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Philosophy and the Sciences - Phil. Sci. Course

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    nductive Logic

    eductive and Inductive Logic

    hat is Reasoning?

    xample: The first theorem Euclids Elementsprovides a good example of the kind of

    asoning that people admire.

    uppose we construct a triangle in the following way: 1. Draw a circle centered at point A.ark a point B on the circumference and draw a line from A to B. Draw a second circleentered at B that passed through A. Mark one of the points at which the circles intersect and draw lines from C to A and from C to B.

    heorem : All the sides of the triangle ABC are of equal length.

    roof: Let |AB| denote the length of the line segments AB, and so on.

    Step 1: |AB| = |AC| because they are radii of the circle centered at A.

    Step 2: |BA| = |BC| because they are radii of the circle centered at B.

    Step 3: |AB| = |BA| because AB and BA denote the same line.

    Step 4: |AC| = |BC| because they are each equal to the same thing (viz. |AB|

    Step 5: Therefore, |AB| = |AC| = |BC| by steps 1 and 4.

    efinition: An argumentis a list of statements, one of which is the conclusion and the reshich are the premises.

    he conclusionstates the point being argued for and the premisesstate the reasons beingdvanced in support the conclusion. They may not be good reasons. There are good and ba

    guments.

    p: To identify arguments look for words that introduce conclusions, like "therefore",onsequently", "it follows that". These are called conclusion indicators. Also look for premidicatorslike "because" and "since".

    emark: Each of the five steps in the proof to Euclids first theorem is an argument. Theonclusions in steps 1 to 4 are called intermediateconclusions, while the conclusion in stepthe mainconclusion.

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    uestion: All arguments, or sequences of arguments, are examples of reasoning, but is evece of reasoning an argument? A perceptual judgment such as "I see a blue square", or t

    onclusions of scientific experts reading in X-rays, or looking through a microscope, may bexamples of reasoning that are not arguments. They are derived from what Kuhn called tacnowledge, acquired through training and experience (e.g., knowing how to ride a bicycle).not easily articulated, and is not stated in any language.

    he Difference between Good and Bad Arguments

    logic, we assume that any reasoning is represented as an argument, and the evaluationn argument involves two questions:

    1. Are the premises true?2. Supposing that the premises are true, what sort of support do they give the conclusio

    nswers to question 2: Compare the following arguments.

    1. All planets move on ellipses. Pluto is a planet. Therefore, Pluto moves on an ellipse.2. Mercury moves on an ellipse. Venus moves on an ellipse. Earth moves on an ellipse.

    Mars moves on an ellipse. Jupiter moves on an ellipse. Saturn moves on an ellipse.Uranus moves on an ellipse. Neptune moves on an ellipse. Therefore, Pluto moves onan ellipse.

    efinition: An argument is deductively validif and only if it is impossiblethat its conclusiolse while its premises are true.

    xamples: Argument 1 is deductively valid, while argument 2 is not.

    emark on terminology: The notion of deductively validity is such a central and importaoncept in philosophy, that is goes by several names. When an argument is deductively vale say that the conclusion follows fromthe premises, or the conclusion is deduced from, oferred from, or proved fromthe premises. Or we may say that the premises imply, or ent

    provethe conclusion. We also talk of deductively valid arguments as being demonstrativl these different terms mean exactly the same thing, so the situation is far simpler than itppears.

    hats possible? The sense of "impossible" needs clarification. Consider the example:

    George is a human being. George is 100 years old. George has arthritis. Therefore, Georll not run a four-minute mile tomorrow.

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    nductive Logic

    uppose that the premises are true. In logic, it ispossible that George will run a four-minutile tomorrow. It is not physicallypossible. But logicians have a far more liberal sense of w"possible" in mind in their definition of deductive validity. Argument 3 is not deductively

    alid on their definition. So, argument 3 is invalid.

    ey idea: In any deductively valid argument, there is a sense in which the conclusion isontainedin premises. Deductive reasoning serves the purpose ofextractinginformation fro

    e premises. In a non-deductive argument, the conclusion goes beyond the premises.ferences in which the conclusion amplifiesthe premises is sometimes called ampliativeference.

    herefore, whether an argument is deductively valid or not, depends on what the premisee.

    Missing premises?: We can always add a premise to turn an invalid argument into a valgument. For example, if we add the premise "No 100-year-old human being with arthritis

    ll run a four-minute mile tomorrow" to argument 3, then the new argument is deductivelyalid. (The original argument, of course, is still invalid).

    efinition: An argument is inductively strongif and only if it is improbablethat its conclusfalse while its premises are true.

    emember: This definition is the same as the definition of "deductively valid" except thatmpossible" is replaced by "improbable."

    he degreeof strength of an inductive argument may be measured by the probability of thae conclusion is true giventhat all the premises are true.

    he probability of the conclusion of a deductively valid argument given the premises is oneeductively valid arguments may be thought of as the limiting case of a strong inductiveguments. Ampliative arguments have an inductive strength less than one.

    he probability of the conclusion given the premises can change from person to person, as

    epends on the stock of relevant knowledge possessed by a given person at a given time.

    ummary: In response to question 2, we may give answers like "the argument is valid", "tguments is inductively strong" or "the argument is inductively weak."

    xercise: Discuss the following examples (all statements are understood to refer to the ye998):

    4. There are multi-celled organisms living on Mars. Therefore, there is intelligent life

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    Mars.

    5. There are multi-celled organisms living on Mars. Therefore, there are single-celledorganisms living on Mars.

    6. There are multi-celled organisms living in Lake Mendota. Therefore, there isintelligent life living in Lake Mendota.

    7. There are multi-celled organisms living in Lake Mendota. Therefore, there are singcelled organisms living in Lake Mendota.

    evertheless, in logic, it is assumed that the answer to question 1 is relevant to the evaluatan argument. But it is a question that needs to be asked in additionto question 2. So, if emises of an inductively strong argument are false, then logicians are forced to say that tgument is not a good one. It is confusing to say that an inductively strong argument is aeak argument, but this is how the terms are defined.

    p: Defined terms must be used as defined. You cant use the term differently just becausou dont agree with the definition.

    fferent Kinds of Ampliative Argument

    efinition: Any argument that is not deductively valid, or deductively invalid, is called anmpliative argument. The term refers to the fact that the conclusion of such argument goes

    eyond, or amplifies upon, the premises.

    emark on terminology: Again the notion ofinvalid is so common and central, that it gy many names. Other terms commonly used are inductive and non-demonstrative. I prefempliativebecause it reminds us that the conclusion goes beyond the premises, and it doot have the bad reputation that sometimes goes along with the word induction.

    ere are a variety of examples of ampliative arguments:

    mple enumerative induction goes from a list of observations of the form "this A is a Bthe conclusion "All As are Bs". The example Hume made famous is like this:

    8. Billiard ball 1 moves when struck. Billiard ball 2 moves when struck. Billiard ball 3moves when struck Billiard ball 100 moves when struck. Therefore, all billiard ballsmove when struck.

    ome ampliative arguments go from general statements to general statements:

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    9. All bodies freely falling near the surface of the Earth obey Galileos law. All planetsobey Keplers laws. Therefore, all material objects obey Newtons laws.

    thers go from general statements to specific statements:

    10. All emeralds previously found have been green. Therefore, the next emerald to bfound will be green.

    onclusion : To understand empirical science we need to understand ampliative inference.

    wo Kinds of Science? A Priori and Empirical?

    1. A prioriscience, like Euclids geometry, is where the conclusions are deduced frompremises that appear to be self-evidently true.

    2. In empirical science, like physics, conclusions are based on observational data.

    This is similar to the distinction between pure mat

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