Charles Dodgson, Modus Ponens, Achilles, and the Tortoise

Charles Dodgson, probably better known to most by his pen name Lewis Carroll and his books Alice In Wonderland and Through the Looking Glass, was a logician and mathematician at Oxford University.  The Alice books are actually wonderfully full of logical puzzles and paradoxes, and I have heard the claim made that the reason that everyone in Wonderland is insane, is precisely because they are all perfectly logical, within their own parameters.

I want to talk about something else today, though.  At one point, Dodgson wrote a short dialogue between swift-footed Achilles and the Tortoise, sometime after Achilles, impossibly, has caught up with the Tortoise and is riding on his back (like Darwin in the Galapagos).  For some reason, the conversation has turned to a discussion of the modus ponens, the logical validity of which Achilles is trying to persuade the Tortoise.

Modus ponens is one of the most basic valid argument schemata in logic.  It goes as follows:

  1. P ⇒ Q
  2. P
  3. ∴ Q

Or “If P then Q ; P ; therefore, Q”.

It is a very elementary form of inference.  So elementary, in fact, that it presents an interesting problem, which Dodgson brings to our attention.


Achilles has proposed to the Tortoise that modus ponens is a valid form of inference. The Tortoise, however, has an objection.

“Does premise 1, that P implies Q, prove that Q?” asks the Tortoise.

Achilles replies that premise 1 alone does not prove Q.

“Does premise 2, P, just by itself, prove that Q?” ask the Tortoise.

Achilles again admits, correctly, that premise 2, P just by itself, does not prove Q.

Achilles explains with great patience to the (seemingly) slow Tortoise, that it is by having premise 1, P implies Q, together with premise 2,  P, that one may validity infer that Q.  Both premises are needed to validly infer Q. Or, to put in another way, only the two premises together logically entail Q.

“I see,” replies the Tortoise. “You are saying that neither premise 1 nor premise 2 can allow you to validly infer Q.  You need both. But this ‘both’ is contained neither in premise 1 nor in premise 2.  But if it is necessary for the argument to be valid, then it must be a part of the argument.  So, really, you are proposing a premise 3, which would state that “premise 1 together with premise 2 imply Q”, or ((P ⇒ Q) & P) ⇒ Q; giving us:

  1. P ⇒ Q
  2. P
  3. ((P ⇒ Q) & P) ⇒ Q
  4. ∴ Q

Achilles agrees to this.  What is needed to arrive at Q is premise 1, premise 2, AND their conjunction, which is now premise 3.

The Tortoise says, “This is very good, but it seems to me I see a problem. You see, as we have already established, neither premise 1 nor premise 2 prove that Q.  But premise 3 doesn’t do it either, since it is an hypothetical implication (like premise 1). Premise 3 can only prove that Q in conjunction with premise 1 and premise 2.”

Achilles is forced to agree.

“Thus,” continues the Tortoise, “it is necessary to add a fourth premise, which will be the conjunction of the first three implying Q, namely “premise 1 and premise 2 together with premise 3 imply Q”, or ((P ⇒ Q) & P & ((P ⇒ Q) & P) ⇒ Q) ⇒ Q. Which yields:

  1. P ⇒ Q
  2. P
  3. ((P ⇒ Q) & P) ⇒ Q
  4. ((P ⇒ Q) & P & ((P ⇒ Q) & P) ⇒ Q) ⇒ Q
  5. ∴ Q

“But,” says the Tortoise, “it seems to me I still see a problem…”

Achilles groans, realizing that, unlike his catching up to the Tortoise, he cannot by mere speed escape this infinite regress of premises.


So what is Dodgson on about here?

I think the moral is a simple one. Dodgson is reminding us of one of the limits of what the Greeks call διάνοια, which we can translate as discursive reason, discursive rationality, discursivity, or simply as logical reasoning. This is the kind of reasoning that proceeds in a step by step fashion through a series of premises to reach a conclusion.

Normally, we say that if the premises are true, and the conclusion validly follows from the premises, then the the conclusion has been proven.  This is, in deductive reasoning, what a proof is.  And it is the gold standard of proof, since it yields certainty, unlike, say an argument which provides only proof beyond a reasonable doubt or by the preponderance of evidence.

However, Dodgson is showing us that any discursive argument can have its validity challenged by demanding yet another discursive link to “prove” that the offered discursive links are really connected in such a way that anything follows from them. Because a discursive proof is made up of discrete steps, one can always, in principle, demand that the “gap” between any two steps be proven by the addition of additional “bridging” premises—which of course by adding steps, create more gaps, which one can then insist be filled with yet more premises.  The very nature of a discursive proof allows this, since such a proof is not a continuous quantity, like a line, but a series of discrete steps.  This procedure can be repeated mechanically and indefinitely.  In one way, it is merely a trick on the order of endlessly repeating “But why?”

But it is more than that.  It shows us something the Greeks knew very well, which we moderns have largely forgotten (but which Dodgson knew, since he studied the Greeks), namely, that the discursive reason, the διάνοια, depends on a higher kind of reason or understanding, namely the νοῦς or νόησις (in its verbal form).  It is only by means of νοῦς/νόησις that we are able to directly apprehend the truthfulness of something evident. The vid- in “evidence” stems from seeing, and the Greek word for “knowledge” is the past participle of “to see”, so “to have seen.”  One knows when one is in a state of having seen.

  1. P ⇒ Q
  2. P
  3. ∴ Q

How do we know modus ponens is valid? We cannot prove it is, discursively. We simply see that it is, by means of our intellectual sight: νοῦς/νόησις.  Any normal human being sees that,

  • IF it is true that
  • 1 IF P is true, THEN Q is true
  • AND
  • 2 P is true
  • THEN
  • 3 Q is true.

This is almost a paradigmatic case of the self-evident. Something is self-evident on the condition that anyone who understands it, understands at the same time it is necessarily true.  (The self-evident should not be confused with the obvious; some things are very hard to understand, but have the character such that, once understood, they are also understood to be necessarily true—e.g. the identity of essence and existence in God).

The apprehension of self-evident truth by means of νοῦς/νόησις applies to all the ἀρχαὶ or first principles.  The basic laws of logic, for example, cannot be discursively proven.  Not because they are questionable, but because they are too basic to question, because any and every possible discursive proof already rests implicitly upon them, such as:

  • The law of noncontradiction: ~(A & ~A). “A thing cannot both be and not be the same at the same time and in the same respect.”
  • The law of identity: A = A. “A thing is what it is and not anything else.”
  • The law of difference: A v ~A.  “A thing and its contradictory divide reality exhaustively.”

This is why Aristotle says: “It is a sign of uneducatedness (ἀπαιδευσία) to demand a demonstration [discursive proof] of everything.”

It is not a bad thing to ask for demonstrations where demonstrations or proofs are possible, but it is nonsense to demand proofs of things which are not susceptible to proof by virtue of their simplicity and self-evident nature.  If you don’t believe me, try to prove in a non-question-begging way that anything can be proven, ever.  Or try to “prove” anything to an interlocutor who simply asks you to prove every premise you make use of in any proof.  It should be clear that both are impossible: to “prove” proof would be circular, and hence, no proof at all, even if it could be done; and the second results in an infinite regress, such that the proof never comes to an end, so nothing is ever proven.

Since the Enlightenment, modernity has tended to reject νοῦς/νόησις (in ever decaying steps: Descartes’ “clear and distinct perception” is his version of it; Kant denies we have an “intellectual intuition” but seems remarkably able to “see” the structures of pure reason which precede all experience—not to mention his uncanny ability to discover such things as the thing-in-itself and the transcendental unity of apperception).

But without its grounding in νοῦς/νόησις, διάνοια becomes ungrounded; discursive rationality breaks down into pure discursivity, inevitably becoming a kind of free-floating talk about talk about talk—which is to say, sophistry.  Which, as we would say it today, is postmodernism.

We moderns have picked up a bad habit of thinking that God’s creation of the cosmos was an event that happened “back then” in time, maybe the first event, but this is false in two ways.  Firstly, time itself is a creature; it belongs to the created world as part of its intrinsic structure.  The “in the beginning” or ΕΝ ἀρχῇ of the Scriptures does not refer to a temporal beginning, but an absolute ontological beginning. God is the absolute source, ἀρχή, of all things.  And it is also the case, as many theologians and philosophers have pointed out, that God’s creation of the world is not a “one and done” event, but an ongoing event. Without God’s constant donation of being to all things, they could not, of their own power, remain in being; they lack the power to do so.  God’s creation of the world is not like a mechanic’s fabrication of a machine (as Enlightenment deism tended to conceptualize it), but much more like a musician’s playing a song, or a poet’s recitation of a poem, or a storyteller’s telling his story: when the musician ceases playing, the music stops; when the poet ceases speaking, the poem stops; when the storyteller ceases telling, the story stops.

Why this little foray into theology? Because the real point Dodgson was making, and that I am making in this post is this: that νοῦς/νόησις has a relation to διάνοια/discursive rationality which in analogous to that between God and creation. νοῦς/νόησις, noētic sight, provides διάνοια/discursive reasoning not only with the ἀρχαὶ/first principles of logic and reasoning, but is present at every step taken in the course of dianoēic/discursive reasoning: every step must be, at each point, noētically seen to be valid; otherwise, it remains a logical leap, and therefore not reasoning at all.

This twofold nature of reasoning  yields the twofold nature of philosophy.  As Socrates enjoined us, we must always follow the λόγος, that is, our dianoēic/discursive reasoning; but at the same time, Socrates’ forever asks the question “What is it?”, a question that can only be answered, as he tells Menon, by looking: οὕτω δὴ καὶ περὶ τῶν ἀρετῶν: κἂν εἰ πολλαὶ καὶ παντοδαπαί εἰσιν, ἕν γέ τι εἶδος ταὐτὸν ἅπασαι ἔχουσιν δι᾽ ὃ εἰσὶν ἀρεταί, εἰς ὃ καλῶς που ἔχει ἀποβλέψαντα; “And so too surely, about the virtues: even if they are many and of all sorts, still they all have some one and the same form [ ἕν γέ τι εἶδος ] through which they are virtues and upon which one would somehow do well to focus one’s gaze.”

Philosophy is a dialectic of seeing and saying, that is, of νόησις of the εἰδή and of λόγον διδόναι, giving a λόγος or a rational account.  First, one must see. Then one tries to say what it is one has seen. In trying to translate sight into speech, problems emerge that force one to look again.  And then to try to speak better.  But the ultimate goal is to see adequately.

If we cannot see what is there to be seen, we cannot say anything meaningful about reality. Our talk will be ungrounded in reality, and thus without truth value.  Speech which aims to be a true account of reality and not simply a story or narrative, a λόγος rather than a mere μῦθος, needs to be an account grounded in true sight of reality, νόησις.  This is the meaning of “evidence.” To have e-vid-ence (video , “to see”, cognate with the Greek ἰδέᾳ/εἶδος ) is to have “that which can be seen.”  Discursive arguments require evidence, and evidence is ultimately always a matter of sight, of νόησις.

And some see better than others.  So it is wise to study the thinking of the wise.

One comment on “Charles Dodgson, Modus Ponens, Achilles, and the Tortoise

  1. Reblogged this on Chris Lansdown and commented:
    Eve explains why requiring all proof to be recursively, discursively priced is a form of skepticism that is simply a denial of reason. There is a real sort of skeptical attitude which is the thirst for truth, but there is also simply the refusal to believe something which lazily uses the impossible standard Eve very clearly describes.


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