LOGIC AND REASONING

the science of reasoning, proof, thinking or inference

Logic

allows us to analyze arguments and determine whether it is valid or invalid

Logic

a tool used in mathematical proofs

Logic

Includes: a language for expressing complicated compound statements, a concise notation for writing them, and a methodology for objectively reasoning about their truth or falsity.

Logic

Who tried to advance the study of logic from a merely philosophical subject to a formal mathematical subject. He never completely achieved this goal.

Gottfried Wilhelm Leibniz (1646 - 1716)

They contributed to the advancement of symbolic logic as a mathematical discipline

Augustus de Moegan (1806 - 1871) and George Boole (1815 - 1864)

Boole published this in 1848.

The Mathematical Analysis of Logic

Boole published this in 1854 as more extensive work

An Investigation of the Laws of Thought

a declarative sentence that is either true or false, but not both true and false

statement

1. Yes

2. No

3. Yes

4. Yes

5. No

6. Yes

7. Yes

8. Yes

is a statement that conveys a single idea

simple statement

is a statement that conveys two or more ideas

compound statement

Connecting simple statements with words and phrases such as and, or, if...then, and if and only if creates a

compound statement

“not p”

Give the:

• connective

• symbolic form

• type fo statement

• not

• ~p

• negation

“p and q”

Give the:

• connective

• symbolic form

• type fo statement

• and

• p^q

• conjunction

“p or q”

Give the:

• connective

• symbolic form

• type fo statement

• or

• pVq

• disjunctions

“if p then q”

Give the:

• connective

• symbolic form

• type fo statement

• if..then

• p—>q

• conditional

“p if and only if q”

Give the:

• connective

• symbolic form

• type fo statement

• if and only if

• p < —> q

• biconditional

What do you call if a compound statement is written in symbolic form, then parenthesis are used to indicate which simple statements are grouped together

Grouping symbols

either true (T) or false (F).

truth value of a simple statement

depends on the truth values of its simple statements and its connectives.

truth value of a compound statement

is a table that shows the truth value of a compound statement for all possible truth values of its simple statements

A truth table

Two statements are equivalent if they have the same truth value for all possible truth values of their simple statements

Equivalent Statements

Equivalent Statements are denoted by

≡

• ~(pVq) ≡ ~p^~q

• ~(p^q) ≡ ~p V~q

De Morgan’s Law for Statements

shows an implication; that is, given that a situation p will happen

Conditional Statement

It is part of a conditional statement that talks about situation p will happen

antecedent or hypothesis

It is part of a conditional statement that talks about another situation q will happen

consequence or conclusion

True or false: Most theorems in mathematics are in the form of a conditional

true

• If p, q.

• p implies q.

• q, if p.

• q when p.

• p is sufficient for q.

• q is necessary for p.

• p only if q.

• q whenever p.

• q follows from p

Are examples of

other ways to express p—>q

two-way conditional statements

Biconditional Statements

is a statement that is always true.

tautology

is a statement that is always false

self-contradiction

consists of a set of statements called premises and another statement called the conclusion.

argument

if the conclusion is true whenever all the premises are assumed to be true

argument is valid

if it is not a valid argument.

argument is invalid

is the process of reaching a general conclusion by examining specific examples

Inductive reasoning

The conclusion formed by using inductive reasoning is often called a

conjecture

true or false: Inductive reasoning is used when we predict the next number in a sequence.

true

Galileo Galilei used inductive reasoning to discover that the time required for a pendulum to complete one swing, called the __ depends on the length of the pendulum.

period of the pendulum

Is it true that Galileo did not have a clock, so he measured the periods of pendulums in “heartbeats.

true

A statement is a true statement provided that it is true in all cases.

Counterexamples

If you can find one case for which a statement is not true then the statement is a false statement.

counterexample

is the process of reaching a conclusion by applying general assumptions, procedures, or principles

Deductive reasoning

Is it true that the conclusions reached by deductive reasoning are valid and can be relied on?

true

“During the past 10 years, a tree has produced plums every other year. Last year the tree did not produce plums, so this year the tree will produce plums”

is this inductive or deductive?

inductive