[MMW] M3 - Problem Solving and Reasoning

REASONING

REASONING

• Essence of reasoning: search for truth

• We can consider something to be true if the available evidence seems to verify it. The more evidence we have, the stronger our conclusion can be

• “You can’t prove the truth, but using deductive and inductive reasoning, you can get close.”

search for truth

essence of reasoning

deductive

inductive

types of reasoning

inductive reasoning

• Drawing conclusions from facts using logic

• If an inductive argument is strong, the truth of the premise would mean the conclusion is likely

• If the inductive argument is weak, the logic connecting the premise and conclusion is incorrect

  • Observations and real life experiences

• Specific to general

  • AKA Scientific Method

scientific method

inductive reasoning is also known as ______

generalized

statistical

sample

analogous

predictive

casual inference

5 types of inductive reasoning

generalized

draws a conclusion from a generalization

• “All the swans I have seen are white; therefore, all swans are probably white.”

statistical

• Draws a conclusion based on statistics

• Generalization

  • “95% of swans are white; therefore, a randomly selected swan will probably be white”

sample

• Draws a conclusion about one group based on a different sample group

• 1 group

  • “There are ten swans in this pond and are all white; therefore, the swans in my neighbor’s pond are probably also white”

analogous

• Draws a conclusion based on shared properties of two groups

• 2 groups

  • “All Aylesbury ducks are white. Swans are similar to Aylesbury ducks. Therefore, all swans are probably white.”

predictive

• Draws a conclusion based on a prediction made using a past sample

• Past sample

  • “I visited this pond last year and all the swans were white. Therefore, when I visit again, all the swans will probably be white.”

causal inference

• Draws a conclusion based on casual connection

• Cause and effect

  • “All the swans in this pond are white. I just saw a white bird in the pond. The bird was probably a swan.”

deductive reasoning

• Begins with a broad truth (the major premise), followed by the more specific statement (the minor premise), then the conclusion

• Black and white; a conclusion is either true or false, and cannot be partly true or partly false

  • General to specific

inductive - predictive

Terry is always with James. Terry came to the party today so James must also be at the party.

is this inductive or deductive?

inductive

I see fireflies in my backyard every summer. This summer, I will probably see fireflies in my backyard.

deductive

Everytime I see Terry, James is with him, therefore Terry and James must always be together.

\n is this inductive or deductive?

intuition

• Representation, an explanation, or an interpretation directly accepted by us as something normal, self-evident, intrinsically meaningful, like a simple, given fact

proof

• An inferential argument for a mathematical statement

• In the argument, other previously established statements, such as theorems, can be used

• In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference

certainty

• Perfect knowledge that has total security from error, or the mental state of being without doubt

• Objectively defined, certainty is total continuity and validity of all foundational inquiry, to the highest degree of precision

• Something is certain only if no skepticism can occur

understand the problem

devise a plan

carry out the plan

look back

Polya’s 4 Steps in Problem Solving