Games for Your Mind

The History and Future of Logic Puzzles

Jason Rosenhouse

• Publisher: Princeton University Press
• Year: 2020
• ISBN:
  9780691200347
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

A lively and engaging look at logic puzzles and their role in mathematics, philosophy, and recreation Logic puzzles were first introduced to the public by Lewis Carroll in the late nineteenth century and have been popular ever since. Games like Sudoku and Mastermind are fun and engrossing recreational activities, but they also share deep foundations in mathematical logic and are worthy of serious intellectual inquiry. Games for Your Mind explores the history and future of logic puzzles while enabling you to test your skill against a variety of puzzles yourself.In this informative and entertaining book, Jason Rosenhouse begins by introducing readers to logic and logic puzzles and goes on to reveal the rich history of these puzzles. He shows how Carroll's puzzles presented Aristotelian logic as a game for children, yet also informed his scholarly work on logic. He reveals how another pioneer of logic puzzles, Raymond Smullyan, drew on classic puzzles about liars and truthtellers to illustrate Kurt Gödel's theorems and illuminate profound questions in mathematical logic. Rosenhouse then presents a new vision for the future of logic puzzles based on nonclassical logic, which is used today in computer science and automated reasoning to manipulate large and sometimes contradictory sets of data.Featuring a wealth of sample puzzles ranging from simple to extremely challenging, this lively and engaging book brings together many of the most ingenious puzzles ever devised, including the "Hardest Logic Puzzle Ever, " metapuzzles, paradoxes, and the logic puzzles in detective stories.

PART II

Lewis Carroll and Aristotelian Logic

CHAPTER 3

Aristotle’s Syllogistic

Lewis Carroll pioneered the field of recreational logic by publishing, in 1886, a short book called The Game of Logic. Noticing that logic could be pursued just for fun was no small accomplishment, given the scholarly climate at the time he was writing. The logic of his day was still dominated by the traditional logic of Aristotle and his followers, and this was primarily what was taught to university students. We cannot appreciate Carroll’s work until we have a solid grounding in Aristotelian logic.

Aristotle wrote six works related to logic. Today these works are known collectively as the Organon. Only one of these works was devoted primarily to issues in formal logic, and that work will be our focus here. Section 8.1 has some discussion of the Organon as a whole.

3.1 The Beginning of Formal Logic

The Greek philosopher Aristotle inaugurated the study of formal logic sometime during the 300s BCE, in a work referred to as the Prior Analytics.

For centuries prior to Aristotle, there were philosophers who not only pondered deep questions but also wrote down their thoughts. One wonders, then, why none of them felt it worthwhile to undertake a formal study of the principles of correct reasoning. Certainly the work of Plato and other pre-Aristotelian philosophers included intimations of logical ideas, but they made no systematic attempt to study logic in its own right (Kneale and Kneale 1962, 1–22).

A plausible answer is that Aristotle wrote at a time when Greek democracy still represented a relatively novel form of government. For the first time, political change could only be affected by persuading others to go along, and this put a premium on skill in rhetoric and argumentation. However, arguments can be persuasive without being correct. Skillful debaters can appeal to people’s emotions or biases, as opposed to their reason. When such arguments can direct the course of society, the development of methods for clear thinking becomes imperative (Shenefelt and White 2013, 33–48).

We can only speculate about Aristotle’s motives, however, since he does not pause to tell us why he chose to address this topic. He just dives right in. The Prior Analytics opens with:

We must first state the subject of our inquiry and the faculty to which it belongs: its subject is demonstration and the faculty that carries it out demonstrative science. We must next define a premise, a term, and a syllogism, and the nature of a perfect and of an imperfect syllogism; and after that, the inclusion or noninclusion of one term in another as in a whole, and what we mean by predicating one term of all, or none, of another.

A premise, then, is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite. By universal I mean the statement that something belongs to all or none of something else; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark to show whether it is universal or particular. (Aristotle 2001, 65)

It seems that Aristotle also inaugurated the tradition that logic texts should be turgid and dull.

The Prior Analytics addressed a particular kind of argument, referred to as a “syllogism.” Aristotle writes: “A syllogism is a discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so” (Aristotle 2001, 66). Deducing the correct conclusion of a syllogism can be seen as bringing out and making explicit information that was contained in disguised form in the premises.

Aristotle noticed that some arguments are made valid purely by their form, and not because of the empirical content of their premises. Hence the term “formal logic.” An example is the argument with which I opened the book:

All cats are mammals.

All mammals are animals.

All cats are animals.

The conclusion follows from the premises, but not because of anything we know about cats, mammals, or animals. Rather, it is valid because it is a concrete instance of the abstract form:

All Ss are M.

All Ms are P.

All Ss are P.

and this form is immediately recognized as valid. Let us use the letters S and P to distinguish the subject of a proposition from its predicate. The letter M denotes the middle term, as we shall discuss momentarily.

Let us now consider the following question: If you are given two statements of this form involving the terms S, M, and P with M appearing in both statements, what conclusion, if any, can be drawn relating S and P?

Sometimes this is an easy question, as in the example just noted. Here is another example where the conclusion is very natural:

All Ss are M.

No Ms are P.

No Ss are P.

In both of these examples, the conclusion follows very naturally from understanding the meanings of the words involved.

Other times, however, it is less simple:

No Ss are M.

All Ps are M.

If you are unaccustomed to this sort of thing, then it might take a moment to see what follows. Matters become more straightforward if you recognize that the first statement is equivalent to, “No Ms are S.” If you then reorder the statements you have

All Ps are M.

No Ms are S.

We recognize this as another instance of the form we previously considered, and quickly deduce the conclusion “No Ps are S.” This is equivalent to “No Ss are P.”

We could also have reached this conclusion by employing a Venn diagram, as shown in Figure 3.1.

In still other cases, no conclusion of the desired sort is possible:

No Ss are M.

No Ps are M.

There is nothing to be said here regarding the relationship of S to P.

Figure 3.1. A Venn diagram representing the premises “No Ss are M” and “All Ps are M.” The correctness of the conclusion “No Ss are P” is now re...