Inductive Reasoning. The Nature of Inductive Reasoning What is an inductive argument? What is an inductive argument? 1. Any argument which is not deductive!

Inductive ReasoningInductive Reasoning

The Nature of Inductive The Nature of Inductive ReasoningReasoning

What is an inductive argument?What is an inductive argument?

1.1. Any argument which is not deductive!Any argument which is not deductive! I.e., any argument which does not I.e., any argument which does not

provide a guarantee of the truth of the provide a guarantee of the truth of the conclusion if the premises are true.conclusion if the premises are true.

Inductive arguments are probabalistic.Inductive arguments are probabalistic.


What else?What else?

2.2. Inductive arguments are ampliative, while Inductive arguments are ampliative, while deductive arguments are non-ampliative.deductive arguments are non-ampliative.

An argument is ampliative (defn) iff there is An argument is ampliative (defn) iff there is information contained in the conclusion that information contained in the conclusion that is not already contained in the premises.is not already contained in the premises.

That’s the trade-off! You lose the That’s the trade-off! You lose the guarantee of the truth of the conclusion for guarantee of the truth of the conclusion for amplification.amplification.


3.3. The logical strength of inductive The logical strength of inductive arguments is arguments is notnot dependent on the dependent on the form of the argument, but rather form of the argument, but rather the content of the premises. (It’s the content of the premises. (It’s the opposite for deductive the opposite for deductive arguments.)arguments.)


However, there are 4 forms of However, there are 4 forms of inductive arguments that usually inductive arguments that usually regarded as logically strong so long regarded as logically strong so long as certain conditions are met.as certain conditions are met.

1.1. Inductive generalizationInductive generalization

2.2. Statistical syllogismStatistical syllogism

3.3. Induction by confirmationInduction by confirmation

4.4. Analogical reasoningAnalogical reasoning

Inductive GeneralizationInductive Generalization

Has the following form:Has the following form:

Z percent of observed F’s are GZ percent of observed F’s are G

It is probable, therefore, that Z percent It is probable, therefore, that Z percent of all F’s are G.of all F’s are G.


E.g.E.g.

60% of students at STFX who were 60% of students at STFX who were questioned believe in God. It is questioned believe in God. It is probable, therefore, that 60% of probable, therefore, that 60% of students at STFX believe in God.students at STFX believe in God.


When assessing these arguments, When assessing these arguments, ask:ask:

1.1. Is the sample representative?Is the sample representative?

2.2. Is the sample large enough?Is the sample large enough?

Statistical SyllogismStatistical Syllogism

Has the following form:Has the following form:

Z percent of all F’s are GZ percent of all F’s are G

x is an Fx is an F

Is it probable to the degree 0.Z that x Is it probable to the degree 0.Z that x is Gis G


What’s the difference between an What’s the difference between an inductive generalization and a statistical inductive generalization and a statistical syllogism?syllogism?

Inductive generalizations reason Inductive generalizations reason fromfrom particularparticular observations observations toto a a generalgeneral claim about a class.claim about a class.

Statistical syllogisms reason Statistical syllogisms reason from a from a generalgeneral claim about a class claim about a class to a claim to a claim about a particular individualabout a particular individual..


E.g.,E.g.,

60% of students at STFX believe in 60% of students at STFX believe in God.God.

Bob is a student at STFX.Bob is a student at STFX.

Therefore, there is a .6 degree of Therefore, there is a .6 degree of probability that Bob believes in God.probability that Bob believes in God.

Statistical SyllogismStatistical Syllogism When assessing these arguments, ask:When assessing these arguments, ask:

1.1. Is there any additional information about Is there any additional information about x that has not been included in the x that has not been included in the premises?premises?

E.g. Bob is President of Catholic League E.g. Bob is President of Catholic League of Students (prob that he believes in God of Students (prob that he believes in God increases).increases).

E.g., Bob is President of the Atheists for E.g., Bob is President of the Atheists for the Environment Society (Prob that he the Environment Society (Prob that he believes in God decreases).believes in God decreases).

Induction by ConfirmationInduction by Confirmation

Induction can be used to support a Induction can be used to support a hypothesis or theory by providing hypothesis or theory by providing confirming instances of that hypothesis or confirming instances of that hypothesis or theory.theory.

When we propose a theory or hypothesis, When we propose a theory or hypothesis, there are certain things that ought to be there are certain things that ought to be observed if it is actually true (or probable).observed if it is actually true (or probable).

These are called observation statements. These are called observation statements. If we do observe what the theory predicts, If we do observe what the theory predicts, then we have confirmed the theory.then we have confirmed the theory.


Induction by Confirmation then has the Induction by Confirmation then has the following form:following form:

If h then oIf h then o

oo

It is probable that hIt is probable that h

NB: similar to the formal fallacy of “affirming NB: similar to the formal fallacy of “affirming the consequent”the consequent”


E.g.E.g.

If the theory of general relativity is true, then If the theory of general relativity is true, then it follows that light rays passing near the it follows that light rays passing near the sun will bend.sun will bend.

During the solar eclipse of 1919 it was During the solar eclipse of 1919 it was observed that light rays passing near the observed that light rays passing near the sun did bend.sun did bend.

It is probable therefore that the theory of It is probable therefore that the theory of general relativity is true.general relativity is true.


When assessing these arguments, ask:When assessing these arguments, ask:

1.1. Is the number of confirming instances Is the number of confirming instances relatively high?relatively high?

• In general, the more confirming instances In general, the more confirming instances the better the theory.the better the theory.

2.2. Are there any disconfirming instances?Are there any disconfirming instances?• Any disconfirming instance Any disconfirming instance refutesrefutes the the

theory.theory.

Induction by ConfirmationInduction by Confirmation Disconfirming instances are regarded as Disconfirming instances are regarded as

refutations of a theory because such a refutation refutations of a theory because such a refutation takes this form:takes this form:

If h then oIf h then oNot-oNot-oTherefore not-hTherefore not-h

That is, a disconfirming instances refutes a theory That is, a disconfirming instances refutes a theory because we are dealing with a deductively valid because we are dealing with a deductively valid argument form: Modus Tollens (denying the argument form: Modus Tollens (denying the consequent).consequent).

Analogical ReasoningAnalogical Reasoning

Analogical reasoning works by Analogical reasoning works by comparing things which are similar comparing things which are similar (analogous) and concluding that (analogous) and concluding that properties or relations that one thing properties or relations that one thing has must also be present in the has must also be present in the other.other.


E.g.,E.g.,

Last year I put some fertilizer on my Last year I put some fertilizer on my strawberries and in the fall got about strawberries and in the fall got about 20 per cent more strawberries. You 20 per cent more strawberries. You should do the same with your should do the same with your strawberries, since you got the same strawberries, since you got the same kind of soil. You’ll probably get more kind of soil. You’ll probably get more strawberries too.strawberries too.


Analogies compare two cases: the Analogies compare two cases: the subject casesubject case, and the , and the analogue analogue casecase..

The subject case is the case about The subject case is the case about which we are trying to derive a which we are trying to derive a conclusion (fertilizer on your soil)conclusion (fertilizer on your soil)

The analogue case is the case about The analogue case is the case about which we are more familiar (fertilizer which we are more familiar (fertilizer on my soil).on my soil).


The conclusion in an analogy makes The conclusion in an analogy makes a claim about the subject case, and a claim about the subject case, and in particular states that the subject in particular states that the subject case will (probably) have the case will (probably) have the target target featurefeature..

The target feature (increase in The target feature (increase in strawberry production) is the feature strawberry production) is the feature that is present in the analogue case, that is present in the analogue case, and it is being concluded that it and it is being concluded that it (probably) is in the subject case.(probably) is in the subject case.


There are two kinds of analogical There are two kinds of analogical arguments:arguments:

1.1. Analogical Argument by PropertiesAnalogical Argument by Properties

2.2. Analogical Argument by RelationsAnalogical Argument by Relations

Analogical Argument by Analogical Argument by PropertiesProperties

Analogical Argument by Properties Analogical Argument by Properties has the following form:has the following form:

x has A, B, C.x has A, B, C. [analogue case][analogue case]

y has A, B.y has A, B.[subject case][subject case]

It is probable therefore that y has C It is probable therefore that y has C [target feature][target feature]


E.g.,E.g.,

Canada geese are water birds that Canada geese are water birds that nest in Canada in the early spring nest in Canada in the early spring and migrate south to warmer and migrate south to warmer climates for the winter months. climates for the winter months. Ducks are also water birds that nest Ducks are also water birds that nest in Canada in early spring. Therefore, in Canada in early spring. Therefore, ducks probably migrate south for the ducks probably migrate south for the winter, too.winter, too.


P1 [analogue case]: Canada geese are water P1 [analogue case]: Canada geese are water birds that nest in Canada in the early birds that nest in Canada in the early spring and migrate south to warmer spring and migrate south to warmer climates for the winter months. climates for the winter months.

P2 [subject case]: Ducks are also water birds P2 [subject case]: Ducks are also water birds that nest in Canada in early spring. that nest in Canada in early spring.

Conclusion: Therefore, ducks probably Conclusion: Therefore, ducks probably migrate south for the winter [target migrate south for the winter [target feature], too.feature], too.

Analogical Argument by Analogical Argument by RelationsRelations

Analogical Argument by Relations Analogical Argument by Relations has the following form:has the following form:

x is to y [analogue case] as a is to b x is to y [analogue case] as a is to b [subject case].[subject case].

x is R to y.x is R to y.It is probable therefore that a is R to b It is probable therefore that a is R to b

[target feature][target feature]


E.g.,E.g.,

The proposal to give clean needles to The proposal to give clean needles to prison inmates to stop the spread of prison inmates to stop the spread of AIDS from the use of dirty needles is AIDS from the use of dirty needles is ridiculous. It is like giving bank ridiculous. It is like giving bank robbers normal bullets to stop them robbers normal bullets to stop them from using dum-dum bullets, which from using dum-dum bullets, which are much more damaging to the are much more damaging to the victim.victim.


P1: Dum-dum bullets are to normal bullets (as used by P1: Dum-dum bullets are to normal bullets (as used by bank robbers) [analogue case] as dirty needles are to bank robbers) [analogue case] as dirty needles are to clean (as used by prison inmates) [subject case].clean (as used by prison inmates) [subject case].

P2: Although dum-dum bullets are much more P2: Although dum-dum bullets are much more damaging to the victim, normal bullets still kill their damaging to the victim, normal bullets still kill their victims. Further, the role of police officers is to stop victims. Further, the role of police officers is to stop bank robbers, not prevent the harms they cause.bank robbers, not prevent the harms they cause.

Conclusion: Although dirty needles are more damaging Conclusion: Although dirty needles are more damaging to the victim (addicts are likely to get HIV, etc.), clean to the victim (addicts are likely to get HIV, etc.), clean needles can be just as damaging (e.g., overdoses). needles can be just as damaging (e.g., overdoses). Further, the role of prison officials is to stop drug use, Further, the role of prison officials is to stop drug use, not prevent the harms caused by it.not prevent the harms caused by it.

Analogical ReasoningAnalogical Reasoning When assessing these arguments, ask:When assessing these arguments, ask:

1.1. How many entities are we comparing?How many entities are we comparing?2.2. What is the variety of dissimilarity?What is the variety of dissimilarity?3.3. In how many respects are the entities In how many respects are the entities

similar?similar?4.4. Are the respects in which the compared Are the respects in which the compared

entities are similar relevant to the entities are similar relevant to the conclusion?conclusion?

5.5. In what ways are the entities under In what ways are the entities under consideration dissimilar?consideration dissimilar?

6.6. How bold or modest is the conclusion?How bold or modest is the conclusion?

Documents

Inductive Reasoning. The Nature of Inductive Reasoning What is an inductive argument? What is an inductive argument? 1. Any argument which is not deductive!