Searching for a Middle Term

“But nothing, I think, prevents one from in a sense understanding and in a sense being ignorant of what one is learning” (Aristotle, Posterior Analytics; Complete Works revised Oxford edition vol. 1, p. 115). The kind of understanding spoken of here involves awareness “both that the explanation because of which the object is is its explanation, and that it is not possible for this to be otherwise” (ibid). To speak of the “explanation because of which” something is suggests that the concern is with states of affairs being some way, and the “not… otherwise” language further confirms this.

Following this is the famous criterion that demonstrative understanding depends on “things that are true and primitive and immediate and more familiar than and prior to and explanatory of the conclusion…. [T]here will be deduction even without these conditions, but there will not be demonstration, for it will not produce understanding” (ibid). The “more familiar than” part has sometimes been mistranslated as “better known than”, confusing what Aristotle carefully distinguishes as gnosis (personal acquaintance) and episteme (knowledge in a strong sense). I think this phrase is the key to the whole larger clause, giving it a pragmatic rather than foundationalist meaning. (Foundationalist claims only emerged later, with the Stoics and Descartes.) The pedagogical aim of demonstration is to use things that are more familiar to us — which for practical purposes we take to be true and primitive and immediate and prior and explanatory — to showcase reasons for things that are slightly less obvious.

Independent of these criteria for demonstration, the whole point of the syllogistic form is that the conclusion very “obviously” and necessarily follows, by a simple operation of composition on the premises (A => B and B => C, so A=> C). Once we have accepted both premises of a syllogism, the conclusion is already implicit, and that in an especially clear way. We will not reach any novel or unexpected conclusions by syllogism. It is a kind of canonical minimal inferential step, intended not to be profound but to be as simple and clear as possible.

(Contemporary category theory grounds all of mathematics on the notion of composable abstract dependencies, expressing complex dependencies as compositions of simpler ones. Its power depends on the fact that under a few carefully specified conditions expressing the properties of good composition, the composition of higher-order functions with internal conditional logic — and other even more general constructions — works in exactly the same way as composition of simple predications like “A is B“.)

Since a syllogism is designed to be a minimal inferential step, there is never a question of “searching” for the right conclusion. Rather, Aristotle speaks of searching for a “middle term” before an appropriate pair of premises is identified for syllogistic use. A middle term like B in the example above is the key ingredient in a syllogism, appearing both in the syntactically dependent position in one premise, and in the syntactically depended-upon position in the other premise, thus allowing the two to be composed together. This is a very simple example of mediation. Existence of a middle term B is what makes composition of the premises possible, and is therefore what makes pairings of premises appropriate for syllogistic use.

In many contexts, searching for a middle term can be understood as inventing an appropriate intermediate abstraction from available materials. If an existing abstraction is too broad to fit the case, we can add specifications until it does, and then optionally give the result a new name. All Aristotelian terms essentially are implied specifications; the names are just for convenience. Aristotle sometimes uses pure specifications as “nameless terms”.

Named abstractions function as shorthand for the potential inferences that they embody, enabling simple common-sense reasoning in ordinary language. We can become more clear about our thinking by using dialectic to unpack the implications of the abstractions embodied in our use of words. (See also Free Play; Practical Judgment.)

Aristotelian Demonstration

Demonstration is literally a showing. For Aristotle, its main purpose is associated with learning and teaching, rather than proof. Its real objective is not Stoic or Cartesian certainty “that” something is true, but the clearest possible understanding of the substantive basis for definite conclusions, based on a grasping of reasons.

Aristotle’s main text dealing with demonstration, the Posterior Analytics, is not about epistemology or foundations of knowledge, although it touches on these topics. Rather, it is about the pragmatics of improving our informal semantic understanding by formal means.

For Aristotle, demonstration uses the same logical forms as dialectic, but unlike dialectic — which does not make assumptions ahead of time whether the hypotheses or opinions it examines are true, but focuses on explicating their inferential meaning — demonstration is about showing reasons and reasoning behind definite conclusions. Dialectic is a kind of conditional forward-looking interpretation based on consequences, while demonstration is a kind of backward-looking interpretation based on premises. Because demonstration’s practical purpose has to do with exhibiting the basis for definite conclusions, it necessarily seeks sound premises, or treats its premises as sound, whereas dialectic is indifferent to the soundness of the premises it analyzes in terms of their consequences.

We are said to know something in Aristotle’s stronger sense when we can clearly explain why it is the case, so demonstration is connected with knowledge. This connection has historically led to much misunderstanding. In the Arabic and Latin commentary traditions, demonstration was interpreted as proof. The Posterior Analytics was redeployed as an epistemological model for “science” based on formal deduction, understood as the paradigm for knowledge, while the role of dialectic and practical judgment in Aristotle was greatly downplayed. (See also Demonstrative “Science”?; Searching for a Middle Term; Plato and Aristotle Were Inferentialists; The Epistemic Modesty of Plato and Aristotle; Belief; Foundations?; Brandom on Truth.)

Three Logical Moments

The “Logic Defined & Divided” chapter of Hegel’s Encyclopedia Logic contains some brilliant, relatively popular aphorisms from his lectures, and provides a nice introduction to his views. Having recently treated with approval Kant’s denunciation of speculation in the usual sense, I’m turning to this now because among other riches, it contains Hegel’s recovery of an alternative, much more positive sense for “speculation”. As Aristotle would remind us, things are said in many ways, and it is wise to give heed to the differences.

Hegel says that every notion and truth involves three moments that are all essential and cannot really be separated from one another: Understanding, Dialectic, and Speculation.

In other places, Hegel frequently polemicizes against the narrowness and rigidity of mere Understanding. Here, he rounds out the picture, noting that “apart from Understanding there is no fixity or accuracy in the region of theory or of practice” and that knowledge begins “by apprehending existing objects in their specific differences”. He cites examples of how Understanding contributes to science, mathematics, law, practical life, art, religion, and philosophy.

Preparing the transition to dialectic, he notes “It is the fashion of youth to dash about in abstractions — but the man who has learnt to know life steers clear of the abstract ‘either-or’, and keeps to the concrete”. Dialectic for Hegel if viewed separately is the moment of “negative” Reason or criticism. He says that dialectic subordinated to Understanding’s mode of thought leads to skepticism, but dialectic freed from this subordination builds on distinctions developed by the Understanding, even while “the one-sidedness and limitation of the predicates of understanding is seen in its true light”. Dialectic studies things “in their own being and movement”. He goes on to expound Plato’s use of dialectic, and its difference from sophistry. (See also Contradiction vs Polarity; Aristotelian and Hegelian Dialectic.)

Speculation in Hegel’s special sense is the “positive” moment of Reason, which if considered separately begins from a kind of faith in reasonableness in the world. He implicitly connects it with a charitable reading of the long religious tradition of faith seeking understanding, construed in such a way as to be not incompatible with a charitable version of Enlightenment criticism. He notes that “the true reason-world, so far from being the exclusive property of philosophy, is the right of every human being [of] whatever grade of culture or mental growth”, adding that “experience first makes us aware of the reasonable order of things… by accepted and unreasoned belief”. Once this rational order becomes an object of thought rather than mere belief, we have speculative Reason proper.

Speculative Reason builds on both Understanding and dialectic. “A one-sided proposition… can never even give expression to a speculative truth.” He notes a connection between this and basic intuitive fairness. Starting from a simple faith in the reasonableness of the world and advancing through various stages of criticism, speculative Reason ultimately realizes substance as subject, and overcomes the dichotomy of subject and object.

Dialectic undid the abstract, atomistic, foundationalist, “either-or” tendencies of isolated Understanding. Speculative Reason in Hegel’s sense turns this into a new affirmation. In many places, Hegel talks about Reason or dialectic in ways that subsume both the dialectical and the speculative moment described here.

I read Hegelian speculative Reason — or dialectic incorporating the speculative moment — as just ordinary reason moving forward without the crutches of foundationalism and dogmatic claims of certainty. Reason without foundationalism is concerned with the very same open-ended work of interpretation I have attributed to Aristotle. Ultimately, Hegelian Reason is defeasible rational interpretation of experience, optimistically doing the best we can with the resources we have, and always on the lookout for something better. Thus, it too can be reconciled with Kantian discipline. (See also “Absolute” Knowledge?)

Epistemic Conscientiousness

I see something like epistemic conscientiousness as almost the highest value, only potentially surpassable by what I will broadly call concern for others. This principally involves a commitment to understanding, which means always seeking deeper and better and more nuanced understanding (see Objectivity of Objects). But equally, it involves taking strong personal responsibility for our acceptance of claims. (See also Assumptions; Error.)

If I have accepted a claim — especially if I have acted on the basis of that acceptance, or encouraged others to accept it — and then encounter reason to question that claim, I have a responsibility to resolve the matter in some appropriate and reasonable way. If I accept a claim, I have a responsibility to also accept its consequences. I have a responsibility not to accept materially incompatible claims.

An epistemically conscientious person will also naturally care about the acceptance of claims by others. Particular concern for particular others will naturally tend to accentuate this. We also want to treat those others with respect and kindness, and combining this with questioning claims they have accepted can be delicate. (See also Things Said; Honesty, Kindness; Intellectual Virtue, Love.)