Logic and Mathematics

The Art and Science of Mathematics

The 4 Color Theorem• In 1879, Kempe produced a famous proof of the

4 color theorem: any map of countries can be colored in such a way that no 2 bordering countries have the same color, using only 4 colors.

• In 1890, Heawood showed the proof not to be a proof at all!

• When is a proof a proof, and when is it not a proof?

• Logic to the rescue!

Logic

• Logic is the study of reasoning• In particular, logic studies the conditions

under which we can say that a piece of reasoning is valid, i.e. that something (the conclusion) can be said to follow from something else (the premises, givens, assumptions).

Deduction and Induction

• If the conclusion has to be true assuming the truth of the premises, we call the reasoning deductive.

• If the conclusion is merely more likely to be true than false given the truth of the premises, we call the reasoning inductive.

• Logic studies both deduction and induction, but does tend to focus on deduction, especially formal logic.

Normative and Descriptive Theories of Reasoning

• Psychology of reasoning is a scientific study of how humans reason:– What do humans infer from what?– What is the mechanism behind human reasoning?

• As such, psychologists come up with descriptive theories of reasoning: hypotheses as to how humans reason based on empirical studies.

• Logicians, however, try to come up with normative theories of reasoning:– What actually follows from what?

• Question: But if not empirical, what is the basis for such theories? (Human!) reason alone?

Implication and Truth• Logic tells us about implication, not truth.• Example: “All flurps are toogle, but not all flems

are toogle, so not all flems are flurps” is perfectly logical, but tells us nothing about what-is-the-case.

• One exception: Implication itself can be seen as a kind of (necessary) truth. So, logic can tells us that certain statements of the form “If <premises> then <conclusion>” are necessarily true (i.e. true in all possible worlds), and hence true in our world as well.

Logic and Science• Of course, if I do know that my premises are

true, then if the reasoning is (deductively) valid I know the conclusion to be true as well.

• But that’s just science: science combines observation (facts) with logic (reasoning), to get to truth (laws of physics, chemistry, etc).

• Of course, scientific reasoning is inherently inductive: a finite set of data is always compatible with multiple theories. Hence: scientific theories can change over time.

Logic and Mathematics• Most of what I just said for logic is true for mathematics

as well!– Scientists use mathematics to help figure out (calculate,

compute, etc) what-is-the-case but mathematics alone does not tell us what-is-the-case

– Like logic, mathematical theorems are proven from a set of definitions or axioms: if those axioms or definitions don’t apply to our world, then the theorem doesn’t say anything about our world either.

– The only thing we can claim to be certain of is a statement of the form “If <bunch of definitions/axioms> then <some claim>”.

– So, theorems like “There is no greatest prime number” are really expressions of “If we define ‘number’ to be ..., and ‘prime’ as … and ‘greater than’ as …, then there is no greatest prime number.”

Further Similarities Between Logic and Mathematics

• Both logic and mathematics have been around for thousands of years

• Both logic and mathematics study abstractions that can be applied to any subject matter

• Formal logic is probably best seen as a branch of mathematics

• Mathematics can be applied to formal logic (mathematical logic)

• Formal logic can be applied to mathematics (theorem proving)

Formal Logic• We can determine that “All flurps are toogle, but

not all flems are toogle, so not all flems are flurps” is a valid inference because of the abstract form of the reasoning: “All P’s are Q’s, but not all R’s are Q’s, so not all R’s are P’s”.

• Formal logic is just that: studying the validity of reasoning by looking at its abstract form: – Just as in mathematics:

• 1) expressions of abstract symbols are assigned the objects of study, and

• 2) by manipulating these expressions of abstract symbols, we can figure something out about these objects.

Little History of Formal Logic• Formal logic goes back at least to Aristotle, probably earlier• In Medieval Times work was being done on categorical

syllogisms like the one on previous page (that one would be classified as AOO-2)

• ‘Modern’ formal logic was developed in mid 19th century by people like Georges Boole and Augustus DeMorgan. They developed the system of propositional or truth-functional logic.

• The much more powerful system of first-order logic (or predicate logic or quantificational logic) was completed by the turn of the 20th century.

• Many other systems of logic have been developed since; just as with mathematics, different systems have different applications.

Truth-Functional Logic

• Applies to reasoning dealing with compound sentences built from truth-functional operators like ‘and’, ‘or’, ‘not’, and ‘if … then’.

• An operator is truth-functional in that the truth-value of a sentence like “P and Q” is a function of the truth-values of the sentences P and Q.

Truth-Tables

P P

TTF

F

P P Q

T

TF

F

QT

FF

F

FFTT

P P Q

T

TF

T

QT

FF

T

FFTT

De Morgan’s Laws

P (P Q)

T

TF

F

QT

FF

F

FFTT

T

F

T

T

P Q

T

F

T

FF

F

T

TT

F

T

T

↑ ↑

(P Q) P Q

T

T

F

TF

F

T

FT

F

T

FF

F

T

TF

F

T

F

↑ ↑

Modus Ponens

P P Q

T

TF

T

QT

TF

F

FFTT

P Q

PPQ Q

T

TF

FFFTT ←

Ladies and Tigers

• The prisoner is told that if there is a lady in room I, then its sign is true, but if there is a tiger in the room, then its sign is false. For room II, it is exactly the opposite.

• Sign I says: ‘Both rooms contain ladies’• Sign II says: ‘Both rooms contain ladies’• Question: Which room should the prisoner

pick?

Truth-Table Solution

L1 L1 L2

T

TF

F

L2

T

FF

F

FFTT

T

FF

FTTFT

←

L1 (L1 L2) L2 (L1 L2)

Boolean Algebra SolutionL2 (L1 L2) (Equivalence)

(L2 (L1 L2) (L2 → (L1 L2) (Transposition)

(L2 → (L1 L2) ((L1 L2) → L2) (Implication)

(L2 → (L1 L2) ((L1 L2) L2) (Double Neg.)

(L2 → (L1 L2) ((L1 L2) L2) (Subsumption)

(L2 → (L1 L2) L2 (Implication)

(L2 (L1 L2)) L2 (De Morgan)

(L2 (L1 L2)) L2 (Commutation)

(L2 (L2 L1)) L2 (Association)

((L2 L2) L1) L2 (Idempotence)

(L2 L1) L2 (Distribution)

(L2 L2) (L1 L2) (Contradiction)

(L1 L2) (Contradiction)

L1 L2

Formal Proof Solution

L2

L2

L1 (L1 L2) L2 (L1 L2)

L2

L2

(L1 L2)

L2

(L1 L2)

L1 L2

1.

2.

3.

4.

5.6.

7.

8.

9.

10.11.

12.

Assumption

1.

Assumption

Assumption

Assumption

Elim 2, 4

Intro 3, 5

Intro 4-6

Elim 7

Elim 8

Intro 3, 9 Intro 3-10 Elim 11

The Dreadsbury Mansion Problem

• Someone who lived at the Dreadsbury Mansion killed Aunt Agatha. Agatha, the butler, and Charlie were the only ones living at the Dreadsbury Mansion. A killer always hates its victim but is never richer than its victim. Whomever aunt Agatha hated, Charlie did not hate. Agatha hated everybody but the butler. The butler hated everybody not richer than aunt Agatha, as well as everyone that aunt Agatha hated. No one who lived at the mansion hated everybody who was living at the mansion. Finally, aunt Agatha was not the butler. Who killed aunt Agatha?

Automated Theorem Proving• Formal proofs seem perfect for automation:

proofs require tediously many applications of precisely defined rules: just something a computer would be good at!

• Problem: the rules of logic are like the rules of chess: they tell you what you can do, but not what you must do.

• In Automated Theorem Proving (a branch of Artificial Intelligence) researchers try and come up with algorithms to create formal proofs (to be exact: they come up with algorithms to check whether some inference is valid or not).

Axiomatization

• In fact, why not do this: Express basic axioms and definitions about various branches of mathematics in logic, and simply run an ATP to automatically prove all theorems about any branch of mathematics!

• This ambitious project was called the Hilbert Program, after the mathematician David Hilbert who proposed it at the turn of the 20th century.

The MetaMath Project

• Check it out!

• www.metamath.org

Godel’s Incompleteness Result• Unfortunately (or fortunately?), the Hilbert

Program failed, and necessarily so.• In 1931 Kurt Godel proved that the idea couldn’t

work for simple arithmetic, let alone all of mathematics.

• Godel showed that for any finite set of axioms for arithmetic (just dealing with addition and multiplication for natural numbers) there is an arithmetical truth that cannot be derived from those axioms.

Arithmetic is Undecidable

• In fact, based on computability theory (starting with Turing in 1936) we have excellent reasons to believe that arithmetic is undecidable, i.e. that there is no systematic procedure (computer program) that can decide for any first-order logic arithmetical statement whether it is true or false!

So what does this mean?• Well, if one believes that human mathematicians

are able to figure out (in principle of course!) all arithmetical truths, then it is false that human information processing can be captured by a computer program. In short, AI is a pipe dream!

• On the other hand, if you do believe that human thought can be captured by some kind of computer program (as many cognitive scientists believe), then it seems that there will be truths (in simple arithmetic!) that we won’t be able to figure out.

How good are Automated Theorem Provers?

• Frankly, they stink!• In 1956, the Logic Theorist was able to prove theorem

2.16 of Russell and Whitehead’s book Principia Mathematica: (P → Q) → (Q → P) given basic axioms about propositional logic

• 50 years later, the best ATP’s around still can’t prove that P() = {} given basic set theory axioms

• Some researchers see this as evidence that human thought cannot be captured through computation, but others say it’s too early to tell.

• For practical matters though, the best approach would be to combine human intellect with computational effort: Assisted Theorem Proving: the human lays out the ‘proof plan’, and the ATP will fill in the details.

The 4 Color Theorem• In fact, this is exactly how the 4 color theorem was finally proven in

1976: Appel and Haken used a computer running for 1200 hours to finally settle this conjecture.

• Or did they?• At first, the proof was regarded with much skepticism, as it wasn’t

verifiable by hand in any reasonable manner.• In fact, the proof is of course only reliable as far as the computer

program was reliable, i.e. did what it was supposed to do. So, you’d need to prove that the computer program was reliable.

• Also, the program didn’t produce anything that we would recognize as a formal proof.

• In 2004, Werner and Gonthier produced a proof using Coq, an Automated Theorem Prover, finally producing a ‘proof-like’ object for the 4 color theorem.

• So, assuming Coq has been verified for working as it should be, that’s it then, right?

Are my Logic Rules Logical?

• Well, one tiny problem.• Suppose I am a real stickler, and

challenge the validity of the logical inference rules as defined.

• In other words, in order for me to accept a formal logic proof of some theorem, I need to be able to accept the validity of the inference rules used in that proof.

Modus Bogus andHokus Ponens

P Q

Q

P P

Modus Bogus Hokus Ponens

Clearly, Modus Bogus and Hokus Ponens are not acceptable!

But what about Modus Ponens?• Earlier I demonstrated the validity of Modus

Ponens using a truth table.• However, I thus reasoned as follows: If the truth-

table of some argument reveals that there is no row where the premises of that argument are true, while its conclusion is false, then that argument is valid. The truth-table for Modus Ponens shows that there is no row where its premises are true and the conclusion is false. Hence, Modus Ponens is valid.

• But what logic did I just use? Modus Ponens!• Hmm…

Interested in Logic?• PHIL 2140 – Introduction to Logic (I think a must

for all Math majors)• PHIL 4XXX – Intermediate Logic (applying logic

to mathematics)• PHIL 4420 – Computability and Logic (applying

mathematics to logic, including a proof of Godel’s incompleteness theorem)

• Also: PHIL/PSYC 2xxx – Methods of Reasoning (more informal logic and psychology of reasoning; good course to improve your everyday life reasoning skills!)