USC2170 Doing ScienceLecture 2: Deductive Reasoning

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email matwml@nus.edu.sghttp://www.math.nus.edu.sg/~matwml/courses/Undergraduate/SPS_Doing/2006/

Tel (65) 6516-2749

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Reference [1] Rudolph Carnap, An Introduction to the Philosophy of Science, Dover, N.Y., 1995. Discuss [1], Chapter 1. The Value of Laws: Explanation and Prediction

Distribute material for and review previous lecture

Introduce the logic of probabilities in preparation for next week’s lecture: Inductive Reasoning and the Experimental Method

Assign writing a 3-5 page Socratic Dialogue that addresses the assigned reading topic.

LECTURE PLAN

REGULARITY

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The observations made in everyday life exhibit regularity.

Question: what are some examples ? Question: if these examples were not regular, could humans adapt ? If yes, then how ?

The laws of science are statements that express thisregularity. They are divided into universal laws and statistical laws. We first address universal laws because they are easier to understand.

STATEMENTS

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are classified into universal and singular

Question: in what class is each statement below ?

1. all ice is cold2. solids expand when they are heated3. ripe apples are red4. the small child is an excellent swimmer5. the elephant is an excellent swimmer6. mammals are warm blooded7. the SPS class started today at 7:30PM

AMBIGUITY

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Ordinary word language is ambiguousThis ambiguity causes misunderstandingSymbolic logic provides clarity

))(( QxPxx

A universal conditional statement

A singular statement Pa says that aparticular object a has property P

says that for any object x If x has property P

)(xThe expression is called a universal quantifier

Then x has property Q

The expression is a connective that means

“implies” or “If …. Then”

UNAMBIGUOUS STATEMENTS

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1. all ice is cold Px = x is ice, Qx = x is cold2. solids expand when they are heatedPx = x is a solid and x is heated, Qx = x expands3. ripe apples are red Px = x-is a ripe apple, Qx = x is red4. the small child is an excellent swimmera = the particular small child, Pa= a is an excellent swimmer5. the elephant is an excellent swimmerPx = x is an elephant, Qx = x is an excellent swimmer6. mammals are warm bloodedPx = x is a mammal, Qx = x is warm blooded7. the SPS class started today at 7:30PM a = the SPS class that started today,Pa = started at 7:30PM

FACTS AND LAWS

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Observations are always of particular objects, therefore observations yield knowledge that is expressed using singular statements. Such itemsof knowledge are called facts. Universal Laws are knowledge that are expressed using universal conditional statements.[1], p. 5. “One of the big, perplexing questions in the philosophy of science is how we are able to go from such singular statements to the assertions of universal laws.”Question: what ancient philosophical work struggles with this mind-boggling problem in the theory of knowledge (epistemology) ?

EXPLANATIONS

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typically consist of asserting facts.Example 1. Question: “How is it that my watch, which I left here on the table before I left theroom, is no longer here?” Answer: “I saw Jones come into the room and take it.”Example 2. Question: “Tommy (a little boy), why are you crying?” Answer: “Jimmy hit me on the nose.”

Question: Are these explanations logically complete? If not, are there implicit ‘parts’ of the explanation that are understood by both people but not mentioned? If so, what are these implicit parts?

Example 3. Question: “Why does this iron rod no longer fit into the apparatus?” Answer: “I heated the rod.”

FORMAL EXPLANATIONS

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always require laws, not mere facts

The general schema for deductive explanations is

))((.1 QxPxx Pa.2

Qa.3

Questions: How can the ‘explanations’ on the previous page be explained using this deductive schema?

FORMAL EXPLANATIONS

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Example 1. Question: “How is it that my watch, which I left here on the table before I left theroom, is no longer here?” Answer: “I saw Jones come into the room and take it.”

Example 2. Question: “Tommy (a little boy), why are you crying?” Answer: “Jimmy hit me on the nose.”

Law: Px = x is a watch and somebody takes x from a table, Qx = x is no longer on the table

Law: Px = x is a little boy and someone hits x on the nose, Qx = x cries.

Example 3. Question: “Why does this iron rod no longer fit into the apparatus?” Answer: “I heated the rod.”Law: Px = x is a body and x is heated, Qx = x expands

LOGICAL LAWS

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Examples 1. if p and q, then p. Example 2. if p, then p or q

like the laws of pure mathematics only describe relationships between concepts and therefore never provide scientific laws.

Euclid of Alexandria (325-265BCE) formulated an axiomatic system for the geometry of his day, Hilbert formulated it Rigorously, see Appendix

An Axiomatic System consists of a collection of statements, called axioms (or postulates), about undefined concepts (points, sets, numbers), and a collection of statements about those concepts that are deduced from these axioms.

19th century mathematicians formulated axiomatic systems for non-Euclidean geometries in which Euclid’s 5th Postulate is replaced by other Postulates. One such geometry is called Riemannian geometry.

STATISTICAL LAWS

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Example 1. ([1], p. 8) a type of slightly poisonous mushroom causes certain symptoms in 90% of people who eat it. If a doctor finds these symptoms in a patient who says that he ate this type of mushroom yesterday, the doctor will consider the eating of the mushroom to explain the symptoms

are often applicable even when universal laws do not apply

Example 2. What if only 5% of people who eat a certain type of food become ill. If a person becomes ill is the eating of this food a ‘good’ explanation?

Question: in example 2, does it follow that only 5% of people who become ill become ill because they ate the certain type of food ?

PREDICTIONS

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follow the same deductive schema as explanations

If Qa is known (from observation), we can explain Qa by showing how it can be deduced from statements 1 and 2

))((.1 QxPxx Pa.2

Qa.3

If Pa is known (from observation), we can predict Qa by showing how it can be deduced from statements 1 and 2

PREDICTIONS

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1. can be future events – a future eclipse2. can be now events – rods expand as they are heated3. can be past events – eclipses, glaciers, historical eventspredicted from psychological laws and historical factsderived from documents. Thucydides (455-400BCE) (History of the Peloponnesian War), who attempted to explain events, not merely attribute them to supernatural forces, concluded “the growth of Athens’ power and Sparta’s fear was, in my view, the cause which propelled them to go to war”.Question: how would Thucydides, 1st scientific historian and former general, explain the wars in 1st decade, 3rd millenium? “It was also my fate to be an exile from my country for twenty years after my command at Amphipolis; and being present with both parties, and more especially with the Peloponnesians by reason of my exile, I had leisure to observe affairs more

closely.”

PHYSICAL GEOMETRY

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In his attempt to explain certain facts about gravity, Einsteinfound the existing universal laws inadequate and developed a theory, consisting of new universal laws, to explain these facts

means the observable properties of physical space. These are scientific – not mathematical – questions.

These laws asserted that physical space is described by(non-Euclidean) Riemannian geometry and predicted phenomena that could be observed (falsifiable theory)

His theory has withstood extensive experiments/observations but is contradicted by predictions (speed of light not constant) of Loop Quantum Gravity that may be observed by the new satellite observatory GLAST planned for 2007.

Theorizing is the art of formulating universal laws, it uses, but goes way beyond, mere deductive reasoning

APPENDIX: EUCLID’s POSTULATES

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http://mathworld.wolfram.com/EuclidsPostulates.html1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius

and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner

angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai  and Nicolai Lobachevsky 

independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

APPENDIX: HILBERT’s AXIOMS

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http://www.math.umbc.edu/~campbell/Math306Spr02/Axioms/Hilbert.htmlUndefined Terms Points, Lines, Planes, Lie on, contains, Between, Congruent Axioms of Incidence Postulate I.1. For every two points A, B there exists a line a that contains each of the points A, B. Postulate I.2. For every two points A, B there exists no more than one line that contains each of the points A,

B. Postulate I.3. There exists at least two points on a line. There exist at least three points that do not lie on a

line. Postulate I.4. For any three points A, B, C that do not lie on the same line there exists a plane α that contains

each of the points A, B, C. For every plane there exists a point which it contains. Postulate I.5. For any three points A, B, C that do not lie on one and the same line there exists no more than

one plane that contains each of the three points A, B, C. Postulate I.6. If two points A, B of a line a lie in a plane α then every point of a lies in the plane α. Postulate I.7. If two planes α, β have a point A in common then they have at least one more point B in

common. Postulate I.8. There exist at least four points which do not lie in a plane. Axioms of Order Postulate II.1. If a point B lies between a point A and a point C then the points A, B, C are three distinct

points of a line, and B then also lies between C and A. Postulate II.2. For two points A and C, there always exists at least one point B on the line AC such that C lies

between A and B. Postulate II.3. Of any three points on a line there exists no more than one that lies between the other two. Postulate II.4. Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which

does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of the segment BC.

APPENDIX: HILBERT’s AXIOMS

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Axioms of Congruence Postulate III.1. If A, B are two points on a line a, and A' is a point on the same or on another line a' then it

is always possible to find a point B' on a given side of the line a' such that AB and A'B' are congruent.

Postulate III.2. If a segment A'B' and a segment A"B" are congruent to the same segment AB, then segments A'B' and

A"B" are congruent to each other. Postulate III.3. On a line a, let AB and BC be two segments which, except for B, have no points in

common. Furthermore, on the same or another line a', let A'B' and B'C' be two segments which, except for B', have no points in common. In that case if AB≈A'B' and BC≈B'C', then AC≈A'C'.

Postulate III.4. If ∠ABC is an angle and if B'C' is a ray, then there is exactly one ray B'A' on each "side" of line B'C' such that ∠A'B'C'∠ ABC. Furthermore, every angle is congruent to itself.

Postulate III.5. (SAS) If for two triangles ABC and A'B'C' the congruences AB≈A'B', AC≈A'C' and ∠BAC ≈ ∠B'A'C' are valid, then the congruence ∠ABC ≈ ∠A'B'C' is also satisfied.

Axiom of Parallels Postulate IV.1. Let a be any line and A a point not on it. Then there is at most one line in the plane that

contains a and A that passes through A and does not intersect a. Axioms of Continuity Postulate V.1. (Archimedes Axiom) If AB and CD are any segments, then there exists a number n such that

n copies of CD constructed contiguously from A along the ray AB willl pass beyond the point B. Postulate V.2. (Line Completeness) An extension of a set of points on a line with its order and congruence

relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence (Axioms I-III and V-1) is impossible.