How do we make sense of the paradoxes of philosophical logic in the larger picture of philosophy as a whole? In particular, we will consider here how to make sense of incompleteness theorems in the context of the philosophical perspective presented in this series. In this series we take contradiction to be the very essence of all discernment, philosophical or otherwise. To see where issues of completeness and consistency in philosophical logic support and are supported by this understanding we will look at these issues with an eye to the instincts that gave rise to them. It might fairly be said that as it stands they do not require support—all the surprising and counterintuitive results here can be phrased in a form that is perfectly sensible—but still the objections of our initial instincts have not been fully satisfied. We identify here our acts of distinction with our process of reason to see where distinction as understood in this series is related to the reason of philosophical logic and where contradiction relates the two. The entry finds us briefly stepping into the mathematical world of philosophical logic.
Often drawn is the conclusion that all reality cannot be reasoned about without contradiction—that things are not all either true or false. We say the use of the word ‘all’ ranging over the universe of all objects of reason will inevitably lead to contradiction, as per Russell's vicious circle. We say this because certain paradoxes arise in any systematized logic that aims to represent all reason. To characterize in modern terms what can be safely reasoned about we ask for collections of formulae Γ and individual formulae φ appropriate to a given consistent logical system L whether Γ ⊢L φ ≡ Γ ⊢Model φ; that is whether our criterion of truth for L is complete. Now the results in this area are uncontroversial under the standard interpretation of consequence and biconditional, but the study arises from a more primordial question. We not only want our theory to be consistent and definitive for every formula in it, but we want the full collection of formulae to encompass all reason as fully as possible. We want our left hand system to be substitutable for the right hand one and the right hand one to itself be all reason.
The standard reference of completeness is our semantical system, above called Model. Standard semantics may or may not encompass all that can be reasoned about, but we will leave the specific nature of Model aside and refer to a third system C that is assumed to be consistent and capable of interpreting at least every formula Model is and also all Model falls short of. That there may be more than one such C will be left aside as more detailed a point than inspection of our instinct requires, and we will pick one, all results following equally for any such one. So, our instinct is to find the largest systems of logic L such that for all Γ and φ in not just L but C, Γ ⊢L φ ≡ Γ ⊢C φ. The observation here is that this system can only meet our instinctual conditions of consistency and completeness if this is a fully tautological statement—that is, without any content or sense. If L is really capable of definitively expressing all C can express, the statement is no more than C = C. If there is some difference between L and C then the primordial instinct is contradictory—they are not substitutable. It is only once we create that contradiction that L is somehow distinct from C yet substitutable with C that we give sense to our systems of reason L. And without contradiction, we have only a contentless tautology that tells us nothing. The nature of this concept of nonequivalence corresponds well with Wittgenstein in his picture concept, and with his Tractatus statement 6.3751 demonstrating the that a thing cannot be two different colors, “... a particle cannot have two velocities at the same time; that is to say, it cannot be in two places at the same time; that is to say, particles that are in different places at the same time cannot be identical.” Though he might stop investigating our line of reasoning at this point of impossibility, we can still continue on since we do not restrict ourselves to seeing only facts that are the case as part of the world. We in fact want L ≠ C but still ∀Γ, φ Γ ⊢L φ ≡ Γ ⊢C φ. This is contradictory, but the resolution of this instinctual overstep is where the subtlety of our systems of logic come from.
That it is only when our concept of reason falls short of all reason that it makes a useful definition of reason is well supported with evidence from both first order logic and higher order logic. No system of reason can be complete and consistent yet first order logic is. Γ ⊢FOL φ ≡ Γ ⊢C φ. The resolution of apparent impossibility of the completeness of first order logic is that it is not complete, certainly not of all reason. We have this from cardinality arguments and the Löwenheim-Skolem theorems. It is even easier to resolve than that because clearly the first order formulae do not have sufficient richness to encompass all predicates of thought until we add enough axioms to create a higher order system such as set theory. That is, we weaken our quantification for which the equivalence holds to a class of formulae significantly lacking in expressiveness. Higher order logic is explicitly incomplete without the pretense of full exhaustiveness of its class of formulae. Gödel showed there are Γ and φ such that Γ ⊢HOL φ ≡ Γ ⊢HOL ¬φ. So even if higher order logic did admit arbitrary formulae, some of them cannot be given meaning so that we have ¬( ∀Γ, φ Γ ⊢HOL φ ≡ Γ ⊢C φ ). If it were complete it would induce inconsistency or incompleteness on C. But again, we really don't want it complete because that fact would become contentless at that time.
One might wonder if these contradictions, Gödel's incompleteness theorems and our instinctual conflicts, are really related. Could we have Gödel's result without identifying HOL with C? No. Without a reference to be incomplete against HOL isn't incomplete. That is, if we're going to restrict the formulae over which its judgments are quantified, we aren't going include those it doesn't assign a value. To put it unabashedly informally, it appears it is exactly our insistence that we can discuss objects of reason with reason that leads to Tarski's universality of language and the feature of contradiction in our larger contentless picture that allows us to create a smaller much richer picture characterized by vast tracts of consistency.