Deductive reasoning

Inductive reasoning

involves starting from specific premises and forming a general conclusion, while deductive reasoning involves using general premises to form a specific conclusion.

deductive reasoning

involves conclusion reached which cannoy be incorrect if the premises are true. That’s because the conclusion doesn’t contain information that’s not in the premises.

a conclusion reached via inductive reasoning

is a generalization

Premise: all known fish species in this genius have yellow fins.

Conclusion: any newly discovered species in the genius is likely to have yellow fins.

premise: this volcano has erupted about every 500 years for the last 1 million years. It erupted 499 years ago.

Conclusion: It will erupt again soon.

Validity:

a deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid.

How is an argument valid? written

If the truth of premises logically guarantees the truth of the conclusion, it would be impossible for the premises to be true and the conclusion false.

Soundness

A deductive argument is sound if and only if it is both valid and all of its premises are actually true.

valid argument example

Anyone who lives in the city Honolulu, HI also lives on the island of Oahu.

Kanoe does not live on the island of Oahu.

Therefore, Kanoe does not live in the city Honolulu, HI.
 

invalid argument ex

Anyone who lives in the city Honolulu, HI also lives on the island of Oahu.

Kanoe lives on the island of Oahu.

Therefore, Kanoe lives in the city Honolulu, HI.

validity argument outline

If A, then B.
Not B.
So, not A.

If A, then B.
A
So, B.

only is valid

invalid argument example

If A, then B.
B
So, A

If A, then B.
Not A.
So, not B.

soundness argument valid

All A are B;
No B are C;
Therefore, No A are C.

All tigers are mammals.
No mammals are creatures with scales.
Therefore, no tigers are creatures with scales.

All spider monkeys are elephants.
No elephants are animals.
Therefore, no spider monkeys are animals.

invalid sound arguments

All A’s are F;
X is F;
Therefore, X is an A.

All basketballs are round.
The Earth is round.
Therefore, the Earth is a basketball.

All popes reside at the Vatican.
John Paul II resides at the Vatican.
Therefore, John Paul II is a pope.

modus ponens

says that if one thing is true, then another will be. It then states that the first is true. The conclusion is that the second thing is true. It is shown below in logical form.

modus ponen template

If P, then Q P
Therefore Q

1. If Kelly does not finish his homework, he will not go to class

2. Kelly did not finish his homework

3. Therefore, Kelly will not go to class

modus tollen written

In this form, you start with the same first premise as with modus ponens. However, the second part of the premise is denied, leading to the conclusion that the first part of the premise should be denied as well.

modus tollen template

If P, then Q
Not Q Therefore not P.

With actual content, it looks like below:

1. If the Saints win the Super Bowl, there will be a party in New Orleans that night

2. There was no party in New Orleans that night

3. Therefore, the Saints did not win the Super Bowl

hypothetical syllogism

If P, then Q
If Q, then R Therefore if P, then R

1. If it rains today, I will wear my rain jacket

2. If I wear my rain jacket, I will keep dry

3. Therefore if it rains today, I will keep dry

disjunctive syllogism

Either P or Q

Not P

Therefore Q

When P and Q are replaced with real life with actual content it looks like below:

1. Either you will see Joe in class today or he will oversleep

2. You did not see Joe in class today

3. Therefore Joe overslept

dilemma

Either P or Q
If P then R
If Q then S
Therefore either R or S

With actual content, it looks like below:

1. Bill will either take the stairs or the elevator to his room

2. If he takes the stairs, he will be tired when he gets to his room

3. If he takes the elevator, he will miss the start of the football game on TV

4. Therefore Bill will either be tired when he gets to his room or he will miss the start of the

   football game