What’s Wrong with Lewis Carroll’s Tortoise?
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’?1placeholder 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.2placeholder 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.3placeholder 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.4placeholder 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.5placeholder 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.
Logic and (Human) Belief
What we have basically spent most of our time on so far has been the tortoise to the neglect of Achilles. Achilles is the image of force. And this second character shows the problem from another perspective. What is the tortoise doing in asking Achilles to force him to accept the conclusion of the argument? In the first place he is showing us the ability to mistake a psychological or pragmatic problem for a logical one. And this itself points to the deeper lesson that human life and logic always exhibit a fundamental separation.
Forcing someone to accept or forcing them to believe something can be understood as a psychological relation of causation. In this regard, it is the question of certain physical or mental states bringing about other mental states. These cases of force focus on psychological connections between believing one thing and then another. The person is psychologically forced to believe something when certain psychological connections and the surrounding environment cause them to believe it. This is the idea of the spontaneous mirage, optical illusions, psychological priming or suggestion, and the premise of physically manipulating a subject’s brain to produce mental states. We are forced to see the following symbols… …as two groups of three dots by virtue of how our psyches organize visual experience. Psychologically we are forced to believe this; it ‘comes to mind.’
But this psychological force is not logical force or necessity. And we know this because we direct the very research of psychology by virtue of a grounding in ‘correct’ and ‘incorrect beliefs’. What would bias or prejudice be if logical implication were psychological causation? The tortoise cannot be meaning that logical force is psychological force.
There is a second meaning of forcing to accept or forcing to believe, understood as a pragmatics of belief or rules for belief (cf. Carnap 1950). The force we mean in these cases are not based on natural relations of causation, but on a voluntary adherence to a standard of rationality. We recognize there are rules for belief. And insofar as you wish to think logically, you are forced to accept these rules for belief. This is the old song of empiricism, positivism, logical empiricism, and its progeny. There are logical rules for forming new beliefs (particularly, theories in the sciences), and we call this inductive logic. These are cases of forming new beliefs based on our evidence, which is always restricted in some way. So it becomes a matter of pragmatics. What is best to believe or accept given our fatty human brains, current technology, funded research programs available to us, etc.? Cases of shortsightedness and rash decision-making, especially militarily, have often been phrased as being compelled to think x, or being forced to believe something based on what is most rational to believe in those circumstances.
The two basic ways to understand inductive rules for belief are either as (1) based on evidence itself — and then you get Hume’s problem of induction, akin to the problem here discussed — or (2) as the application of a rule, which is not itself another rule (Winch 1958). This second line is inaugurated by Wittgenstein’s later work but it is most clearly given by Peter Winch when he says, “learning to infer is not just a matter of [seeing] explicit logical relations between propositions; it is learning to do something” (Winch, p. 57). The tortoise fails to accept the conclusion of the syllogism not because it needs more premises, which again it would accept but no more accept the logical implication; the tortoise fails to accept the logical implication because it has not yet learned how to do this.
But again this puts the emphasis on the wrong thing (psychology) instead of logic. Is logical implication a rule for belief? Let’s turn back to the logical implication of the CI: if As are Bs and if this c is an A, then this c is a B. Or, consider again the Law of Contradiction: it is not the case that A and not-A are both true. The point is that the laws of logic are features of truth, or the connections between truths; they say nothing about belief or acceptance. Unlike psychological laws or pragmatic rules, they bear no relation to a thinker at all. Of course, we can rephrase a logical law into a normative statement about beliefs (or a hypothetical proposition as the tortoise does), and say ‘If you believe A, then you should not believe not-A’. But the normative force of this statement is not thereby logical. And this is why the phrasing of the scenario by the tortoise already prevented Achilles from escaping the regress. Logical laws are in themselves not normative; they do not force any thinker to believe anything. That is why persuasion has nothing to do with truth (Schopenhauer Die Kunst, Recht zu behalten; Bernays Propaganda; etc.).
All of this is to say that logic and human life will always exhibit a separation. Carroll’s puzzle is not a problem for logic per se as it is a problem for the tortoise qua conscious individual. It is the problem we all have of wanting to be forced or compelled; of wanting something in lieu of the indifferent void that meets us at every present; hence magic words; hence prayer; hence language. The rabbis of the Talmud believed in the mysterious power of language to create reality. Consider the narrative of the Golem in Jewish folklore: a creature sculpted from mud is brought to life by inscribing a word on it, such as emet (Hebrew for “truth”). The creature becomes unruly, its master removes the word, and the monster returns to dust. Or consider John 1:1 from the New Testament, “In the beginning was the Word, and the Word was with God, and the Word was God.” And consider magic words. “Abracadabra” is Arabic for, “I create as I speak.” This phrase was written on the clothing of Roman soldiers in the 3rd century to prevent malaria (Champlin 1981). And again it was put on the doors of Londoners in the 17th century to resist the plague (Defoe 1911). We have a very deep-seated desire as human beings to have a connection between world and word. We want words to change reality. And we don’t have to go very far for a proof — albeit a far shallower boon than magic promises. This essay is its own proof. It’s the very idea of communication. And for the sake of science, politics, and everyday life, we hope that we can actually change peoples’ minds by reasoned argument. But, nevertheless, as we have shown, logical arguments can no more force a rock to fly upwards than they can force a person to think anything. As Dostoevsky’s underground man exclaims in the heart of duress, “It may be the law of logic, but not the law of humanity” (Ch. 9).6placeholder
So what did the troubled tortoise teach us? In this quick, three-page scene we are exposed to two very important lessons in the nature of logic. First, that logical implication is something quite different than a proposition to be accepted. And second, that human reality can always ask “ and why must I?”
Aristotle, C. D. C. Reeve, and R. McKeon. The Basic Works of Aristotle. Paw Prints, 2008.
Carnap, R.Logical Foundations of Probability. Chicago: University of Chicago Press, 1950.
Carroll, L. “What the tortoise said to Achilles.” Mind 4.14 (1895): 278–280.
Champlin, E. “Serenus Sammonicus,” Harvard Studies in Classical Philology 85 (1981): p. 193.
Defoe, D. A Journal of the Plague Year. London: Dent, 1911.
Dostoevsky, Fyodor. Notes from Underground. Open Road Media, 2014.
Winch, Peter. The Idea of a Social Science and its Relation to Philosophy. Oxford: Oxford Press. 1958
I am following Carroll’s exposition itself in construing logical consequence as a matter of truth (semantic consequence), and not provability (syntactic consequence).
A syllogism is a logical argument that proves from true premises a true conclusion on the basis of logical validity. Syllogism-focused logic begins with Aristotle’s Organon; the most common form being the categorical syllogism Barbara: “If all A’s are B’s and all B’s are C’s, then all A’s are C’s.” How we should understand logical validity is the focus of the present paper.
Achilles’ hands have already been tied since the idea of accepting logical validity. This will be my point below.
Aristotle says as much in the Prior Analytics, “A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that they produce a consequence, and by this, that no further term is required from without in order to make the consequence necessary” (Bk 1, 24b18–22).
This name was inspired by the work of Richard Lee at the University of Arkansas. If we wished to rephrase the argument it would be a Modus Ponnens: If things are equal to the same, then they are equal to each other, and the two sides of this Triangle are things that are equal to the same. Therefore, the two sides of this Triangle are equal to each other. Whether or not we should rephrase this argument is unimportant to this essay.
In light of the magazine title Epoché it would be wrong of me to fail to mention the connection with this notion and that of the ability to suspend the world, or doubt, or the epoché, as utilized in 20th century phenomenology. The reason I do not explore this further is that although this separation between human life and logic smacks of the epoché, the notion of non-transgressable logical laws alluded to with Aristotle’s help above would not sit nicely in a Sartrean ontology, for example.