Life Is Non-Boolean

George Boole (1815-1864) invented what we now call Boolean logic, which effectively assumes that all propositions are classifiable as either “true” or “false”, with no gradations of evidence or undecided cases. This provided the basis for formally defining logical operators in terms of how they transform what are called truth values, which are just the Boolean values of “true” and “false”. The use of so-called truth tables to define logical operators by cases is characteristic of what is today called “classical” logic.

Though it certainly has its uses in technical contexts, this kind of approach has been criticized as tacitly presupposing what has been called logical omniscience or the assumption of a closed world. For example, in computer science Boolean data types are used to represent things that are stipulated to have one of exactly two values. Then if we can rule out one, we can simply assume the other. This creates a closed world. “Omniscience” is implied by this kind of assumption.

The great developer of mathematical logic Gottlob Frege (1848-1925), at least at one point in his career, went so far as to argue that there are really only two distinct logical propositions: “true” and “false”. Depending one’s point of view, what I call real-world meanings are either reduced to nothing by this or are irrelevant to it.

From the point of view of “constructive” logic, on the other hand, there are infinitely many distinct propositions, distinguishable by the combination of what they presuppose and what they imply. It begins to be possible to reconstruct real-world meanings within logic instead of only outside of it.

In real life there are countless distinctions that are relevant and meaningful, and countless things that we simply don’t know. (See also Logic for People.)

Verificationism?

If something is true, it ought to make a difference, and in some sense that difference ought to be verifiable. Generically, this is the space inhabited by logical positivism, but the logical positivists had rather specific, foundationalist notions of verifiability that I would not wish to follow. (Moritz Schlick, the founder of the Vienna Circle, spoke passionately of verification against “the Given”, which was supposed to be a bedrock of pure, uninterpreted empirical fact that would anchor the whole enterprise of science. He also literally talked about foundational “pointing”. But he had a good critique of epistemic claims for intuition and images; emphasized conceptual development, form, and structure; made interesting use of relations; and reportedly spoke of laws of nature as inference rules.)

Logic by itself will not reform the world. However, the analysis of illogic is generally salutary.

The kind of verification that seems most applicable to the sorts of meta-ethical theses I am mainly interested in would be pragmatic. I imagine general pragmatic verifiability as just extensive openness to rational examination, with a responsibility for due diligence. Obviously, this is a loose criterion, but as Aristotle would remind us, we should not seek more precision than is appropriate to the subject matter.

In principle, material-inferential things can be verified “as far as you like” by a sort of recursive expansion of material consequences and material incompatibilities.

Purely formal-inferential things can be rigorously verified by mathematical construction or something resembling it. In constructive logic, proof comes before truth, so verifiability is built in.

Constructive

Brandom’s inferentialist alternative to representationalism stresses material, meaning-oriented over formal, syntactic inference. Prior to the development of mathematical logic, philosophers typically used a mixture of reasoning about meanings with natural language analogues of simple formal reasoning. People in ordinary life still do this.

Where Brandom’s approach is distinctive is in its unprecedentedly thorough commitment to the reciprocal determination of meaning and inference. We don’t just do inference based on meanings grasped ready-to-hand as well as syntactic cues to argument structure, but simultaneously question and explicitate those very meanings, by bracketing what is ready-to-hand, and instead working out recursive material-inferential expansions of what would really be meant by application of the inferential proprieties in question.

For Brandom, the question of which logic to use in this explicitation does not really arise, because the astounding multiplication of logics — each with different expressive resources — is all in the formal domain. It is nonetheless important to note that formal logics vary profoundly in the degrees of support they offer for broad representationalist or inferentialist commitments.

Michael Dummet in The Logical Basis of Metaphysics argued strongly for the importance of constructive varieties of formal logic for philosophy. Constructive logics are inherently inference-centered, because construction basically just is a form of inference. (Dummet is concerned to reject varieties of realism that I would call naive, but seems to believe the taxonomy of realisms is exhausted at this point. This leads him to advocate a form of anti-realism. His book is part of a rather polarized debate in recent decades about realism and anti-realism. I see significant overlap between non-naive realisms and nonsubjective idealisms, so I would want to weaken his strong anti-realist conclusions, and I think Brandom helps us to do that.)

Without endorsing Dummet’s anti-realism in its strong form, I appreciate his argument for the philosophical preferability of constructive over classical logic. It seems to me that one cannot use modern “classical” formal logic without substantial representationalist assumptions, and a lot of assumed truth as well. If and when we do move into a formal domain, this becomes important.

As used in today’s computer science, constructive logic looks in some ways extremely different in its philosophical implications from Brouwer’s original presentation. Brouwer clouded the matter by mixing good mathematics with philosophical positions on intuition and subjectivity that were both questionable and not nearly as intrinsic to the mathematics as he seemed to believe. The formal parts of his argument now have a much wider audience and much greater interest than his philosophizing.

Constructive logic puts proof or evidence before truth, and eschews appeals to self-evidence. Expressive genealogy puts the material-inferential explicitation of meaning before truth, and eschews appeals to self-evidence. Both strongly emphasize justification, but one is concerned with proof, the other with well-founded interpretation. Each has its place, and they fit well together.