1. DEDUCTIVE vs. INDUCTIVE REASONING
Section 1.1

2. Problem Solving Logic – The science of correct reasoning.
Reasoning – The drawing of inferences or conclusions from known or assumed facts.
When solving a problem, one must understand the question, gather all pertinent facts, analyze the problem i.e. compare with previous problems (note similarities and differences), perhaps use pictures or formulas to solve the problem.

3. Two basic categories of human reasoning
Deduction: reasoning from general premises, which are known or presumed to be known, to more specific, certain conclusions.
Induction: reasoning from specific cases to more general, but uncertain, conclusions.
Both deductive and inductive arguments occur frequently and naturally…both forms of reasoning can be equally compelling and persuasive, and neither form is preferred over the other (Hollihan & Baske, 1994).

4. Deduction Vs. Induction
commonly associated with “formal logic.”
involves reasoning from known premises, or premises presumed to be true, to a certain conclusion.
the conclusions reached are certain, inevitable, inescapable.
Induction
commonly known as “informal logic,” or “everyday argument”
involves drawing uncertain inferences, based on probabalistic reasoning.
the conclusions reached are probable, reasonable, plausible, believable.
Premise: a basis for reasoning
Inference: a conclusion based on evidence

5. Deductive Versus Inductive Reasoning
Deduction
It is the form or structure of a deductive argument that determines its validity
if the premises are true, then the conclusion necessarily follows.
The conclusion is said to be “entailed” in, or contained in, the premises.
example: use of DNA testing to establish paternity
Induction
By contrast, the form or structure of an inductive argument has little to do with its perceived believability or credibility, apart from making the argument seem more clear or more well-organized.
The receiver (or a 3rd party) determines the worth of an inductive argument

6. Deductive Reasoning
Deductive Reasoning – A type of logic in which one goes from a general statement to a specific instance.
The classic example
All men are mortal. (major premise)
Socrates is a man. (minor premise)
Therefore, Socrates is mortal. (conclusion)
The above is an example of a syllogism.

7. Deductive Reasoning
Syllogism: An argument composed of two statements or premises (the major and minor premises), followed by a conclusion.
For any given set of premises, if the conclusion is guaranteed, the arguments is said to be valid.
If the conclusion is not guaranteed (at least one instance in which the conclusion does not follow), the argument is said to be invalid.
BE CARFEUL, DO NOT CONFUSE TRUTH WITH VALIDITY!

8. Deductive Reasoning Examples: All students eat pizza.
Claire is a student at LCMS.
Therefore, Claire eats pizza.
2. All athletes work out in the gym.
Kellen Moore is an athlete.
Therefore, Kellen Moore works out in the gym.

9. Deductive Reasoning 3. All math teachers are over 7 feet tall.
Mr. Skeesuck. is a math teacher.
Therefore, Mr. Skeesuck is over 7 feet tall.
The argument is valid, but is certainly not true.
The above examples are of the form
If p, then q. (major premise)
x is p. (minor premise)
Therefore, x is q. (conclusion)

10. Venn Diagrams
Venn Diagram: A diagram consisting of various overlapping figures contained in a rectangle called the universe U
This is an example of all A are B. (If A, then B.) B A

11. Venn Diagrams
This is an example of No A are B. U A B

12. Venn Diagrams
This is an example of some A are B. (At least one A is B.)
The yellow oval is A, the blue oval is B.

13. Example
Construct a Venn Diagram to determine the validity of the given argument.
#14 All smiling cats talk.
The Cheshire Cat smiles.
Therefore, the Cheshire Cat talks.
VALID OR INVALID???

14. Example Valid argument; x is Cheshire Cat
Things
that talk
Smiling cats x

15. Examples #6 No one who can afford health insurance is unemployed.
All politicians can afford health
insurance.
Therefore, no politician is unemployed.
VALID OR INVALID?????

16. Examples X=politician. The argument is valid. X Unemployed Politicians
People who can afford
Health Care.
Politicians X
Unemployed

17. Example #16 Some professors wear glasses. Mr. Einstein wears glasses.
Therefore, Mr. Einstein is a professor.
Let the yellow oval be professors, and the blue oval be glass wearers. Then x (Mr. Einstein) is in the blue oval, but not in the overlapping region. The argument is invalid.

18. Inductive Reasoning
Inductive Reasoning, involves going from a series of specific cases to a general statement. The conclusion in an inductive argument is never guaranteed.
Example: What is the next number in the sequence 6, 13, 20, 27,…
There is more than one correct answer.

19. Inductive Reasoning Here’s the sequence again 6, 13, 20, 27,…
Look at the difference of each term.
13 – 6 = 7, 20 – 13 = 7, 27 – 20 = 7
Thus the next term is 34, because 34 – 27 = 7.
However what if the sequence represents the dates. Then the next number could be 3 (31 days in a month).
The next number could be 4 (30 day month)
Or it could be 5 (29 day month – Feb. Leap year)
Or even 6 (28 day month – Feb.)

20. Sample Deductive and Inductive Arguments
Example of Deduction
major premise: All tortoises are vegetarians
minor premise: Bessie is a tortoise
conclusion: Therefore, Bessie is a vegetarian
Example of Induction
Boss to employee: “Biff has a tattoo of an anchor on his arm. He probably served in the Navy.”

21. Deduction Versus Induction ---continued
Inductive reasoning enjoys a wide range of probability; it can be plausible, possible, reasonable, credible, etc.
the inferences drawn may be placed on a continuum ranging from cogent (clear, logical and convincing)at one end to fallacious (made-up) at the other.
Deductive reasoning is either “valid” or “invalid.” A deductive argument can’t be “sort of” valid.
If the reasoning employed in an argument is valid and the argument’s premises are true, then the argument is said to be sound.
valid reasoning + true premises = sound argument
fallacious
cogent

22. Deduction Versus Induction --still more
Deductive reasoning is commonly found in the natural sciences or “hard” sciences, less so in everyday arguments
Occasionally, everyday arguments do involve deductive reasoning:
Example: “Two or more persons are required to drive in the diamond lane. You don’t have two or more persons. Therefore you may not drive in the diamond lane”
Inductive reasoning is found in the courtroom, the boardroom, the classroom, and throughout the media
Most, but not all everyday arguments are based on induction
Examples: The “reasonable person” standard in civil law, and the “beyond a reasonable doubt” standard in criminal law

23. Inductive or deductive reasoning?
A sample of fifty motorists who were stopped by the CHP at a sobriety checkpoint on a Saturday at midnight revealed that one in four drivers were either uninsured, intoxicated, or both. Thus, if you get involved in an accident on the freeway there is a 25% chance the other motorist will be drunk or uninsured.
The Law of the Sea treaty states that any vessel beyond a 12 mile limit is in international waters. The treaty also states that any vessel in international waters cannot be legally stopped or boarded. Therefore, when the U.S. Coast Guard intercepts boats coming from Cuba or Haiti more than 12 miles from the U.S. coast, it is violating the Law of the Sea.