In the past couple of weeks, we've been looking at Tarski's work on truth and, relatedly, work on the Liar Paradox.
This week, and next, we'll look at work by Tarski and others on LOGICAL CONSEQUENCE and LOGICAL CONSTANTS.
(Owing to the strike, there was no lecture last week, and will probably be no lecture next week. But I'll put slides online.)
The truth definitions we've been looking provide definitions of truth for INTERPRETED languages, whose sentences have meanings that determine truth-values.
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 also provide definitions of truth in a MODEL for UNINTERPRETED languages, whose sentences don't have meanings that determine truth-values.
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 non-logical 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.
And using this, we can go on to define the notion of LOGICAL CONSEQUENCE (and related notions such as LOGICAL VALIDITY and LOGICAL TRUTH).
A sentence φ is a LOGICAL CONSEQUENCE of a set Γ of sentences IFF φ is true in every model in which every member of Γ is true.
To better appreciate the merits of this, let's think a little bit about what we want from a definition of the notions of logical consequence.
What is it for a conclusion, φ, to be a logical consequence of a set of premises, Γ?
ARGUMENT 1
The premise of this argument might not be true, but one thing we seem to be sure of is that, if it is 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: the conclusion of an argument is a logical consequence of its premises IFF 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.
But: the conclusion of an argument is a logical consequence of its premises IFF the argument is, in what sense, necessarily truth preserving?
One idea: the conclusion of an argument is a logical consequence of its premises IFF there is no POSSIBLE WORLD in which the premises are true and the conclusion is false.
This marks a difference between arguments 1 and 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
There is no possible world in which the premise of argument 3 is true and its conclusion is false, but its conclusion is not a logical consequence of its premises.
Notice that, while 'water' and 'H2O' necessarily refer to the same substance, it is not part of their meanings that they refer to the same substance.
So perhaps: the conclusion of an argument is a logical consequence of its premises IFF it is not CONCEPTUALLY possible for the premises to be true and the conclusion false.
Yet the relevant notion of conceptual possibility is murky. It depends on the analytic-synthetic distinction, famously attacked by Quine.
An alternative: the conclusion of an argument is a logical consequence of its premises IFF it is KNOWABLE A PRIORI that the argument is truth-preserving.
But 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, it's unclear this fares any better.
Moreover, while appeals to conceptual necessity and a priori knowability may help to distinguish argument 1 from arguments 2 and 3, they don't help with...
ARGUMENT 4
There is presumably no conceptual possibility in which the premise of argument 4 is true and its conclusion is false.
Moreover, that the conclusion of argument 4 is true if its premise is true seems to be something that we can know a priori.
Yet the conclusion of argument 4 does not seem to be a logical consequence of its premise.
In sum: appeal to the notion of necessity doesn't seem to get us any closer to sufficient conditions on a conclusion's being a logical consequence of a set of premises.
A different thought: while arguments like ARGUMENT 4 are truth-preserving, they are truth-preserving, not in virtue of their form, but rather in virtue of their matter.
To see the thought here, notice 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:
And intuitively, other instances of this pattern are truth-preserving.
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.
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:
A worry: it may be that every substitution instance of an argument is truth-preserving because of expressive limitations of the language in which it is formulated.
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 count as a logical truth.
Another worry: 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, will count as 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 explain logical consequence in terms of THE ABSENCE OF COUNTER-EXAMPLES. But they offer different accounts of the range of potential counter-examples.
On the substitutional conception, a counter-example is 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 (or structure) 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, for example, there are models in which the sentence '∃x∃y x≠y' comes out as false.
Tarski's model-theoretic account of logical consequence is appealing, then. But it is also faces a number of problems.
First, it relies on a distinction between the logical and non-logical expressions of a language. (A model pairs non-logical expressions with appropriate values.)
But how, exactly, are logical expressions or constants to be distinguished from non-logical ones?
This is THE PROBLEM OF LOGICAL CONSTANTS. It's the topic of next week's lecture, so I won't say any more about it today.
John Etchemendy famously offers two objections designed to show that Tarski's account of logical consequence is theoretically inadequate.
Etchemendy's first objection is that the model-theoretic account of logical consequence is CONCEPTUALLY inadequate.
On the model-theoretic account, an argument is logically valid IFF there are no models in which its premises are true and its conclusion is false.
According to Etchemendy, this leaves something essential out of account: the logical validity of an argument provides a GUARANTEE that the argument is truth-preserving.
It perhaps follows from the fact that an argument is logically valid that there are no models in which its premises are true and its conclusion is false.
But its logical validity does not consist in there being no models in which its premises are true and its conclusion is false.
According to Etchemendy, the model-theoretic account of logical consequence thus makes a mistake akin to that of mistaking the symptoms of a disease for the disease itself.
Etchemendy's second objection is that the model-theoretic account of logical consequence is EXTENSIONALLY inadequate.
He thinks the model-theoretic account both OVERGENERATES, i.e. declares as logically valid arguments that are not logically valid ...
... and that it UNDERGENERATES, i.e. declares as logically invalid arguments that are not logically invalid.
Etchemendy's focus is on overgeneration. But he does not think that the model-theoretic account overgenerates in first-order logic.
Thanks to an argument from George Kreisel (1967), known as the SQUEEZING ARGUMENT, it can be shown that the model-theoretic account does not overgenerate in first-order logic.
In order to find examples of arguments which are truth-preserving in all models but not logically valid, Etchemendy therefore focuses on second-order logic.
The argument turns on the CONTINUUM HYPOTHESIS. This is the hypothesis that there is no set whose cardinality is between that of the integers and the real numbers.
It is possible to use nothing but logical expressions of second-order logic to formulate a sentence which is true in all second-order models IFF the continuum hypothesis is true.
Call this sentence S. Its negation, ¬S, is true in all second-order models IFF the continuum hypothesis is false.
Now consider the following arguments:
ARGUMENT A
ARGUMENT B
If the continuum hypothesis is true, then S is true in all models, and ARGUMENT A is declared logically valid.
If the continuum hypothesis is false, then ¬S is true in all models, and ARGUMENT B is declared logically valid.
Either way, one of the two arguments gets declared logically valid. But, Etchemendy claims, neither of them is in fact logically valid.
Why not? The thought seems to be that they can only be logically valid if either the continuum hypothesis or its negation is a logical truth.
But it is not the case that either the continuum hypothesis or its negation is a logical truth.
We've looked at the intuitive notion of logical consequence, and seen that it seems to involve the notion of FORMALITY.
This is nicely captured by Tarski's model-theoretic account of logical consequence. But Tarski's account faces various problems.
Next week we'll focus on the problem of logical constants, but along the way look at an alternative to the model-theoretic account.