of the class K' - that is, sentences (Kj) and (KJ) - are true, then sentence X' must be true as well. And clearly exactly the same result will be obtained for any other interpretation - any other replacement of the expressions in K and X. The substitution requirement is therefore met.

While developing his account, Tarski was particularly concerned with the formal aspect of logical consequence. In fact, for him, logical consequence and formal consequence amount, basically, to the same concept. The logical consequence relation is one that holds in virtue of the form of the sentences in question. As Tarski points out:

Since we are concerned here with the concept of logical, i.e., formal, con­sequence, and thus with a relation which is to be uniquely determined by the form of the sentences between which it holds, this relation cannot be influenced in any way by empirical knowledge, and in particular by know­ledge of the objects to which the sentence X or the sentences of the class K refer. The consequence relation cannot be affected by replacing the designa­tions of the objects referred to in these sentences by the designations of any other objects. (Tarski 1936b, 414-15; emphasis in original)

By highlighting the formal component of the consequence relation, Tarski is in a position to articulate, in this passage, a significant additional feature of logic. Since logical consequence depends on considerations of logical form, it becomes entirely independent of empirical matters. In fact, each particular interpretation of the non-logical vocabulary could be thought of as one way in which empirical (non-logical) considerations could impinge on the consequence relation.

With these considerations in place, we now have all of the resources to present Tarski's analysis of the concept of logical consequence. As noted above, Tarski's proposal is formulated in terms of models. A model is an interpretation - a systematic replacement of the non-logical expressions satisfying the substitution requirement - of a class of sentences in which all of these sentences are true. In this case, Tarski notes: “The sentence X follows logically from the sentences of the class K if and only if every model of the class K is a model of the sentence X” (Tarski 1936b, 417). Clearly, the truth-preservation component is in place here. After all, the truth of the sentences of K, which is exemplified in each model of K, guarantees the truth of X. Moreover, as we saw, the substitution requirement underlies the formulation of the models under consideration, since each model presupposes the satisfaction of that requirement. And given that, as we also saw, the universality, formality, and a priori character of logic emerges from this requirement, it is no surprise that the concept of logical consequence, as characterized by Tarski, also inherits these features.

2.3 Modal matters

Tarski's analysis of logical consequence also has an intriguing feature. Recall that when we first introduced the concept of logical consequence above, we did it in modal terms: an argument is valid if, and only if, it is impossible for the premises to be true and the conclusion false. This means that if the premises of a valid argument are true, the conclusion must be true as well. There is a necessity involved in the connection between premises and conclusion. However, prima facie, we do not seem to find such a necessity explicitly stated in Tarski's analysis. We simply have a universal claim about models: in a valid argument - that is, one in which the conclusion follows logically from the premises - every model of the premises is a model of the conclusion. Where does the necessity come from?

There are two ways of answering this question. At the time Tarski presented his analysis, there was considerable skepticism, at least in some quarters, about modal notions. These notions were perceived as being poorly understood and genuinely confusing, and it would be better to dispense with them entirely. Much of the early work of logical positivists, particu­larly Rudolph Carnap, can be seen as an attempt to avoid the use of modal notions. In fact, even W.V. Quine's well known distrust of modality can be interpreted as the outcome of this positivist hangover. In light of this historical background, it is not surprising that Tarski would try to avoid using modal terms explicitly in his analysis of logical consequence.

But there is a second, stronger response. Tarski did not think he needed to use modal concepts in his characterization of logical consequence given that the models he was invoking already had the relevant modal character. A mathematical model represents certain possibilities. If certain statements can be true together, that is, if there is no inconsistency in their conjunc­tion, and the latter is, hence, possible, then there will be a model - a particular interpretation of the non-logical expressions involved - in which these sentences are all true. If the sentences in question are inconsistent and, thus, their conjunction impossible, no such model will be available. Thus, models are already modal in the relevant way. That is why math­ematicians can replace talk of what is possible (or impossible) with talk of what is true in a model (or in none).

Of course, this response presupposes the adequacy of models to repre­sent possibilities, so that to every way something can be, there is a model that is that way. Perhaps this is not an unreasonable assumption when we are dealing only with mathematical objects. But since the scope of logic goes beyond the domain of mathematics - after all, we use logic to reason about all sorts of things - one may wonder whether the Tarskian is really entitled to this assumption.^[30]

2.4 Logical consequence and logical truth

But whether we adopt an explicitly modal characterization of logical consequence, or a model-theoretic one (in terms of models), once the concept of logical consequence is formulated, the notion of logical truth can be characterized as well. A sentence X is logically true if, and only if, X is a logical consequence from the empty set. In other words, any inter­pretation of a logical truth X is a model of X; that is, any such interpret­ation makes X true. So, it does not matter which set of sentences K we consider, X will be true in any interpretation. This includes, of course, the case in which K is empty.

There are those who have insisted, or at least have implied, that logical truth, rather than logical consequence, is the fundamental concept of logic. In the very first sentence of his elementary textbook, Methods of Logic, W.V. Quine notes: “Logic, like any science, has as its business the pursuit of truth. What are true are certain statements; and the pursuit of truth is the endeavor to sort out the true statements from the others, which are false” (Quine 1950, 1). This suggests a certain conception of logic accord­ing to which its aim is to determine the logical truths and to distinguish them from the logical falsities. In fact, Quine continues: “Truths are as plentiful as falsehoods, since each falsehood admits of a negation which is true. But scientific activity is not the indiscriminate amassing of truths; science is selective and seeks the truths that count for most, either in point of intrinsic interest or as instruments for coping with the world” (Quine 1950, 1). So, it seems that we are after interesting logical truths, even though it is not entirely clear how such truths can be characterized. What exactly would be an uninteresting logical truth? Quine, of course, does not pause to consider the issue. And he does not have to. After all, as he seems to acknowledge a few paragraphs after the quotation just presented,^[31] the cen­tral concept of logic is not logical truth, but logical consequence (see also Hacking 1979, Read 1995, Beall and Restall 2006).

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