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propositional logic

the branch of deductive reasoning that deals with the logical relationships among statements. The use of symbols to represent and clarify these relationships.

Takes this symbolization to another level by using symbols to stand not just for statements, but also for the relationships between statements—relationships that we specified previously with logical connective words, or connectives, such as if–then and or.

Truth tabling

A table that specifies the truth values for claim variables and combinations of claim variables in symbolized statements or arguments. which is just a graphic way of displaying all the possibilities.

Truth values

A statement that is either true or false. A true statement has a truth value of true, and a false statement has a truth value of false.

inference

The steps in the proof—the valid arguments

Modus Ponens (MP)

p → q

p

∴ q

Modus Tollens (MT)

p → q

~q

∴ ~p

Hypothetical Syllogism (HS)

p → q

q → r

∴ p → r

Simplification (SIMP)

p & q

∴ p

or

p & q

∴ q

Conjunction (CONJ

p

q

∴ p & q

Addition (ADD)

p

∴ p v q

or

q

∴ p v q

Disjunctive Syllogism (DS)

p v q

~p

∴ q

or

p v q

~q

∴ p

Constructive Dilemma (CD)

(p → q) & (r → s)

p v r

∴ q v s

Destructive Dilemma (DD)

(p → q) & (r → s)

~q v ~s

∴ ~p v ~r

propositional enthymeme

an argument where one or more premises are implied rather than explicitly stated, creating a kind of incomplete syllogism

Enumerative Induction

arguing from premises about some members of a group to a generalization about the entire group. Can fail to be strong by having a sample that’s to small or not representative. Becomes an error which is seen as hasty generalization

Enumerative Induction

For example, all the swans you have ever seen are white, so all swans must be white.

Analogical Induction

we reason that since two or more things are similar in several respects, they must be similar in some further respect. (1) the number of relevant similarities between things being compared, (2) the number of relevant dissimilarities, (3) the number of instances (or cases) of similarities or dissimilarities, and (4) the diversity among the cases.

Analogical Induction

Ex. (1) since humans can move about, solve mathematical equations, win chess games, and feel pain, and (2) since computers (or robots) are like humans in that they can move about, solve mathematical equations, and win chess games, it’s therefore probable that computers can also feel pain. Analogical induction, like all inductive reasoning, can establish conclusions only with a degree of probability.

Causal Argument

inductive argument whose conclusion contains a causal claim. May include the method of agreement, the method of difference, the method of agreement and difference, and the method of concomitant variation. Errors in cause-and-effect reasoning are common. a statement about the causes of things/ an inductive argument whose conclusion contains a causal claim.

inference to the best explanation

It’s the essence of scientific thinking and a mainstay of our everyday problem-solving and knowledge acquisition. If the explanations in these arguments really are the best, then the arguments are inductively strong. And if the premises are also true, then the arguments are cogent.

inference to the best explanation

Ex. Phenomenon Q.

E provides the best explanation for Q.

Therefore, it is probable that E is true.

Ex. The new quarterback dropped the ball again. The best explanation for that screw-up is that he’s nervous. So he’s definitely nervous.

Statement (claim

An assertion that something is or is not the case

Premise

A statement given in support of another statement

Conclusion

A statement that premises are intended to support

Argument

A group of statements in which some of them (the premises) are intended to support another of them (the conclusion)

Indicator words

Words that frequently accompany arguments and signal that a premise or conclusion is present

Deductive argument

An argument intended to provide conclusive support for its conclusion

Inductive argument

An argument intended to provide probable support for its conclusion