Induction and Deduction


Professor Fisher's notes on this topic in PDF
Professor Fisher's slides on this topic in PDF (password required)
Old slides from another professor in PDF.

Induction

    Note: Philosophers (including Prof. Fisher) distinguish induction (generalizing the traits of some sample to other things) from abduction (inference to the best explanation), whereas the content below blurs both of these together under the single heading of "induction". When taking this class with Professor Fisher, it's better to rely upon his notes or slides on this topic, rather than the content below.

    Induction is the creative part of science. The scientist must carefully study a phenomenon, then formulate a hypothesis to explain the phenomenon. Scientists who get the most spectacular research results are those who are creative enough to think of the right research questions.

    Natural sciences (physics, chemistry, biology, etc.) are inductive. Evidence is collected. The Scientific Method is applied. Start with specific results and try to guess the general rules. Hypotheses can only be disproved, never proved. If a hypothesis withstands repeated trials by many independent researchers, then confidence grows in the hypothesis. All hypotheses are tentative; any one could be overturned tomorrow, but very strong evidence is required to overthrow a "Law" or "Fact".

    Specific -> General

    Here's an example of induction: Suppose I have taken 20 marbles at random from a large bag of marbles. Every one of them turned out to be white. That's my observation - every marble I took out was white. I could therefore form the hypothesis that this would be explained if all the marbles in the bag were white. Further sampling would be required to test the hypothesis. It might be that there are some varicolored marbles in the bag and my first sample simply didn't hit any.

    Incidentally, this is one case where we could prove the hypothesis true. We could simply dump out all the marbles in the bag and examine each one.

Abduction: Inference to Best Explanation

    This is a method of reasoning that is not rigorous like Deduction. Suppose you have observations of an interesting occurrence. You are not immediately sure what explains the observations. You think of all the hypotheses you can that could plausibly explain the observations. This is a creative process, unlike deduction. Given all that you know, you select the hypothesis that appears most reasonable and likely. You have chosen that apparent best explanation, although it might be wrong.

    This kind of reasoning is the forte of Sherlock Holmes. Conan Doyle's famously hyper-observant detective notices even very tiny clues that the police inspector usually misses. Holmes combines these clues with the general knowledge he has built up about the case to come up with the best explanation - whodunit! He also explains how the crime was done. In a nice touch for the reader, Holmes often explains the reasoning that led him to the solution. It seems simple when Holmes explains it...

Deduction

    Mathematics is a deductive science. Axioms are proposed. They are not tested; they are assumed to be true. Theorems are deduced from the axioms. Given the axioms and the rules of logic, a machine could produce theorems.

    General -> Specific

    Start with the general rule and deduce specific results. If the set of axioms produces a theorem and its negation, the set of axioms is called INCONSISTENT.

    Suppose we have the following known conditions.

    • We have a large bag of marbles.
    • All of the marbles in the bag are white.
    • I have a random sample of 20 marbles taken from the bag.
    From these, I can deduce that all the marbles in the sample are white, even without looking at them. This kind of reasoning is called modus ponens (more about this in Schick and Vaughn, chapter 6).

    How about this?

    • We have a large bag of marbles.
    • All of the marbles in the bag are white.
    • I have a sample of 20 marbles of mixed colors.
    From this I quickly deduce that the sample was not taken from the bag of white marbles. This kind of reasoning is called modus tollens (more about this in Schick and Vaughn, chapter 3, where they spell it modus tolens).

Deductive Arguments

    Let's look at the idea of deductive reasoning (derived from Prof. Fisher's notes). Here we are talking about making an argument for some idea or conclusion based of a set of premises (facts, ideas, etc). Prof. Fisher notes that philosophers usually show this as
    1.  Premise number 1
    2.  Premise number 2
    ...
    ------------------------
    C.  Conclusion to be reached
    
    This structure constitutes an argument. This sometimes gets written as

    Premise 1, Premise 2, Premise... => Conclusion.

    Let's abbreviate this into the syllogism like this:

    P => C (Premises imply Conclusion)

    This simply means that if the premises are true, then the conclusion is also.

    "To be good, an argument must have true premises and the premises must offer support for the conclusion. The strongest possible support would provide an absolute guarantee that the conclusion will be true (presuming, of course, that the premises are true). We'll consider that sort of support first, but then move on to consider some weaker sorts of s upport as well." (from Prof. Fisher)

    One especially useful sort of argument is a deductively valid argument. (This is often abbreviated as "valid argument" or sometimes as "deductive argument.") Deductively valid arguments are arguments in which the premises, if true, would be the strongest possible evidence that the conclusion is true. Indeed these arguments provide the following guarantee: if the premises are true, then the conclusion must be true as well.

    An argument which appears to be deductive but has premises which do not support the conclusion (no guarantee) is an invalid argument.

    Let's repeat the syllogism P => C (Premises lead to conclusion). There are four possible assertions you could make about this:
    1. All the premises are true (P is true) (modus ponens reasoning)
    2. One or more of the premises is false (P is false - not P)
    3. The conclusion is true (C is true)
    4. The conclusion is false (C is false - not C) (modus tollens reasoning)
    Only two of these (#1 and #4) constitute deductively valid reasoning. Let's look at the reason for this, one choice at a time.
    1. P is true (P) - Affirming the antecedent. If all the premises are true the conclusion must be true. This is a very strong argument. (modus ponens)
    2. P is false (not P) - Denying the antecedent. P being false does not guarantee that C is either or false. There could be other causes.
    3. C is true (C) - Affirming the consequent. If C is true it might be for one of several reasons, not this specific P.
    4. C is false (not C) - Denying the consequent. Here, if C is false then P must also be false. If P were true C would be also. (modus tollens)
    In short, #1 and #4 (modus ponens and modus Tollens) are deductively valid while #2 and #3 are not.

    If the premises apply to things in the real world, you still need to do a test fot invalidity. Try to think up some situation in which the premises can be true but the conclusion is not.
    Now we bring up the subject of Venn diagrams. THese can be of great value in checking the premises of an argument.



The Aristotelean Method

Here's the way I see it. Everybody listen to me.





Aristotle (384-322 BCE)



From http://www.rwe.org/images/aristotle.jpg

Some things he said seem reasonable:

All Earthly objects tend to rest -- their natural state.

All celestial objects remain in circular motion forever.







But other things he said make no sense today:

"Males have more teeth than females in the case of men, sheep, goats, and swine; ..."
Aristotle online -- The History of Animals 350 BCE

Heavier objects fall faster than light ones, in proportion to their weight.

Experiment?

Self-consistent?

If your theory is not self-consistent, or your theory disagrees with careful experiments, then your theory is wrong. It doesn't matter how beautiful the theory is; it's wrong.




Galileo Galilei (1564-1642)

From http://helios.gsfc.nasa.gov/galileo.jpg

Often called the "Father of Science"



He did NOT invent the telescope!



He made excellent observations without too much prejudice.



He measured phenomena quantitatively. (E.g. the water stopwatch.)



He used mathematics. (He was professor of mathematics at the University of Padua in Venice.)



Hammer and Feather Gravity Demo


e.g. Euclid's fifth postulate.

(1) Through any two different points, it is possible to draw one line.
(2) A finite straight line can be extended continuously in a straight line.
(3) A circle can be described with any point as center and any distance as radius.
(4) All right angles are equal.
(5) Through a given point, only one line can be drawn parallel to a given line.



The words "point" and "line" have no intrinsic meaning.



One could swap "point" and "line" and still have true theorems.



One could say
(1) Through any two different blargs, it is possible to draw one fleem...



The fifth postulate can be changed in two ways:

(5) Through a given point, no line can be drawn parallel to a given line.

(5) Through a given point, many lines can be drawn parallel to a given line.

Both of these new postulates give rise to different CONSISTENT geometries. Which one is right? They all are! Which one describes this Universe? That's PHYSICS!

Reference for Non-Euclidean Geometry: http://www.cut-the-knot.com/triangle/pythpar/NonEuclid.shtml
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