The keener one’s sense of logical deduction, the less often one makes hard and fast inferences" — Bertrand Russell, 1872–1970
Through the Looking-Glass of Logic
In 1897, Carroll published a curious work designed to teach elementary symbolic logic with a pedagogical wink\(^{2}\). In it, he proposed a method for determining whether the conclusion of a syllogism truly follows from its premises—a method that can be expressed, quite naturally, in the modern language of set theory.
This method invites comparison with contemporary approaches to assessing logical reasoning. While modern cognitive tests frequently employ syllogistic forms, they typically do so under strict time constraints and with reduced structural complexity. Such conditions favor rapid decision-making, superficial pattern matching, and familiarity with test formats over sustained analytical engagement. As a result, these assessments may capture only a limited subset of the cognitive processes involved in formal reasoning.
In some instances, such assessments reduce logical reasoning to the bare recognition of elementary distinctions—verifying, under pressure, that A is not non-A in the same respect and at the same time (e.g., the so-called PI cognitive assessment). While such discrimination is a necessary condition for rational thought, it is hardly a sufficient one (Stanovich, 1999).
Carroll’s works, by contrast, reflect a different conception of logical ability, one that emphasizes careful deliberation, the management of multiple constraints, and the exploration of non-obvious implications. Rather than isolating reasoning into quickly solvable components, his logical puzzles encourage extended reflection and conceptual clarity. In this sense, they align more closely with pedagogical traditions that regard reasoning as a deliberative, structurally complex skill developed through sustained engagement (Pólya, 1945; Piaget, 1950; Bruner, 1960), and they predominantly engage slow, analytic reasoning processes as opposed to the rapid heuristic responses privileged by time-limited cognitive assessments (Stanovich, 1999; Kahneman, 2011).
To illustrate his method, Carroll offered statements that are at once rigorous and slightly madcap. The following examples will show a translation of this method into set-theoretic language. Suppose we are given the following premises:
“All cats understand French;
Some chickens are cats.”
What conclusion can be drawn? (Don't let these ridiculous premises distract you from answering correctly.)
To answer this question, we identify the sets involved: Let \(F\), \(C\), and \(H\) be the sets of all entities that understand French, all cats, and all chickens, respectively. Using set-theoretic concepts, the first premise implies that \(C\subseteq F\). Similarly, the second premise affirms that \(H\cap C\neq \emptyset\), that is, there is at least one element in both \(H\) and \(C\).
Now, suppose \(x\in H\cap C\), then \(x\in H\cap F\) because \(C\subseteq F\) . Hence, \(H\cap C\subseteq H\cap F\neq \emptyset \) by definition of subset. Therefore, both \(H\) and \(F\) have an element in common. In other words, we necessarily deduce that some chickens understand French! This situation may be represented in the Euler diagram below,
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| The black dot represents the fact that \(H\cap F\neq \emptyset\)
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The Train Puzzle and the Illusion of Necessity
Now, consider the following syllogism:
"No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station."
Question: Does it follow that this party of tourists does not need to run?
You may be tempted to conclude that this party of tourists does not need to run if everything you saw was the given premises. Moreover, linguistic confusion can arise from the way language is employed in this example. But, is this a valid conclusion to draw? Can we have absolute certainty about it? To see that the answer is no, let \(S\) be the set of all people who mean to go by the train and cannot get a conveyance. Then, consider the set \(A\subset S\) defined as:
$$A=\{ a|a\in S \wedge a\text{ has enough time to walk to the station}\}$$
And the set \(B\subset S\) defined as:
$$B=\{ b|b\in S \wedge b \text{ needs to run}\}$$
Bear in mind these set definitions, as we analyze the whole argument in terms of set operations and relations. In fact, the people who mean to go by the train but cannot get a conveyance and do not have enough time to walk to the station are all the elements that are in \(S\) but not in \(A\), that is, \(S-A\) (the set difference between \(S\) and \(A\)) whereas the people who can take the train without running are all the people who are in \(S\) but not in \(B\), which simply is \(S-B\).
Hence, the first premise is equivalent to saying that \(\forall x\in S,x\in S-A\implies x\notin S-B\). Note that to say \(x\notin S-B\) is to say \(x\in \left( S-B\right)^{c}\implies x\in B\) by definition of complement. Consequently, the first premise tells us that \(\forall x\in S,x\in S-A\implies x\in B\), i.e, all the people who mean to go by train and cannot get a conveyance but do not have enough time to walk to the station need to run.
If on the other hand, we consider the set \(P\) as being the party of tourists; the second premise tells us that \(P\subseteq S\), and \(\forall x\in S,x\in P\implies x\in A\), ie. "this party of tourists in \(S\) have enough time to walk to the station" The first premise refers to \(A's\) relative complement in \(S\), and the second makes reference to \(A\) itself. It has not been established at any rate that \(A\subset S-B\). On the contrary, we have only been able to show that if \(x\in S-A\) then \(x\in B\) by the first premise, which is the inverse of "if \(x\in A\) then \(x\notin B\) " (what would have been needed to deduce the alleged conclusion).
Nothing so far has excluded the possibility of \(A\cap B\neq \emptyset\). To see this concretely, imagine, as Carroll invites us to imagine, what would happen if there were a mad bull behind them. Or, using a more modern example, imagine what would happen if there were an armed psychotic following the tourists around the station!
The Assizes Puzzle
Carroll often preferred to clothe his logical puzzles in the sober language of courts and committees, perhaps to remind us that fallacious reasoning is not confined to Wonderland. The final example, drawn from such a setting, appears straightforward enough at first glance, yet rewards a slower and more cautious reading:
“Of the prisoners who were put on their trial at the last Assizes, all, against whom the verdict ‘guilty’ was returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also sentenced to hard labour”.
Question: Does it follow that some, against whom the verdict ‘guilty’ was returned, were also sentenced to hard labour?
Let \(P\) denote the set of all prisoners who were put on trial at the last Assizes. Define the subsets \(G\subset P\), \(I\subset P\), and \(H\subset P\) as follows,
$$G=\{ g|g\in P \wedge g\:\:\text{the verdict "guilty" was returned} \}$$
$$I=\{ i|i\in P \wedge i\:\:\text{were sentenced to imprisonment} \}$$
$$H=\{ h|h\in P \wedge h\:\:\text{were sentenced to hard labour} \}$$
The first premise is equivalent to saying that \(\forall x\in P,x\in G\Longrightarrow x\in I\). In contrast, the second premise states that \(\exists x\in P\:\: \text{such that}\:x\in I\wedge x\in H\). The crucial error here lies in the fact that the statement \(\forall x\in P,x\in G\Longrightarrow x\in I\) is not biconditional. Based on the first premise and the definition of a subset, we have shown only that \(G \subset I\), but not the converse. Therefore, we have not proved that \(I \subset G\). Consequently, we cannot draw any conclusions insofar as it is still possible for \(G\) and \(I\) not to be identical to each other and thereby leaving wide open the possibility of \(G \cap H = \emptyset\).
This seems to defy common sense, as Lewis pointed out, who then were sentenced to imprisonment and were also sentenced to hard labour? They must have had the verdict ‘guilty’ returned against them, or how could they be sentenced? Well, it happened like this, you see. They were three ruffians, who had committed highway-robbery. When they were put on their trial, they pleaded ‘guilty’. So no verdict was returned at all; and they were sentenced at once (Carroll, 1897).
It's worth noting at this point that while it may make more intuitive sense to draw the corresponding Venn diagrams for each of the arguments as a substitution for the above procedure, it is always of great help to define the sets involved explicitly, even if this step is only done mentally, for otherwise it would do you no good to draw an erroneous diagram in the hope of determining the validity of a syllogism. The definitions make clear what we are referring to, and what the actual relations among the sets are.
Timeless Logic
Seen from this angle, Carroll’s logical puzzles are more than period curiosities or amusing sidelines to Alice. They embody a conception of reasoning that is increasingly easy to forget: logic is not a test of reflexes, nor a contest against the clock, but a disciplined form of attention—an art of coherent identification. What matters is not how quickly one answers, but whether one has carefully traced the structure of the problem and resisted the pull of premature conclusions.
The final example makes this especially clear. Its vocabulary is plain, its premises few, and yet the correct judgment depends on distinguishing what is merely suggested from what is strictly entailed. Nothing in the problem rewards haste; everything rewards patience. One must slow down, map the relevant sets, and accept that intuition alone is an unreliable guide.
Carroll understood that genuine logical ability reveals itself not in speed but in restraint—in the willingness to pause, to check, and to think twice. In an age increasingly enamored with fast tests and compressed measures of “ability,” his puzzles offer a quiet reminder that reasoning, like Wonderland itself, often becomes intelligible only when we are prepared to linger there a little longer.
References:
1. Bruner, J. S. (1960). The process of education. Harvard University Press.
2. Carroll, L. (1897). Symbolic logic. Macmillan.
5. Piaget, J. (1950). The psychology of intelligence (M. Piercy & D. E. Berlyne, Trans.). Routledge & Kegan Paul. (Original work published 1947)
7. Stanovich, K. E. (1999). Who is rational? Studies of individual differences in reasoning. Lawrence Erlbaum Associates.
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