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deduction__induction.pptxDeduction & Induction
Two different types of argumentation.
Deduction v. Induction
Deductive Argument (193 in text)
The conclusion follows directly from the premises & nothing more need be added for that to happen.
Think of it as working with a closed set – like a puzzle that is not missing any pieces. From the pieces you deduce that the conclusion follows.
We don’t need to add premises in order to extract the exclusion
A deductive argument is either valid or invalid as regards its structure, and sound or unsound as regards its content & structure together (see next slide)
Proofs in geometry are deductive.
Inductive Argument (194 in text)
The conclusion is said to follow with a degree of probability or likelihood from the premises – but there is always a wedge of doubt about that actually occurring. Unlike a valid deductive argument, the conclusion is not guaranteed and is not certain.
Think of it as working with a puzzle that’s missing at least one piece (sometimes more!). From these pieces, you induce that the conclusion will likely follow, but you can’t be certain.
An inductive argument can be either stronger or weaker, along a continuum (that is, in degrees, rather than either/or).
Statistics is often inductive.
Deductively valid arguments & Deductively sound arguments
Step 1: Validity
See electronic reserves article “Deduction”
If the premises of the argument are assumed true, it is impossible for the conclusion to be false, that is the conclusion is guaranteed as true. This leaves open whether or not its premises are actually true.
Validity refers to the structure of the argument – how the jigsaw puzzle pieces fit together – and not the content of the argument.
This is technical definition of validity, which is different from the everyday use of the term.
Step 2: Soundness
See electronic reserves article “Deduction”
After validity is determined, the second step is to consider the soundness of the argument.
An argument that is both valid and has all and only true premises. (The content is true. However, if the argument isn’t also valid in structure, it does not matter how true the premises are.)
Examples of Deductively valid argument structures: Modus Ponens
See pp. 204 – 206 in text & electronic reserves article, “Deduction”.
Modus ponens: (Latin for mode that affirms):
If A then B.
A is the case.
Therefore, B is true also.
Example:
If the dentist slips while operating [A], then Omar will need stitches [B].
The dentist slipped while operating. [A is the case.]
So, Omar needed to get stitches. [B is true also.]
This conclusion is guaranteed, if we assume the premises to be true. However, for the argument to be sound, the premises must also be actually true. This structure is always valid.
More examples of deductively valid argument structures: Modus Tollens
Modus Tollens (Latin for mode that denies).
If A then B.
B is not the case.
Therefore, A is not true either.
Example:
If Bruce’s mother gets a tatoo of a dragon[A] , his mother will go through the roof. [B]
Bruce’s mother did not go through the roof [not B]
Therefore, Bruce did not get a tatoo of a dragon. [not A]
This conclusion is guaranteed, if we assume the premises to be true. However, for the argument to be sound, the premises must also be actually true. This structure is always valid.
More examples of deductively valid argument structures: Disjunctive Syllogism
Disjunctive syllogism: (either/or)
Either A or B.
A is not the case.
Therefore, B must be true
OR: Either A or B.
B is not the case.
Therefore, A must be true.
Examples:
Either that’s a rainbow trout [A] or a weird-looking salmon. [B]
That’s not a rainbow trout. [A is not the case.]
So, it’s a weird-looking salmon. [So, B must be true.]
OR
Either that’s a rainbow trout [A] or a weird-looking salmon. [B]
That’s not a weird-looking salmon. [B is not the case.]
So, it’s a rainbow trout. [So, A must be true.]
More examples of deductively valid argument structures: Hypothetical Syllogism
Hypothetical syllogism:
If A then B.
If B then C.
Therefore, if A then C.
OR: All A is B.
All B is C.
Therefore, all A is C.
Examples:
If Louie goes to the powwow [A], he’ll miss the ball game.[B]
If Louie misses the ball game [B], he won’t get a chili dog.[C]
Therefore, if Louie goes to the powwow [A], he won’t get a chili dog. [C]
OR
Anyone who enjoys music [A] will like my new Taylor Swift album.[B]
Anyone who likes my new Taylor Swift album [ B]will like Loreena McKennitt.[C]
Therefore, anyone who enjoys music [A] will like Loreena McKennitt.[C]
Deductively valid arguments can be nonsensical in their content.
Example of a deductively valid structure:
All humans are mortal.
Socrates is human.
Therefore, Socrates is mortal.
Example of an argument nonsensical in content, but a deductively valid structure of the same type as above:
All blogs are blue.
Jane is a blog.
Therefore, Jane is blue.
Other examples;
All beagles eat mice.
Amy Bush is a beagle.
Therefore, Amy Bush eats mice.
Examples of deductively invalid argument structures & Formal fallacies
Some structures do not work: they never guarantee the truth of their conclusions.
For example, compare the following 2 structures:
Valid structure:
All wines are beverages.
Chardonnay is a wine.
Therefore, Chardonnay is a beverage.
Invalid structure:
All wines are beverages.
Chardonnay is a beverage.
Therefore, Chardonnay is a wine.
The second example above is invalid because if chardonnay is a beverage it does not necessarily follow that it is a wine. It could be a sparkling water, it could be a soda. The conclusion is not guaranteed.
More examples of deductively invalid argument structures & formal fallacies
See pp. 211 – 213 in your text for invalid structures that are also considered “formal fallacies” (errors in reasoning).
Fallacy of “Affirming the Consequent”:
If A then B.
B is the case.
Therefore, A is also true.
Notice that is the reverse order of Modus Ponens (see slide 4)
Example:
If the dentist slips while operating [A], Omar will need stitches.[B]
Omar will need stitches.[B is the case.]
So, the dentist slipped while operating.[So, A is also true.]
This is invalid because Omar might need stitches for any number of reasons. Maybe Omar slipped out of the dentist’s chair. Maybe Omar fell in a bicycle accident. Therefore, the conclusions is not guaranteed.
More examples of deductively invalid arguments & formal fallacies
See pp. 212 - 213 of your text.
Fallacy of Denying the Antecedent:
If A, then B
Not A.
Therefore, not B.
Notice that this reverses the order of Modus Tollens (see slide 5).
Example of this fallacy:
If Bruce gets a tattoo of a dragon [A], his mother will go through the roof. [B]
Bruce does not get a tattoo of a dragon. [A is not the case.]
Therefore, his mother will not go through the roof. [So, B is not true either.]
However, this conclusion is not guaranteed, because maybe his mother goes through the roof for many other reasons, too. She might go through the roof anyway.
More pointers on Deductive arguments
Value claims can be treated as if they can be true for purposes of determining validity.
Deductively invalid arguments (the examples given on slides 10 & 11), which do not guarantee their conclusions, can be used to set up inductive or non-deductive arguments, in which the conclusions are only likely and are uncertain.
See the exercises on p. 219 of your text. The next graded homework assignment will include one exercise that is like the exercises on this page.
Major types of inductive arguments & understanding the “wedge of doubt”
See p. 224 in your text:
Predictions: an argument is made of the future based on past or present evidence. Predictions always are uncertain, even if it is a prediction that the sun will rise tomorrow, insofar as the fact that the sun rose today and yesterday does not alone assure that it will rise tomorrow.
Retrodictions: arguments about the past based on present evidence: an inference is drawn about what happened at some earlier point in time based on current evidence. There is uncertainty partly because we do not have direct access to the past.
Cause-and-effect Reasoning: This will be discussed in more detail in the second power point presentation for this unit.
Arguments based on Analogy: This involves comparisons, from which an inference is made from similarities in known terms to possible similarities applied to an unknown term. (See the example in the text.) Analogies always remain uncertain because they are only as strong as there are relevant similarities that outweigh differences.
Statistical reasoning: These arguments draw from sample studies or statistical reasoning, from which an inference is drawn about either all or part of the targeted population. This will be discussed in more detail in later power point presentations. They remain uncertain because we generalize from a sample that might not adequately represent the target population.
The importance of entertaining alternative explanations in induction
See the electronic reserves article, “Choice and Chance”, p. 5, the two indented paragraphs. The first paragraph gives the following strong inductive argument:
P1: Black has confessed to killing White. [confession]
P2. Dr. Zed has signed a statement to the effect that he saw Black shoot White. [eyewitness testimony]
P3. A large number of witnesses heard White gasp with his dying breath, “Black did it.” [testimony]
C: Therefore, Black killed White.
However, in the next paragraph, the author gives an alternative scenario, in which the above premises could be true, but the above conclusion false, because we have added premises that change the conclusion. [See the next slide.]
In inductive arguments, adding premises (that is, additional information) may change the conclusion, or strengthen or weaken the argument.
The fallacy of “Suppressed evidence” occurs when a relevant premise or information is omitted from an argument in such a manner that the conclusion is significantly changed.
Alternative argument with additional premises to the one in previous slide.
The second paragraph on p. 5 of “Choice and Chance” provides the following revised inductive argument:
P1. Black is insane and confesses to every murder he ever heard of, but that this fact was generally unknown because he had just moved into the neighborhood. [his confession now becomes unreliable.]
P2. This peculiarity was, however, known to Dr. Zed, who was White’s psychiatrist. [so Dr. Zed was in a position to take advantage of this knowledge]
P3. For his own malevolent reasons, Dr. Zed decided to eliminate White and frame Black. [He could frame Black, because of P1. & P2 above, & thus, when Dr. Zed signed a statement to the effect that he saw Black shoot White, he was lying.]
P4. Dr. Zed convinced White under hypnosis that Black was a homocidal maniac bent on killing White. [This would account for why White gasped with his dying breath, “Black did it.”]
C: Dr. Zed shot White from behind a potted plant and fled.
It may be somewhat difficult to put the above in argument form, but it is an alternative explanation of the events that transpired, which lends itself to a different conclusion. Although this explanation is less plausible, it is not impossible, and shows how the conclusions of inductive arguments are never certain.
Another example of how adding premises to an argument may change the conclusion
Consider the following argument:
The sand on the beaches in Cape Cod clean.
The surf on the beaches in Cape Cod is great.
The weather in Cape Cod in the summer is warm
Therefore, go to Cape Cod to swim and surf.
Now consider whether or not you would change the conclusion if we added the following premise:
There are sharks at the beaches in Cape Cod.
Fine differences between deduction & induction;
See electronic reserves article, “Choice and Chance”, p. 8:
The following is an inductive argument because the factual claim made by the conclusion is not implicit in the premises. Inductive arguments venture beyond the factual claims made by the premises, which is why they are only probable.
George is a man.
George is 100 years old.
George has arthritis.
Therefore, George will not run a four-minute mile tomorrow.
However, by adding another premise to the above argument we can convert it to a deductively valid argument:
George is a man.
George is 100 years old.
George has arthritis.
No 100-year-old man with arthritis can run a four-minute mile.
Therefore, George will not run a four-minute mile tomorrow.
By adding the last premise to this argument, we have included all the puzzle pieces, so that the conclusion is now implicit in the premises. That is why it is now deductive and it is valid, because, if the premises are assumed to be true, the conclusion now necessarily follows.
However, how do we come up with general claims like “No 100-year-old man with arthritis can run a four-minute mile”? Do we infer from observations, so that we make inductive generalizations?
