The profound thinker known as Lewis Carroll made enormous strides in both the imagination and in mathematics and logic. In an 1895 issue of Mind we are exposed to a masterful display of both at once. Carroll writes a three-page scene in which Aesop’s famous tortoise questions the Homeric hero Achilles on the foundations of logic. In effect, he asks: if I accept as true that ‘All As are Bs’, and that ‘This is an A’, why must I accept that ‘It is a B’? [1] I will discuss what I believe are the two important lessons that the troubled tortoise taught us: one about logic and one about ourselves.

We begin with Carroll flouting Zeno’s racetrack paradox. Clearly he accepts an implicit argument against it and must so for we see Achilles and the Tortoise at the end of the race. The hero is asked by the Testudine to consider a subtler ‘race course’ that, unlike what many believe, is in fact impossible to traverse. This infinite series is the logical syllogism. [2] If we accept the truth of the premises of an argument, it does not “force” or logically necessitate us to accept its conclusion. That is, it does not by itself; for this you need to accept that it is logically valid. “And what forces me to accept this?” asks the Tortoise.

Consider an argument from Euclid (Elements, Bk. I, Prop. 1):

(A) Things that are equal to the same are equal to each other

(B) The two sides of this Triangle are things that are equal to the same

(Z) The two sides of this Triangle are equal to each other

What the tortoise says in effect is this: ‘Imagine I accept A and B as true, but don’t accept Z. I am not logically necessitated to accept the conclusion. What would you say?’ Achilles replies, ‘I would say if A and B are true, Z must be true.’ ‘But why must I accept this?’ says the Tortoise. Let’s call this new proposition C:

(C) If A and B are true, then Z is true.

The point the tortoise first makes is that without accepting a further proposition stating the relation of logical implication or validity — our example is C — we are under no logical compulsion or necessity to accept the conclusion of a logical syllogism, like the argument above. The devastating second point is that even if we accept as true this added proposition, we are still under no force to accept the conclusion of the argument: I can accept A and B and C, but not Z. “If you accept A and B and C, you must accept Z,” Achilles cries out (p. 692). The tortoise goes on, “and why must I?” (ibid). By now Achilles’ hands are already tied. [3] His new appeal becomes just another proposition to be added to the syllogism, accepted, and then shown to still fall short of forcing the tortoise to accept the conclusion of the argument. Endlessly, ‘why must I accept this?’ If the logical validity of an argument amounts to accepting another proposition, and this proposition is related to other propositions as premises in an argument, then this new argument must again show its validity, and so on ad infinitum. Accepting that some set of true premises logically implies a conclusion depends on accepting another proposition, which depends on accepting another one, and another, and so on. There is a regress problem in logical implication. Foreseeing all of the additional propositions needed to conclude Z on the basis of A and B (Euclid’s original argument), Carroll’s tortoise looks to Achilles’ notebook and declares, “Plenty of blank leaves, I see! […] We shall need them all!” (p. 693).

The mighty Achilles cannot force the meager tortoise to accept the conclusion of a logical argument. We must leave the question open whether or not Carroll himself is the voice of the tortoise. The important question is what does the tortoise asking Achilles to force him to accept the conclusion of the syllogism show us? There are two important lessons to be drawn from the puzzle.

Taking Logical Validity as a Proposition Amongst Propositions

As I said, from the very first elucidation of the problem by the tortoise Achilles had his hands tied. This is because the tortoise is treating the ground of logical validity as equivalent to that of the truth of propositions or statements. He wants that A and B proves Z to be proven. Calling this request absurd goes back to Aristotle and I will touch on this below (Metaphysics 1006a12). But without this we can still give a clear definition of absurdity by saying it is a statement or argument that denies its own possibility; or the grounds of its own truth. Two popular examples are those sweeping epistemological chestnuts of empiricism and post-modernism: “Every truth is a matter of fact,” and “There are no truths,” respectively. Is it true that there are no truths?

Likewise, the turtle asking that logical proof be logically proven is absurd. If it had to be, it could never be. If the logical implication of the syllogism has to be itself proven, it could never be. Carroll is showing us through the tortoise’s lesson that logical implication is not the same as accepting the truth of a proposition (contra Conventionalism, Psychologism, etc.). We do not accept the conclusion of a valid logical argument in the same way that we accept the truth of ‘It is raining outside today here’ or ‘The King of France is headless.’ In other words, the validity of an argument is not a premise amongst other premises to be thought true or false. [4] It is something else. In the first place, we know that it is absurd to say logical proof is provable.

Let us bring back the original argument:

(A) Things that are equal to the same are equal to each other

(B) The two sides of this Triangle are things that are equal to the same

(Z) The two sides of this Triangle are equal to each other

The logical implication between the truth of A and B and the truth of Z does not itself need to be proven (or logically implied), and could not. And yet if A and B are true in Euclid’s above argument, Z must be true. What is the ground of logical implication then if not an acceptance of another claim? What makes Euclid’s argument valid?

The form of this argument is a categorical instantiation. [5] The argument makes a claim about the relation of the members of two categories and brings that relation to bear on a specific instance (‘the two sides of this Triangle’). We know that a categorical instantiation (CI) is valid when true and we know this not by argument, but because it is self-evident. CI is a logically valid syllogism. It states that if As are Bs and if this c is an A, then this c is a B. It does not matter what we refer to by the schematic variables: birds, nations, quasars, etc., the logical implications holds between premises of this form. And that the two premises logically imply the conclusion must be known some other way than through another logical implication. It is self-evident.

What does self-evidence mean? This question itself is a bad question, since it wants to reduce self-evidence to something else. Take for example the idea that a statement is self-evident if by knowing its meaning we know its truth. That is, we know x is true by understanding what it means. Similarly, some have defined self-evidence as something known to exist by the perception of certain marks or symptoms in experience. In both cases we deny the self-evidence its self-evidence. We cannot say that we know logical implication by virtue of something else (meaning, marks, etc.) and still call it self-evident. All we can say is, ‘if A and B are true, then Z is true’, is valid because if A and B are true, then Z is true. Self-evidence is opaque to analysis. One thing suggested to us is that if the validity of CI in some sense rests on just what it is to share class membership. If these two classes overlap and x is a member of one, x is a member of the other. That’s it. Not only is the request for a proof of logical provability in the first place absurd, but in the second place, logical implication or proof is self-evident.

To get closer to what this self-evidence means, let us consider another case of logical implication: the Law of Contradiction. It is always false that both A and not-A are simultaneously true. Both E and F:

(E) It is raining on my window right now; and,

(F) It is not raining on my window right now,

cannot be simultaneously true. Of course, like the tortoise, “some [may] demand that even this shall be demonstrated, but this they do through want of education” says Aristotle (Metaphysics 1006a4–6). As we have shown there is no positive way to demonstrate laws of logical implication; they are self-evident. Aristotle does give us what he calls a negative demonstration however, and this is the idea that in saying anything whatsoever you prove these logical laws. We can prove this by saying anything (1006a12). “And if [our opponent] says nothing,” Aristotle continues, “it is absurd to seek to give an account of our views to one who cannot give an account of anything […] For such a man, as such, is from the start no better than a vegetable” (ibid. 1006a13–15). In some sense discourse itself is constituted by these logical laws. To say anything is to be logical. And we cannot deny these, i.e. discourse cannot pull it off. It is true that if As are Bs and this is an A, it is a B. We cannot coherently deny this. As we have seen, logical implication is not itself shown through implication. Its grounds lie somewhere other than the grounds of accepting the truth of a proposition or statement. We do not accept it or prove it. It just is, viz. self-evident. We can now see a clearer idea of its self-evidence. We might say logical implication is true if anything is true. What the tortoise’s request teaches us is this.