The Game of Logicby

Habib Bin Muzaffar

Dictionary Definiton of Logic (Merriam-Webster Dictionary)

1. A proper or reasonable way of thinking about or understanding something

2. A particular way of thinking about something3. The science that studies the formal process

used in thinking and reasoning

The beginning of Mathematics• Should one start with the first methodical deductions in

geometry traditionally credited to Thales of Miletus around 600 B. C.? Or should one go back further and start with the empirical derivation of certain mensuration formulas made by the pre-Greek civilizations of Mesopotamia and Egypt? Or should one go back even further and start with the first groping efforts made by prehistoric man to systematize size, shape and number?........Or was mathematics, as Plato believed, always in existence, merely awaiting discovery? ( An Introduction to the History of Mathematics by Howard Eves)

Aristotle’s statement in his Metaphysics

• When all the inventions had been discovered, the sciences which are not concerned with the pleasures and necessities of life were developed first in the lands where men began to have leisure. This is the reason why mathematics originated in Egypt, for there the priestly class was able to enjoy leisure. ( The History of Mathematics, An Introduction by David M. Burton)

Thales of Miletus (circa 625-547 B. C)

1. First known person with whom specific mathematical discoveries are traditionally associated

2. Credited with introducing proofs in mathematics3. Credited with lots of wise statements4. When asked what was the strangest thing he had

ever seen, he is said to have answered “ An aged tyrant”

The Flourishing of Greek Mathematics (600-200 B.C.)

1. Extensive discoveries especially in geometry2. Discovery of irrational numbers3. Organization of mathematics, especially

geometry, into an ordred list of theorems proved in a “logical” manner

4. Exact calculation of the areas and volumes of several geometric objects (with proofs)

5. Essentially discovered the modern mathematical concept of “limits”

The most important contributors

1. Thales of Miletus (~625-547 B.C.)2. Pythagoras of Samos(~580-500 B.C.)3. Theaetetus of Athens(~415-369 B.C.)4. Eudoxus of Cnidos (~408-355 B.C.)5. Euclid of Alexandria (~300 B. C.)6. Archimedes of Syracuse (~287-212 B.C.)7. Appolonius of Perga (~262 B.C.-190 B.C.)

The Pythagorean school1. Pythagoras (~580-500 B.C.) formed a school at Crotona

when he was about 50 years old. 2. The aims of the school were political, philosophical and

religious.3. The community had the character of a secret society with

initiations, rites and prohibitions.4. Pupils concentrated on four subjects of study: arithmetic (in

the sense of number theory), music, geometry , and astronomy.

5. The school continued to exist for several centuries after the death of Pythagoras.

Commensurable quantities and rational numbers

• The Pythagoreans (the followers of Pythagoras) believed that any two line segments are commensurable, i.e there is a third segment, perhaps very small, that could be marked off a whole number of times on each of the given segments.

• In symbols, this says that given line segments of lengths r and s, there is a third line segment of length t such that and for some positive integers a and b. It follows that . In other words the ratio of the lengths of any two line segments is a rational number, i.e. a ratio of two integers.

The discovery of irrational numbers

• The Pythagoreans proved (most likely in the 5th century B.C.) that the diagonal of a square is incommensurable with any of its sides. It follows that is an irrationalnumber, i.e. it cannot be written as a ratio of two integers.

s d

s

The first crisis in mathematics (~5th century B.C.) and its resolution

1. The discovery of irrational numbers was a shock to the Pythagoreans.

2. They had used their wrong belief that any two line segments are commensurable in some of their proofs. Thus, a new approach was needed to salvage many of their theorems.

3. Around 370. B.C., Eudoxus (~408-355 B.C.) devised a theory of proportions which resolved the crisis by making it possible to give correct proofs of the aforementioned theorems.

Aristotle (384-322 B.C.)

1. Student of Plato2. Tutor of Alexander the great 3. Contributed to logic, philosophy, metaphysics,

mathematics, physics, biology, ethics, politics, agriculture, medicine and theatre

4. Wrote more than 200 treatises of which only 31 survive

5. Systematized the science of logic which was already in use by the Greeks

Aristotle’s logic

1. The heart of Aristotle’s logic is the syllogism.2. An example of a syllogism is:

“All men are mortal. Socrates is a man. Therefore Socrates is mortal.” Note that as long as the premises are true, the conclusion must be true.

3. The syllogistic form of logical argumentation dominated logic for more than 2000 years.

The Elements of Euclid (~300 B.C.)

1. A compilation of the most important mathematical facts available at that time

2. Organized into 13 parts or books3. Unified a collection of isolated discoveries into a single

deductive system based on a set of initial definitions, postulates and common notions

4. Was the standard introduction to geometry until the early 20th century

5. Modern introductions to geometry differ from the Elements in logical order and proofs of theorems but little in actual content.

Some definitions from the Elements

1. A point is that which has no parts.2. A line is a being without breadth.3. Parallel lines are straight lines which, being

in the same plane and being produced indefinitely in both directions, do not meet one another in either directions.

The Postulates (axioms) of the Elements

1. A straight line can be drawn from any point to any other point.

2. A finite straight line can be produced continuously in a line.3. A circle may be described with any center and distance.4. All right angles are equal to one another.5. If a straight line falling on two straight lines makes the

interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely meet on that side on which are the angles less than two right angles.

Flaws in the Elements

1. Some definitions are not satisfactory.2. Some terms are used but not defined.3. The postulates are incomplete, i.e. certain

tacit assumptions were used in the deductions (proofs) which should have been included as postulates or derived from them as propositions (theorems).

Fixing the flaws in the Elements

1. The flaws could be fixed by introducing some undefined terms (from which all other definitions must be made) and a complete set of postulates (or axioms). Such a collection would be called an axiom system for Euclidean geometry.

2. From the late 19th century onwards, many mathematicians attempted to give a complete statement of the postulates needed for proving all the familiar theorems of Euclidean geometry.

3. These attempts are regarded as satisfactory.

The most popular of these attempts

1. David Hilbert (1899): 21 postulates , 6 undefined terms, namely point, straight line, plane, on, congruent, and between

2. George David Birkhoff (1932): 4 postulates , 4 undefined terms, namely: point, straight line, distance, and angle

Note: Birkhoff ‘s postulates related geometry to real numbers.

The basic underlying philosophy

• The only assumptions (about the various geometric objects) which may be used in the proof of a theorem are the postulates or the theorems which have already been proved. No, other “tacit assumption” or “intuitively obvious fact” may be used.

Hilbert expressed this by saying “ One must be able to say at all times – instead of points, straight lines and planes – tables, chairs and mugs.” ( Hilbert by Constance Reid)

Symbolic or Mathematical Logic

1. Although the Greeks considerably developed the science of logic and Aristotle systematized the material, the early work was all carried out with the use of ordinary language.

2. Modern mathematicians have found it necessary to have a symbolic language in order to further develop the science of logic.

A dose of Modern Symbolic Logic : Propositional Logic

• Propositional logic deals with propositions.• A proposition is a declarative sentence (i. e.

a sentence that declares a fact) that is either true or false but not both.

• A propositional variable is a variable that represents propositions.

• Propositional variables are usually denoted by letters such as p, q, r, s etc.

Logical operators stands for “Not”, ˅ stands for “or”, ˄ stands for “and”. These are defined using truth tables as follows:

pT FF T

p q p˅qT T TT F TF T TF F F

p q p˄qT T TT F FF T FF F F

More logical operators: Implications

• → stands for “implies” , ↔ stands for “if and only if”

• is also read as “if p then q”

p q p → qT T TT F FF T TF F T

p q p ↔ qT T TT F FF T FF F T

The NOR operator ↓

This is defined as follows:

Note that is the same as .

p q p ↓ qT T FT F FF T FF F T

Some things of interest in propositional logic

1. To find the truth table of a compound proposition (i. e. a sentance made up using propositional variables and logical operators)

2. Given a truth table involving any number of propositional variables, to find a corresponding compound proposition

3. To study and develop relationships between various logical operators

4. To study the notion of functional completeness

Functional Completeness

1. A collection of logical operators is said to be functionally complete provided that every compound proposition can be expressed as a compound proposition involving only these logical operators.

2. It is not hard to show that the collection is functionally complete as is the collection .

3. The collection is fuctionally complete.

The Liar Paradox

1. The earliest attribution is to Eubulides of Miletus (~ 4th century B.C., contemporary of Aristotle) who said “A man says that he is lying. Is what he says true or false?”.

2. Alternate version: “This sentence is false”3. In symbols : L is the sentence “L is false”. This

means that L is true if and only if L is false. 4. In propositional logic, such sentences are not

considered.

The cannibals puzzle

• An explorer visiting an island is captured by a group of cannibals who reside on the island. There are two types of cannibals: those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular cannibal always lies or always tells the truth. What one question can he ask the cannibal to save himself?

Solution to the cannibals puzzle

Ask the cannibal any question whose correct answer is “yes”. A truthful cannibal will answer “yes” and a liar will answer “no”. One possible question is “Are you a resident of this island?”. Another possibility is “ Are you a cannibal?”.

The “villager and ruins” puzzle

Each inhabitant of a remote village always tells the truth or always lies. A villager will only give a “Yes” or a “No” response to a question a tourist asks. Suppose you are a tourist visiting this area and come to a fork in the road. One branch leads to the ruins you want to visit; the other branch leads deep into the jungle. A villager is standing at the fork in the road. What one question can you ask the villager to determine which branch to take?

V

A propositional logic approach to the “villager and ruins” puzzle

• r: The road on the right leads to the ruins you want to visit

• v: The villager standing at the fork is truthful

• S: Statement to be constructed

• A: Answer by villager to the question “ Is S true” • One possible S is

v r S AT T T YesT F F No F T F YesF F T No

Another solution to the villager and ruins puzzle

If I were to ask you if the road on the right leads to the the ruins, would you answer “yes”?

Raymond Smullyan (1919-)

Raymond Smullyan (brief biography)

• Dropped out of school• After moving from one university to another, he

was given an undergraduate degree in mathematics by the University of Chicago in 1955

• Obtained a Ph. D. in logic in 1959 at Princeton• Taught at Dartmouth College, Princeton

University, Yeshiva University, the City University of New York, and Indiana University

• Wrote several books on logic, puzzles and chess

How he got his undergraduate degree

• Took the College Board Exams and was accepted by Pacific University in Oregon

• Later studied at Reed College, University of Wisconsin and the University of Chicago

• Got a teaching position at Dartmouth College based on someone’s recommendation and a paper he had written

• He was a little short of the number of credits required to graduate from the University of Chicago

• The problem was solved by giving him credit for a calculus course he was teaching at Dartmouth but had never taken

His books about puzzles• (1978) What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles -

knights, knaves, and other logic puzzles• (1979) The Chess Mysteries of Sherlock Holmes - introducing retrograde analysis in the game

of chess.• (1981) The Chess Mysteries of the Arabian Knights - second book on retrograde analysis chess

problems.• (1982) The Lady or the Tiger? - ladies, tigers, and more logic puzzles• (1982) Alice in Puzzle-Land• (1985) To Mock a Mockingbird - puzzles based on combinatory logic• (1987) Forever Undecided - puzzles based on undecidability in formal systems• (1992) Satan, Cantor and Infinity• (1997) The Riddle of Scheherazade• (2007) The Magic Garden of George B. And Other Logic Puzzles, Polimetrica (Monza/Italy)• (2009) Logical Labyrinths, A K Peters• (2010) King Arthur in Search of his Dog• (2013) The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs

A puzzle from Raymond Smullyan The Politician Puzzle.

A certain convention numbered 100 politicians. Each politician was either crooked or honest. We are given the following two facts:

• At least one of the politicians was honest.• Given any two of the politicians, at least one of the two was crooked.

Can it be determined from these two facts how many of the politicians were honest and how many of them were crooked?

Solution to the Politician Puzzle

Let A be an honest politician. Let B be any other politician. Since at least one of A and B is crooked, it follows that B is crooked. This applies to any politician other than A. Therefore, there is one honest politician and the other 99 are crooked.

One last puzzle

Consider the following wrong equation:

How can you correct this equation by adding a single straight line segment? The following is not allowed:

550555

550555

Some contributors to the development of symbolic logic

1. Gottfried Leibniz (1646-1716)2. George Boole (1815-1864)3. Augustus De Morgan (1806-1871)4. Charles Sanders Pierce (1839-1914)5. Ernst Schröder (1841-1902)6. Gottlob Frege (1848-1925)7. Giuseppe Peano (1858-1932)8. Bertrand Russell (1872-1970)9. Alfred North Whitehead (1861-1947)

The second crisis in mathematics: The foundations of calculus

1. Calculus was developed in the 17th century and expanded considerably in the 18th century.

2. The mathematicians involved in this process failed to consider sufficiently the solidity of the base upon which the subject was founded.

3. With the passage of time, contradictions arose and it became necessary to properly deal with the foundations of calculus.

Resolution of the second crisis

1. Augustin-Louis Cauchy (1789-1857) , Karl Weierstrass (1815-1897) and his followers resolved the crisis by successfully developing a theory of limits.

2. They showed that all of existing calculus (and related areas) can be logically derived from a postulate set (or axiom system) characterizing the real number system.

Further work on foundations (Late 19th century)

1. It was shown that Euclidean geometry can be based upon the real number system, (i. e. the undefined terms of Euclidean geometry can be defined in terms of real numbers and then the postulates of Euclidean geometry can be proven using the postulates for the real number system).

2. It was shown that the real number system can be based upon a postulate set for the natural number system.

3. In order to carry out these programs, basic set theory is required.

The third crisis in mathematics: Paradoxes in set theory

1. Georg Cantor (1845-1918) systematically developed modern set theory.

2. It was hoped that all of mathematics can be made to rest upon set theory as a foundation.

3. The discovery of several paradoxes in set theory in the late 19th and early 20th century was a shock.

Bertrand Russell (1872-1970)

Russell’s paradox

• Bertrand Russell (1872-1970) discovered the following paradox:

Consider the set of all sets which do not belong to themselves (such a set was allowed in Cantorian set theory). Does this set belong to itself? Either possibility leads to a contradiction.

. Therefore,. have we,set any for Thus,

. that such sets all ofset let symbols, In

NNNNXXNXX

XXXN

Axiomatic set theory: A way to avoid the paradoxes of set theory

1. In the first half of the 20th century, many different axiom systems were developed for set theory which avoided all known paradoxes.

2. Unfortunately, these systems do not provide any guarantee that no new paradoxes will arise.

David Hilbert (1862-1943)

Hilbert’s dream

1. David Hilbert (1862-1943) developed “proof theory” or “metamathematics” in which logical reasoning itself was put into an axiomatic framework.

2. This led to the notion of a formal theory or system (a set of postulates together with a set of rules of inference, i.e. a logic).

3. His “dream” was to prove that any formal system that was rich enough to cover all of mathematics was consistent, i.e. did not produce any contradictions.

Kurt Gödel (1906-1978)

Gödel’s Incompleteness Theorems (1931): End of Hilbert’s dream

1. First incompleteness theoremAny consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.

2. Second incompleteness theoremFor any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself.

A statement of F. De Sua

Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. Quantum mechanics for example would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified.

Non-Aristotelian logics

1. Many valued logic2. Intuitionist logic3. Infinite valued logic4. Fuzzy logic

The intuitionist school1. Started in early 20th century by L. E. J. Brouwer although some of the

main ideas were mentioned earlier by Kronecker (1823-1891).2. Many eminent mathematicians have joined this school.3. The main idea is that any mathematical object must be built in a

purely constructive manner, employing a finite number of steps or operations from the natural numbers.

4. They deny the universal applicability within mathematics of the “fact” that any clear mathematical statement must be either true or false.

5. They have succeeded in building large parts of present day mathematics according to their principles.

6. It is generally conceded (not proven) that their methods do not lead to paradoxes.

Conclusion

1. Mathematical Logic and related subjects such as Set Theory have continued to be subjects of intense research and controversy until today.

2. The Game of Logic that began with the Greeks is still thriving.

Solution to “last” puzzle

5+5+5 = 550

5+5+5 = 550

Thank You