I ) My pet is either a dog or a cat (but not both). If it is a cat, then it meows. If it is a dog, then it barks. The pet must, then, either meow or bark. My pet does not bark. Therefore, my pet must meow.

 

P1) C Ú D ~(C & D)

P2) C®M

P3) D®B

P4) M Ú B

P5) ~B

C)   \ M

 

II) If the Spartans win every game, then they will progress through the tournament. If they progress through the tournament then they could go to the tournament finals. If they win the tournament finals then they would be national champions. The Spartans did not win every game. Therefore the Spartans cannot be national champions.

 

P1) W®P

P2) P®F

P3) F®C

P4) ~W

C)   ~C

 

III) – What’s wrong with the following argument?

 

P1) P Ú Q

P2) B®~P

P3) B®~Q

P4) B

P5) ~P

C)   \Q

      

IV) IF GWB resigned, then Americans would be happy. If Americans are happen, then GWB will be re-elected. Therefore, if GWB resigns, GWB will be elected.

 

 

 

 

 

V) The scowl on your face says it all. Those who read this article on poverty are either very appalled or completely in agreement. Those who agree have unusually experienced some sort of first hand poverty. You must not have been poor growing up.

 

 

 

 

V1) If Bush wins presidency, then Cheney will be VP, Either Bush or Kerry will win.  Kerry will lose. Therefore, Cheney will be VP and Bush will win.

 

P1) A®B

P2) A Ú C

P3) ~C

L1) A                  2, 3

L2) B                  1, L1

C) A & B            L1 & L

 

 

 

Inductive Arguments – Use inductive reasoning to make an inference from known cases to unknown cases. They predict the way the world will be (or sometimes retrodict the way it was).

 

Examples of Inductive Reasoning

·      Belief that your table will remain roughly the same size and shape tomorrow as it is today. (That it won’t spontaneously change size or shape or even disappear entirely.)

·      Belief that whatever is true of a representative sample of the population will be true of the population at large.

 

Problem of Induction (David Hume)

P1) Every day that I can remember the sun has risen.

C) Therefore the sun will rise tomorrow.

 

Problem – However sure we are about matters of experience we cannot prove beyond a shadow of a doubt that the sun will rise tomorrow. Our confidence in this type of specific inductive reasoning rests on the plausibility of the following claim:

 

·      The future will resemble the past.

 

P1) Every day that I can remember the sun has risen.

P2) The future will resemble the past.

C) Therefore the sun will rise tomorrow.

 

This amended argument is deductively valid. If the premises are true, the conclusion has to be true. Now we have to ask whether the premises are true. What does our confidence in (P2) rest on? Answer: We know (P2) on inductive grounds as well. Thus, we can never be certain of either (P2) or any conclusion of an argument that relies on (P2) (either implicitly or explicitly) as a premise.

 

Responses to Hume

1) Fundamental skepticism. There are many things that we don’t know.

2) Try to improve inductive arguments by e.g., developing inductive logic.

3) Reject Hume’s assumption that only deductively valid arguments are cogent.

 

Can’t easily give up induction. We cannot get through even a single hour of life without implicitly or explicitly depending on inductive reasoning. We assume that there are basic regularities and continuities in the world. We naturally classify things and operate on the principle that things resembling each other in some respect are likely to resemble each other in further respects. We assume that there are intelligent patterns in the world and relations of cause and effects.

 

Three Types of Inductive Arguments

·      Inductive Generalizations

·      Causal Inductive Arguments

·      Inductive Analogies.

 

 

Inductive Generalization

The premises describe a number of observed objects or events as having a certain feature and the conclusion asserts, on the basis of these observations, that all or most objects or events of the same type will have that feature.

 

P1) All the children I have known over the past three decades have begun to talk before the age of two.

P2) The seven childcare books I have consulted all indicate that nearly all normal children begin to talk between the first and second year.

L) Nearly all children will begin to talk before the age of two.

C) Your child will begin to talk before the age of two.

 

(P1) is based on direct experience. (P2) is based on indirect experience. 

 

Retrodiction – Reason backwards from present evidence to claims about the past.

Good Samples – For an inductive generalization it is important to find a sample that is representative of the population.

 

Random Samples - A sample is random if every member of the population has an equal change of being chosen for it.

 

Stratified Sample – In a stratified sample the population is divided into large subgroups, then every effort is made to ensure that the sample has the same proportion of representation of each subgroup as the population as a whole.

 

Variability - The smaller the variability the smaller you need your sample size to be.

 

Biased Samples – A sample is biased if it does not adequately represent the population.

 

 

Causal Inductive Arguments

What do we mean by cause? Sometimes a necessary condition, sometimes a sufficient condition and sometimes a contributing factor.

 

We say that A causes B when i) A is a necessary cause of B (dry forest conditions are necessary for forest fires), ii) A is a sufficient cause of B (the lightning that struck the tree was sufficient, given the background conditions, to start the fire) ii) A is both a necessary and a sufficient condition for B (entire causal history is given) or iv) A is a contributing factor of B (forest fired caused by a failure to apply forest management techniques).

 

Correlation vs. Cause

 

Positive Correlation: If a higher proportion of As than non-As are B, then there is a positive correlation between being A and being B.

Negative Correlation: If a smaller proportion of As than non-As are B, then there is a negative correlation between being A and being B.

No Correlation: If about the same proportion of As as non-As are B, then there is no correlation between being A and being B.


Correlation is not the same thing as causation.

·      Correlation has to be significant

·      Can have correlation without causation

 

If A is positively correlated with B, then one of the following will be true:

 

·      A causally contributes to B

·      B causally contributes to A

·      Some other factor, C, is the underlying cause

·      The correlation between A and B is coincidental

 

Causal Argument

P1) C and E are regularly associated events

P2) C regularly occurs before E

P3) The claim that C is a cause of E is consistent with background knowledge about C and E

Therefore, probably

C)     C is a cause of E

 

Fallacies of Inductive Reasoning (pp. 336-341)

 

Hasty Generalization – Making an inference from a hopelessly inadequate sample. For example, generalizing from anecdotal evidence.

 

Post Hot Fallacy (Post hoc ergo propter hoc, after this, therefore because of this) – Inferring that A caused B because A came before B.

 

Fallacies of Composition and Division – Attributing to the whole what is true of the part, or to the part what is true of the whole.

 

Objectionable Cause – Arguing for a causal interpretation on the basis of limited evidence, with no attempt to rule out alternative explanations.

 

P1) A occurred

P2) B occurred

P3) We can plausible connect A to B in a causal relationship

C) Therefore, A caused B

 

Causal Slippery Slope Arguments – A proposed action is wrong because it would set off a series of side effects, ending ultimately in general calamity.