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Numbers and Their Application - Lesson 5

Logical Reasoning

Lesson Overview

René Descartes

René Decartes, the early French mathematician (1591-1650) spent considerable time philosophizing about mathematics and its very existence. To get started he had to assume his very own existence in his famous quote (in Latin): "Cogito ergo sum", which means, "I think, therefore I am."

Hypothesis, Conclusion, Conjecture

A premise (also known as an antecedent or hypothesis) is a tentative assumption made in order to draw out and test its logical or empirical consequences.

A consequence or conclusion is the necessary result of two or more propositions taken as premises.

Sentential logic or propositional logic, consists of a sentential languge, a semantic interpretation of that language, and a sentential derivation system. Predicate logic goes further and builds on sentential logic. We give here the merest overview of this broad field.

Deductive vs. Inductive Reasoning

As stated in the first two lessons, Geometry often deals with proofs. Proofs are based on logical reasoning which follow two basic types.

Deductive (or logical) Reasoning is the process of demonstrating that if certain statements are accepted as true, then other statements can be shown to follow from them.

Inductive Reasoning is the process of observing data, recognizing patterns, and making generalizations from the observations.

Both are important to mathematics in general and to Geometry specifically.

The generalization used in inductive reasoning is called a conjecture.

A statement is a declarative sentence which is either true or false, but not both. Proposition is often used interchangely with the term statement. A paradox is a sentence which is both true and false, such as "I am lying" (cf Titus 1:12). A simple statement is a statement containing no connecting words. Compound or complex statements are formed from simple statements using basic connection. The basic connections are: and, or, if... then..., if and only if, not. Often, other connecting words such as unless, because, either/or, neither/nor, although, nevertheless, except, but (save), only, as, since, etc. are used which can be restated using the basic ones.

Examples:
"Unless he is careful, he will crash." means the same as "If he is not careful, then he will crash."
"Whenever I tell a joke, my students laugh." is equivalent to "If I tell a joke, then my students laugh." except for some circumstance of time.

A web link or mapping of these other connecting words onto the basic connections is worth extra credit and would complete this lecture.

This definition of statement is based on an axiom of Aristotle (ancient Greek philosoper (c. 384 B.C.) called the law of excluded middle. If we reject this axiom, fuzzy logic involving probability is the result. In recent years, fuzzy logic has started to invade your cars and homes (washing machines, etc.), and is "the rage."

Short hand notation is often used when writing logical arguments. Statements such as "I have a job." may be replaced by p and the conditional statement, "If I have a job, then I must work." might be replaced by p=>q, where q in this case is equivalent to "I must work". A conditional is also known as an implication. An if-then statement can be rewritten using the word implies, and in fact, the symbol => is often read that way. Some reasoning is valid, in that it gives correct or truthful results whereas some is faulty or invalid. You may think the old adage: "Watch your p's and q's" is derived from the extensive usage of these symbols. However, it actually is drinking advice to watch ones pints and quarts!

A theorem is a statement that has been proven, or can be proven, from the postulates.

A corollary is a result which follows naturally, or a specific application of a theorem. A lemma is a mathematical statement proven not for its own sake, but for use in proving a more important statement called a theorem.

Modus Ponens (MP) says that if p=>q is true and p is true, then q must be true.
This principle is also known as the Law of Detachment (LD).

Modus Tollens (MT) says that if p=>q is true and q is false (not true), then p must be false.
MT is essentially equivalent to the Law of indirect Reasoning (below) and is the basis for proof by contradiction.

For example, consider the following conditional statement: If the weather is beautiful, then we'll go for a walk. MP implies that if p is true (The weather is beautiful.) q is also true (We'll go for a walk.). MT implies that if p=>q is true (If the weather is beautiful, then we'll go for a walk.) and q is false (It is not the case that we'll go for a walk.) then p is false (The weather is not beautiful.).

Negation/Double Negation

   
The negation, symbollically ~p, ¬p, or p,
of a statement is very useful. If p is "I have a job", then ~p is "I do not have a job".

The double negation, as taught in English (not Spanish!) gives back the original statement! ~(~p) is equivalent to p. If it is not true, that "I do not have a job". Then it must be true "I have a job".

The Converse of p=>q is q=>p.

The Inverse of p=>q is ~p=>~q.

The Contrapositive of p=>q is ~q=>~p.

Law of Contrapositive (LC) states that if a conditional is true, so is its contrapositive.

Continuing the weather example above, the contrapositive would be "If we'll not go for a walk, then the weather is not beautiful." LC tells us this is true if the original statement is true. It should be easy to see that the converse of the inverse is the contrapositive.

Whether the conditional is true does not affect whether the converse is true.

A counterexample is an example of a conditional statement being false.

Sometimes, instead of writing a long proof to determine something is true, many will try to find a counterexample.

An "if and only if" (often abbreviated iff) statement is called a biconditional and combines the statements p=>q and q=>p into p<=>q. To prove a biconditional, one proves the corresponding two conditionals.

A syllogism is composed of a major premise, a minor premise, the resulting conclusion.

A syllogism has three parts. Therefore, this is not a syllogism. (ha ha ha). The consequence is often preceeded by the word therefore which is also often abbreviated by three dots arranged in a triangle pointing up ().

The Law of Syllogism is also called the Law of Transitivity and states:
if p=>q and q=>r are both true, then p=>r is true.

Reasoning and also definitions are sometimes said to be circular.

Direct vs. Indirect Proof

Mathematical proofs come in two basic flavors known as direct and indirect. We already saw an example of an indirect proof in Numbers lesson 3 when we proved by contradiction that primes form an infinite set. Proof by contradiction is also known as using the law of indirect reasoning.

Law of Indirect Reasoning:
If valid reasoning from a statement p leads to a false conclusion, then p is false.

Any proof using the Law of Contrapositive (above) or the Law of Ruling out Possibilities (below) are also classified as indirect proofs.

Law of Ruling out Possibilities:
When statement p or statement q is true, and q is not true, then p is true.

We will see further examples of these five laws of logic in chapter 11 of our Geometry textbook and my associated supplement.

Galileo Galilei

Galileo Galilei (1564-1642), often referred to only as Galileo, was an Italian mathematician, astronomer, and physicist. He made several significant contributions to modern scientific thought. He is considered a founder of the experimental method on which modern science is based. He is especially noted for being the first man to study the skies with the telescope and proving the Earth revolves around the Sun. For thousands of years many men were content to assume heavier things fell faster, but Galileo proved theoretically that falling bodies obey what is now known as the law of uniformly accelerated motion. He gather evidence that proved the Earth revolves around the Sun and that it was not the center of the universe as was then believed. More importantly, he maintained this position despite trial in Rome and church orders to recant. He was forced to spend the last eight years of his life under house arrest. His most far-reaching achievement was the re-establishment of the scientific method against Aristotle's flawed approach. He also insisted that the "Book of Nature" was written in mathematical characters.

Galileo was born at Pisa in 1564. He studied medicine at the university there starting in 1581. Supposedly it was here in the Pisa cathedral during his first year that he observed a lamp swinging and found that it period was constant, independant of the amplitude of the oscillation. Later in life he verified this observation experimentally and suggested that this principle might be used to regulate clocks. A chance Geometry lesson he overheard awakened his interest in mathematics and he began to study Mathematics and Science. In 1585, before he received a degree, he was withdrawn from the univerisity due to lack of funds. Four years later his treatise on center of gravity earned him a post of mathematics lecturer back at Pisa. Galileo spent his childhood and the intervening years in Florence. In 1592 he was awarded the chair of mathematics at Padua where he remained for 18 years and performed the bulk of his work.

Galileo had enough faith in the mathematical model of a moving earth to suffer condemnation by the establishment (Catholic church) until 1992!

The Scientific Method

The scientific method is the accurate observation of facts and the determination of order among the facts. Generally these are framed in mathematical form. This may be done by inductive reasoning, inference from a number of observed facts, or by deductive reasoning from a set of principles. Often, predictions are possible that are open to experimental testing.

The scientific method usually has at least five steps: (i) stating the problem; (ii) forming the hypothesis; (iii) observing the experiment (taking data); (iv) interpretting the data; and (v) drawing the conclusion by developing theory. These steps, however, often don't follow that exact order; unexpected results are often observed! These checkpoints are often used to arrange and write up an experiment. More will be said in this regard in the statistics lectures.

Mathematics itself is seldom in conflict with religion. However, science, scientists, the scientific method, and the scientific theories generated often are. The following statement is included with this in mind: Popes Pius XII in 1951 and John Paul II in 1996 declared that Catholics may accept Evolution as more than a hypothesis and the Big Bang as a "splendid solution" without contradicting their faith. One can only hope that other religious groups will consider a similar position within a few nanohubble times. Nothing further can be said here while scrupulously avoiding either the fact or appearance of inadequate separation of secular and sectarian activities. (include link to dissertation Appendix B).

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