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 a process of trying to figure out the workings of some phenomenon by studying a sample of it. You work with a sample because looking at every component of the phenomenon is not feasible. Induction is a creative process. The scientist must carefully study a sample of a phenomenon, then formulate a hypothesis to explain the phenomenon. The scientific process of testing the hypothesis follows. 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 (a sample) 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. Such a procedure is called a census - look at every one. In most cases where induction is used a census is not feasible.

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. You need to know as much as possible about the situation for this to work well. Given all that you know, you select the hypothesis that appears most reasonable and likely. You have chosen the apparent best explanation, although it might be wrong. There is no guarantee. Further checking of your explanation is usually a good idea if it is possible; you might find more information that could make you reconsider your conclusion.

    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 known to be 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 below and in Prof. Fisher's notes).

    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 below and in Prof. Fisher's notes).

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. It sometimes gets written as

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

    Let's abbreviate this into the syllogism like this:

    P => C (Premises imply Conclusion, just like above)

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

    "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 true 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. There are at least three approaches.
    1. See if you can find a scenario in which the premises are true but the conclusion in false.
    2. Try replacing a noun or adjective in the argument with another one. Replace all instances. If this is obviously not valid, then the original was not valid.
    3. Represent P and C in a Venn diagram.If you can do this so that P is true but C is false, the argument is invalid. We've done these diagrams for you below.

    Now we bring up the subject of Venn diagrams. These can be of great value in checking the premises of an argument.


Venn Diagrams

Venn diagrams are a graphic method of representing logic.

This is about the simplest diagram you will get. The outer oval contains all things having property A, while the inner oval holds all things having property B as well as property A. This is shown as B=>A, which that all things having property B also have property A. You could also say "B implies A." If B is true, then A must be also. This is the guarantee mentioned above.
We will now assert that we have something with property B. We can confidently argue that it has property A also. This is Modus Ponens (affirming the antecedent) reasoning. Any entity with property B obviously must also be in the property A space.
Suppose we now claim that an entity does not have property A. The Venn diagram shows that it cannot have property B either. Anything outside the property A space must also be outside the property B space. This is Modus Tollens (denying the consequent) reasoning and is valid and strong.
Now we will try asserting that our thing does not possess property B. A look at the Venn diagram shows the problem here. An entity not having property B can lie inside the A space or outside of it. The assertion of "not B" tells us nothing about A. This is not valid reasoning, so we called it Modus Bogus. It is Denying the Antecedent and is not valid.
The final possibility is asserting that our thing has property A. The Venn diagram shows the problem. The entity could lie anywhere in the A space, either inside or outside of the B space. Asserting that the thing has property A tells you nothing about whether it also has B. This is also Modus Bogus, or invalid reasoning. It is called Affirming the Consequent.
Here we are dealing with properties A, B, and C. Here we can guarantee only that an entity having B and C also has A. Notice that some of B and C spaces lie outside of A.

  • Assert A: Could be anywhere in A
  • Assert B: Some of B lies outside of A
  • Assert C: Some of C lies outside of A
  • Assert A and B: Might include some C; might not.
  • Assert B and C: The intersection BC lies entirely in A. Arguing that B and C => A works.
  • Assert A and C: Might include some B; might not

What can you do with this? Properties A and B are as above, but what about C? Something having property C might lie within A or it might not. Something in A might also be in C or it might not. You can't do anything with C here. We do have the guarantee that something having B also has A.
How about this. The properties are exclusive; an entity cannot have both at the same time. It can have none, A, or B but not AB. The statements A=> not B and B => not A are valid.
Finally, we want our argument to be sound. For this to happen, the argument must be deductively valid as above and the premises must be true in the real world. This means that all sound arguments are valid but all valid arguments are not necessarily sound.
A sound argument requires that it be deductively valid and have true premises. Having one does not automatically guarantee the other. The region that is inside both T and V is the region of sound arguments.






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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  • "Magic" Demonstrations
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