Is Validity Local or Global?
Once one has the non-disquotational notion of determinate truth on the table (being true and not borderline true) in addition to disquotational truth, one might go on to ask what arguments count as valid.
If one maintains that a valid argument need only preserve determinate truth one gets a distinctive logic: one that agrees with classical logic about all of its theorems, but counts further inferences as valid which classical logic does not (and consequently must relinquish some meta-inferences to compensate—principles stating that if one sort of inference is valid, so is another). An inference that preserves determinate truth is called a globally valid inference, whereas an inference that furthermore preserves disquotational truth is called a locally valid inference. More explicitly, we say that an argument from premises Γ to conclusion A preserves determinate truth if the following material conditional holds of logical necessity:If each of the premises in Γ is determinately true, then A is determinately true.
Similarly, we say that an argument preserves disquotational truth if the corresponding conditional involving disquotational truth holds of logical necessity.
The global account of consequence is typically associated with traditional super- valuationism, where the notion of ‘supertruth’ is intended to replace the usual role of truth in reasoning and in other domains: on this picture the identification of a valid argument with an argument that preserves supertruth is extremely natural. However, there is an ambiguity in the literature: some supervaluationists define a globally valid inference as one that not merely preserves determinate truth (i.e. supertruth), but additionally preserves determinately determinate truth, determinately determinately determinate truth, and so on.
Thus on this alternative conception a globally valid argument Γ entails A when the following conditional holds of logical necessity:If each of the premises in Γ is determinate at all orders, then A is determinate at all orders.
The different ways of understanding global validity can lead to conflicting verdicts. The inference from A to ‘it’s determinate that A’ is taken to be a globally acceptable inference in this alternate sense even though it does not preserve determinate truth. It is immediate that if A is determinate at all orders then so is the claim that A is determinate, so the inference preserves determinacy at all orders. The inference does not preserve determinate truth, however, because A can be determinate without it being determinate that A is determinate. (To deny this would be to endorse the S4 principle for determinacy, an extremely contentious logic for determinacy. We will return to this in chapter 7.)
Here and throughout we will discuss inferences that involve the notion of de- terminacy itself. We represent this formally by introducing a sentential operator, Δ, into our language, where ΔA informally means ‘it’s determinate that A’.When Γ is a set of sentences, and A a sentence of this language, we will write Γ ∣= A to mean that the inference from Γ to A is valid.
The inference from A to ‘it’s determinate that A’ is globally valid on one understanding of‘globally valid’ but not the other. However, there are inferences which are uncontroversially globally valid—globally valid on both disambiguations—that are not locally valid. Thus I can, for the most part, draw the conclusions I need by focusing on these inferences. Here is one example of an inference that is globally valid on either understanding:[28]
(ΔECQ) A,—ΔA ∣= B.
This is effectively a strengthening of the rule of explosion.
If both A and —ΔA were determinate, then by assumption A would be determinate, and by the factiv- ity of determinacy, A would not be determinate. Thus it is simply impossible for the premises of this argument to be determinate, and ipso facto, impossible for the premises of this argument to be determinate at all orders. Thus in either sense of global validity these premises validly entail everything.Accounts of consequence that accept the above inference cannot be closed under certain meta-inference rules commonly associated with classical logic. For example, the inference rule of reductio would allow one to infer from the validity of the above inference and conjunction elimination that —(A ∧ —ΔA). This is equivalent to the absurd principle (A → ΔA), which entails, via excluded middle, that everything is either determinately true or determinately false. This same conclusion could also be inferred using conditional proof with some other uncontroversial logic. Thus conditional proof must be relinquished as well.[29] For this reason people have called global accounts of consequence ‘semi-classical’.
Just as we argued that ΔECQ preserves determinate truth, we can also argue that it doesn't preserve disquotational truth. For if ΔECQ preserved disquotational truth that would mean that A and —ΔA could never be both disquotationally true together, which means in particular that all instances of the problematic schema we just encountered, A → ΔA, would be always be disquotationally true.
Thus we have another question that one must make one’s mind up about:
Validity: Is the consequence relation global or local?
One might be tempted to think that a disagreement about logic is automatically to be taken seriously.
But this is too fast, for it is far from clear that Validity actually corresponds to a disagreement about what the correct logic is.
One could imagine two people who, when restricted to sentences not involving the word ‘valid’, agree about everything and are willing to reason from the same sets of sentences in exactly the way. But one could also imagine that one of the disputants counts no inference as ‘valid’ for spurious philosophical reasons (choose your favourite). This looks like it is purely a disagreement about which inferences to call ‘valid, and not about logic at all. (Since we may even suppose that both disputants are classical logicians: that they both assent to sentences like ‘either Harry is bald or he isn’t’, and they draw exactly the same conclusions from these sentences. It’s just that while one considers this sentence to be a logical validity, the other merely counts it as a truth that is not valid.)Note that there are genuine questions about how best to use the word ‘valid’ Should we apply it to inferences that preserve truth in all models that keep the meanings of the truth functional connectives and the quantifiers fixed, or should we apply it to inferences that preserve truth in models that keep the interpretations of other expressions fixed as well (such as the identity symbol, the modal operators, generalized quantifiers, and so on)? Should we count the theorems of higher-order logic as logical validities? Are contingent but a priori sentences like ‘I’m here now’ valid?
Presumably the answers to these questions won't be resolved by simply reflecting on our pre-philosophical notion of validity, but on the slightly more technical notion that is of interest to philosophers.
Such questions are often not as productive as they might first appear. It could just be that there are lots of different properties that arguments can have, none of which is more important for theorizing about reasoning than any other (see, for example, Beall and Restall [14]). In which case it may just be a matter of preference which to accord the title of ‘valid argument'.
But whether this is merely a matter of preference or not, the important point here is that these disputes are importantly different from the dispute between the classical and non-classical approaches to vagueness, where the truth of certain principles of logic, such as the instance of excluded middle mentioned above, are actually in contention. Those who maintain that the schema ‘if it's necessary thatp thenstyle='font-style:italic'>p, is not a logical truth because they do not consider modal operators to be logical constants, are nonetheless in full agreement with everyone else about the fact that what's necessary is true—they are not endorsing a non-standard logic of necessity, only a non-standard view about which truths are logical. One must take care, then, to distinguish a dispute about logic from a dispute about ‘logic': the classical and non- classical logician are having the former kind of dispute, whereas someone who denies that the truths of modal logic are logical validities is having the latter.Notice, then, that in the dispute about local and global validity both participants of the dispute agree about the following: ΔECQ preserves determinate truth but not disquotational truth (recall that both of these conclusions were derived from assumptions that both sides accept: that the schema ΔA → A is generally true, and that the converse schema A → ΔA is not). Thus the dispute looks initially as though it is merely a dispute about which of these inferences to call ‘valid', and not a dispute about the inferences themselves. In particular, someone endorsing a global account of validity need not be endorsing a revisionary logic.