We've been looking at the significance of Tarski's theory of truth (and Kripke's alternative) for the Liar Paradox.
This week, we will look at its significance for thinking about the concept of LOGICAL CONSEQUENCE.
First, I'll explain the connection between Tarski's definitions of truth and contemporary MODEL THEORY.
Then, I'll explain how this suggests an appealing account of the concepts of LOGICAL CONSEQUENCE and LOGICAL TRUTH.
Next week, I'll look in a bit of detail at two problems for this account, the problem of LOGICAL CONSTANTS and some influential objections raised by John Etchemendy.
I'll also look at alternative account. Where Tarski's account is MODEL-THEORETIC, the alternative is PROOF-THEORETIC.
The Tarski-style definitions of truth that we sketched were definitions of truth for INTERPRETED languages, languages whose sentences have meanings that make them either true or false.
These include languages such as the LANGUAGE OF ARITHMETIC. In these, NON-LOGICAL expressions such as '0', 'S' ('the successor of'), '+', and 'x' have fixed meanings.
In later work, Tarski showed how we can provide definitions of truth in a MODEL for UNINTERPRETED languages, languages whose sentences don't have meanings that make them either true or false.
These include languages such as L1, L2, and L= of first year. In these, non-logical expressions such as 'P', 'Q', 'a', and 'b' do not have fixed meanings.
Roughly, a model for a language specifies just enough information about its nonlogical vocabulary for assigning truth values to each of the sentences of the language.
A bit more precisely, a model for a language L is a nonempty domain D plus an appropriate assignment of denotations from D to the basic non-logical expressions of L.
For example, constants (names) might be assigned objects in D and n-place predicates might be assigned sets of n-tuples of objects in D.
We can then define truth in a model for an uninterpreted language by abstracting from definitions of truth (simpliciter) that we give for interpreted languages with the same vocabulary.
In the case of the uninterpreted language of predicate logic, the result is the definition of truth in a model (or STRUCTURE) that you're familiar with from 1st year.
Using this, we can define the notion of LOGICAL CONSEQUENCE as follows:
A sentence φ is a LOGICAL CONSEQUENCE of a set Γ of sentences — that Γ ⊨ φ — IFF φ is true in every model in which every member of Γ is true.
And we can then go on to define the notions of LOGICAL VALIDITY and LOGICAL TRUTH as follows:
An argument whose premises are the members of a set Γ and whose conclusion is a sentence φ is LOGICALLY VALID IFF φ is a logical consequence of Γ
A sentence φ is a LOGICAL TRUTH (⊨ φ) IFF φ is a logical consequence of ∅ — i.e. IFF φ is true in every model.
Given a proof system for a language, e.g. a set of rules of inference, we can also go on to investigate such metatheoretical questions as:
Whether the proof system is SOUND, i.e. whether Γ ⊢ φ only if Γ ⊨ φ.
Whether the proof system is COMPLETE, i.e. whether Γ ⊢ φ if Γ ⊨ φ.
These sorts of questions had been raised (and in some cases settled) before Tarski. Tarski's achievement, by showing us how to construct precise definitions of truth in a model, was to bring them inside mathematics.
Let's have a closer look at the sorts of issues that arise in providing an account of logical consequence, so as to better appreciate the philosophical merits of Tarski's account.
ARGUMENT 1
The premises of this argument might not both be true, but one thing we seem to be sure of is that, if they are both true, the conclusion is also true.
Otherwise put: it is not the case that the premises are all true and the conclusion is false. We'll say that such an argument is TRUTH PRESERVING.
In order for the conclusion of an argument to be a logical consequence of the premises, it is necessary that the argument be truth preserving. But it is obviously not sufficient.
ARGUMENT 2
So what else is needed? There are broadly speaking two ideas. One appeals to the notion of NECESSITY. The other appeals to the notion of FORMALITY.
The first thought: when the conclusion of an argument is a logical consequence of its premises, the argument is, in some sense, NECESSARILY truth preserving.
That is to say, is in some sense not POSSIBLE for the premises to be true and the conclusion false.
The source of this thought: logic is in some sense INDEPENDENT of how things actually are.
But what sense of 'necessity' is at issue here? Three different ideas are usually suggested.
The first is that it is METAPHYSICAL necessity that is at issue.
In other words: when the conclusion of an argument is a logical consequence of its premises, there is no POSSIBLE WORLD in which the premises are true and the conclusion is false.
This marks a difference between ARGUMENT 1 and ARGUMENT 2. Although both are truth preserving, only ARGUMENT 1 is, in this sense, necessarily truth preserving.
But it does not mark a difference between ARGUMENT 1 and other arguments where, intuitively, the conclusion is not a logical consequence of the premises.
ARGUMENT 3
We might think that the problem here is that, while the words 'water' and 'H2O' necessarily refer to the same substance, this fact was an empirical discovery; it is not part of their meanings that they refer to the same substance.
This that it is CONCEPTUAL or ANALYTIC necessity that is at issue.
In other words: when the conclusion of an argument is a logical consequence of its premises, it is not conceptually possible for the premises to be true and the conclusion false.
This seems to mark a difference between ARGUMENT 1, on the one hand, and both ARGUMENT 2 and ARGUMENT 3, on the other.
But the distinction between ANALYTIC and SYNTHETIC truths is unclear. (And famously attacked by Quine.)
And it is still not sufficient for marking a difference between ARGUMENT 1 and every argument where, intuitively, the conclusion is not a logical consequence of the premises.
ARGUMENT 4
The final suggestion is that is it is A PRIORI KNOWABILITY that is at issue.
In other words: when the conclusion of an argument is a logical consequence of its premises, it is knowable a priori that it is not the case that the premises are true and the conclusion is false.
(This raises a host of issues. What is a priori knowledge? Do we have any? And if so, how is it even possible for us to have it?)
(Insofar as an account of how it is possible to have to a priori knowledge depends on the analytic/synthetic distinction, Quine's criticisms of the latter will have to be addressed.)
But there's a more immediate problem: the appeal to a priori knowability doesn't seem to distinguish ARGUMENT 1 from ARGUMENT 4 either!
In summary, while any of these three notions of necessity may help us to articulate necessary conditions on a conclusion's being a logical consequence of a set of premises, none of them seem to yield a sufficient condition.
What more is needed? One influential idea appeals to the notion of FORMALITY.
The idea is that, while arguments like ARGUMENT 4 are truth-preserving, they are not truth-preserving in virtue of their form, but rather in virtue of their matter.
One way to try to bring this out is to point out that ARGUMENT 4 is an instance of a certain pattern of argument, obtained by replacing its non-logical expressions with schematic letters:
And other instances of the same pattern are not truth-preserving:
ARGUMENT 5
By contrast, ARGUMENT 1 is an instance of a different pattern of argument:
The idea, then, is that if the conclusion of an argument is a logical consequence of its premises, the argument is truth-preserving in virtue of its LOGICAL FORM, where...
the LOGICAL FORM of an argument (or sentence) is the pattern of argument (or sentence) obtained by replacing its non-logical expressions with schematic letters.
There are three slightly different sources for this idea:
Let's try to state the idea a little more precisely.
When an argument (or sentence) is an instance of a certain logical form, we may say that it is a SUBSTITUTION INSTANCE of that form.
In these terms, it seems that the conclusion of an argument is a logical consequence of its premises only if every substitution instance of that argument is truth-preserving.
That is to say, it is a necessary condition on the conclusion's being a logical consequence of the premises that every substitution instance be truth-preserving.
(This will be accepted by anyone who accepts that there is such a thing as the logical form of a sentence, and so of an argument.)
Can we say something stronger? Can we say that it is also a sufficient condition on the conclusion's being a logical consequence of the premises?
This is the SUBSTITUTIONAL conception of logical consequence:
The conclusion of an argument is a logical consequence of its premises if and only if every substitution instance of that argument is truth-preserving.
(The substitutional conception is often associated with the Czech philosopher, logician, and mathematician, Bernard Bolzano (1781-1848), but it is perhaps more accurately associated with certain mediaeval philosophers, such as Buridan.)
One worry is that it may be that every substitution instance of an argument is truth-preserving not because the conclusion is a logical consequence of the premises, but because of the expressive limitations of the language.
For example, in a language that contains just one name, a, which denotes the number 2, and one predicate, F, which denotes even numbers, the sentence Fa will be a logical truth.
Another worry is that it may be that every substitution instance of an argument is truth-preserving because of contingent facts about the cardinality of the universe.
For example, since there are more than two objects, the sentence '∃x∃y x≠y', which contains no non-logical expressions, turns out to be a logical truth too.
Tarski's account of logical consequence can be understood to belong to the same tradition as the substitutional conception, but it is slightly different.
Both can be thought of as explaining logical consequence in terms of THE ABSENCE OF COUNTER-EXAMPLES. But they offer different accounts of the range of potential counter-examples.
For proponents of the substitutional conception, a counter-example is, as we have seen, a substitution instance of an argument's logical form whose premises are all true and whose conclusion is false.
For Tarski, a counter-example is rather a MODEL in which the premises of the argument are all true and the conclusion is false.
Since a model pairs non-logical expressions not with other expressions in the language, but rather with appropriate denotations from the domain, this addresses the first worry.
The translation of 'Two is even', for example, will turn out to be false in some models that pair the translation of 'two' with the number 3.
And since different models have different domains of quantification, with different cardinalities, it also addresses the second worry.
Since there are domains with just one object, there are models in which the sentence '∃x∃y x≠y' comes out as false.
This week, I've sketched Tarski's MODEL-THEORETIC account of logical consequence and other notions.
And we've looked at the place of Tarski's account within the more general context of thinking about these notions.
We've looked at attempts to spell out logical consequence in terms of NECESSARY preservation, where the relevant notion of necessity is understood as:
And we've looked at attempts to spell it out in terms of FORMALITY, finding that Tarski's model-theoretic account belongs to the same tradition as the SUBSTITUTIONAL conception.
Next week, we'll begin by looking at some problems for Tarski's account: