When classical sentential logic is used in a study of a particular subject it is extended by the addition of new axioms and new rules peculiar to that subject. In the previous posts we saw how the extension can take on a new character, because some of the familiar meta-rules, in the natural deduction formulation, become inadmissible.
Can we always understand such an extended logic as applying to statements which can be true or false, and can be used in arguments that are meant to preserve truth?
That is a large question, posed at a high level of generality. Any answer will have to be similarly short on specifics, it can only be more about form than about substance. But we can demand an answer that satisfies the basic criteria of success for a semantic analysis of logic. That means that we should show how to associate to each such extended classical logic a language, for which the logic captures precisely the valid argument relation in that language.
So let us consider such an extended logic, though leaving the specifics only specified in a general way — call it logic ExL.
The story of a world
What we need to ask first of all is this:
according to this logic, what must a true description of the world be like?
Such a description will contain a certain amount of logically contingent information, something that is just asserted to be true as it happens, quite independent of logic itself. Then in addition it has to have everything in it that the logic requires. And finally, of course, it must be consistent.
To have a good word for it, call such a description a Story. So if X is a Story, here is what it contains:
[I] Some (or no) statements which are logically contingent
[II] The axioms of classical sentential logic
[III] The additional axioms proper to the subject
[IV] Everything that follows by the classical rule of Modus Ponens
[V] Everything that follows by the additional rules proper to the subject
Now this is exactly what is normally meant by a theory in the logic ExL. A theory in a logic is a set of statements that is closed under the deductive inference relation of that logic. Using the familiar deducibility symbol:
A set of statements X is a theory of logic ExL if and only if X = {B: X ├ B in ExL}.
So could every consistent theory count as possibly being the One True Story of the World, a description of everything that is the case? On this philosophers differ, and there is choice for the logician of two policies.
The liberal policy says: Any consistent theory could be the entire truth about what there is.
The conservative policy says: Only consistent theories that are as informative as is logically possible could be the entire truth about what there is.
The conservative policy is practically automatic in the case of classical sentential logic, where the language is subject to evaluation by the familiar truth-tables. There each statement is assigned T or F, and its negation is assigned the other value. So it is understood that for any statement A, in the One True Story of the World, either A or its negation ~A appears. As a result, such a Story is a maximal consistent theory, which is to say, it is a consistent theory such that, if you were to add more statements to it, you’d at once make it inconsistent.
Beyond bivalence
However, if we are working in a context where bivalence is not assumed, where some statements may be neither true nor false, then we are not similarly constrained. We just have the option: liberal or conservative. I’m going to just go with the liberal policy here (see Appendix for remarks about whether that makes any difference).
So, let’s start on how we can read X, a consistent Story in ExL, as true. To begin, we know that it is classically consistent, that is, consistent by the criteria of classical sentential logic. Because of the soundness and completeness theorems for that logic, it follows that there are ways of assigning truth values to all the statements in the language which follow the familiar truth-tables, and which assign T to all the statements in X. That sort of assignment I’ll call a classical valuation:
A classical valuation of the the language is function φ that assigns each statement either T or F, subject to the constraint that φ(~A) = T if and only if φ(A) = F and φ(A ⊃ B) = T if and only if either φ(A) = F or φ(B) = T.
I’ll take just ~ and ⊃ as the primitive statement connectives, while the other truth-functional connectives can be defined in terms of them. And I’ll say “satisfy” for “assign T”.
In general there will be many classical valuations that satisfy X. Let us call them the members of class SAT(X). They assign T to the statements in classes [I], [II]. and [III] above, and they will not offer any counterexamples to [IV], Modus Ponens. But since the extra rules in [V] are so far ignored, many of those valuations will violate those rules. However, they offer no counterexamples to those rules among the statements that belong to X — for the simple reason that they satisfy X which is a theory, so that everything that follows from X by those rules is also in X.
What do the members of SAT(X) have in common? That they all satisfy X.
How do they differ from each other? By their assignment to statements that are independent of X.
Now if we think of X as possibly the One True Story of the World, then statements independent of X have no significant role to play. So when it comes to genuine truth and falsity, that is just truth and falsity assigned by all members of SAT(X).
So we define what should count as possibly correct truth-value assignments in two steps:
The function Ψ is a supervaluation induced by set Y of statements if and only if, for any statement A, Ψ(A) = T if and only if for all members φ of SAT(Y), φ(A) = T, and Ψ(A) = F if and only if for all members φ of SAT(Y), φ(A) = F; and Ψ is not defined otherwise.
The function Ψ is an admissible valuation for our language if and only if it is a supervaluation induced by a theory in ExL.
Soundness and completeness
We can note quickly that there is a one-to-one correspondence between theories in ExL and admissible valuations. For if Ψ is a supervaluation induced by theory X, and X is a theory, then the set of statements assigned T by Ψ is a theory that includes X. But in that case if B is not a deductive consequence of X then it is not a classical deductive consequence of X, and so there is a classical valuation that satisfies X but not B. So then B is not in the set of statements assigned T by Ψ.
What about valid arguments? Define A to be a semantic consequence of set X in our language if and only if all admissible valuations that satisfy X also satisfy A. Because of what we just saw, that is the case if and only if A is a member of all theories in ExL that contain X. And that is the same as A being a deductive consequence of X in ExL. In other words, the logic ExL is sound and complete with respect to our language.
That’s it! We have just shown how to read what happens in logic ExL in terms of what can be true and false together, what can be a Story of what the world is like. For every extended classical logic there is a language for which it is sound and complete.
APPENDIX: the liberal and conservative options
There is certainly a diversity in world view signaled by the difference between the conservative and liberal policies. Leibniz, reasoning on behalf of a rationalist God, insisted that only a maximum diversity allowed by overall unity could have sufficient reason to be created. But Kant thought of the world, as well as the past, as ‘ein unendliche Aufgabe’, an incomplete and incompletable problem, as opposed to a completed whole. These views are at least suggestive of the two options as to what could possibly be the One True Story of the World.
When it comes to the logic, and more specifically the proof of soundness and completeness, however, the choice does not make a difference.
To begin, if every theory that contains X also contains A (so that A is a semantic consequence of X according to the liberal policy) then it is certainly also the case that every maximal consistent theory that contains X also contains A.
What about the converse? Suppose that some theory X does not contain A. Then there is a classical valuation φ which satisfies X but assigns F to A. Now think of all the theories Y which are satisfied by this classical valuation, and contain X. That family is partially ordered by set inclusion, and the union of any chain in that family also belongs to the family. So, by Zorn’s lemma, the family contains a maximal member, call it X*. This set contains X, and is satisfied by φ, so it assigns F to A. Thus X does not semantically entail A even on the conservative policy. The logic is the same on both options.
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