Module 5. Mathematical Logic Flashcards
deals with the scientific method of judging the truth or falsity of statements
Logic
body of statements or mathematical sentences
Mathematics
important in understanding mathematics as a logical system
symbolic language
basis of all mathematic reasoning
logic
one of the first mathematicians to make a serious study of symbolic logic
Gottfried Wilhelm Leibniz (1646-1716)
Leibniz tried to advance the study of logic from a merely __ subject to a formal __ subject
- philosophical
- mathematical
Was Leibniz able to achieve the goal of advancing the study from merely a philosophical subject to a formal mathematical subject?
no
(HAHAHAHA taas kaayo question unya ani ra answer)
Several mathematicans who contributed to the advancement of symbolic logic as a mathematical discipline
- Augustus De Morgan (1806-1871)
- George Boole (1851-1864)
- declarative sentence
- either true or false
- but not both true and false
statement/proposition
To determine whether a sentence is a statement, it may not be necessary to determine whether it is __
true
a statement is either true or false but not
both true and false
statement that conveys a single idea
simple statement
statement that conveys two or more ideas
compound statement
denial of a statement
negation
if p is any proposition, what is its negation symbol
~p (not p)
negation of a true statement
false
negation of a false statement
true
truth table for negation
p ~p
T
F
p ~p
T F
F T
some, all, and no (or none)
quantifiers
examples of quantifiers
- some
- all
- no (or none)
Negation of:
all p are q
some p are not q
Negation of:
some p are q
no p are q
- if p and q denotes to propositions
- compound proposition “p and q”
conjunction of p and q
symbol of conjunction of p and q
p ∧ q
a conjunction is true only if
both p and q are true
truth table of conjunction of p and q
p q p ∧ q
T T
T F
F T
F F
p q p ∧ q
T T T
T F F
F T F
F F F
- compound proposition
- formed by joining of two or more propositions by the word “or”
disjunction
symbol for disjunction of p and q
p ∨ q
equivalent to “and/or”
inclusive sense
inclusive sense is equivalent to
and/or
equivalent to “either … or” but not both
exclusive sense
exclusive sense is equivalent to
“either … or” but not both
Logicians have agreed to use the inclusive “or” meaning
one or the other or both
truth table of disjunction of p and q
p q p ∨ q
T T
T F
F T
F F
p q p ∨ q
T T T
T F T
F T T
F F F
proposition “if p, then q”
conditional/implication
symbol for conditional or implication of p and q
p => q
truth table for the conditional or implication
p q p => q
T T
T F
F T
F F
p q p => q
T T T
T F F
F T T
F F T
if p, then q
conditional
conditional
if p, then q
- interchaning the antecedent p with consequent q
- if q, then p
converse
converse
if q, then p
- negating the antecedent p and negating the consequent q
- if ~p, then ~q
inverse
inverse
if ~p, then ~q
- negating both the antecedent p and the consequent q and interchanging them
- if ~q, then ~p
contrapositive
contrapositive
if ~q, then ~p
truth table for converse
p q q => p
T T
T F
F T
F F
p q q => p
T T T
T F T
F T F
F F T
truth table for inverse
p q ~p => ~q
T T
T F
F T
F F
p q ~p => ~q
T T T
T F T
F T F
F F T
truth table for contrapositive
p q ~q => ~p
T T
T F
F T
F F
p q ~q => ~p
T T T
T F F
F T T
F F T
statement of the form
(p => q) ∧ (q => p)
biconditional
symbol for biconditional
p <=> q
how to read the symbol
p <=> q
p if and only if q
abbreviation of “p if and only if q”
p iff q
biconditional means
two conditionals
a biconditional can also be expressed by using __ and __ terminology
- necessary
- sufficient
“p is sufficient for q” can be rephrased as
if p then q
“p is necessary for q” can be rephrased as
if q then p
biconditional “p if and only if q” can also be phrased as
p is necessary and sufficient for q
biconditional combines a __ and its __ into one statement
- conditional
- converse
truth table for biconditional
p q p=>q q=>p q<=>p
T T T T
T F F T
F T T F
F F T T
p q p=>q q=>p q<=>p
T T T T T
T F F T F
F T T F F
F F T T T
words that connect simple statements and creates a compound statement
- and
- or
- if…then
- if and only if
used symbols such as p, q, r, and s to represent statements and symbols ∧, ∨, ~, =>, <=> to represent connectives
George Boole
when someone makes a sequence of statements and draws some conclusion from them, he or she is presenting an __
argument
two components of arguments
- initial statements (hypotheses)
- final statement (conclucsion)
if the conclusion of the argument is guaranteed under its given set of hypotheses
valid
consists of a set of statements called premises and another statement called the conclusion
argument
set of statements in an argument
premises
another statement in an argument
conclusion
an argument is __ if the conclusion is true whenever all the premises are assumed to be true
valid
an argument is __ if it not a valid argument
invalid
Two Argument Types
- inductive argument
- deductive argument
uses a collection of specific examples as its premises and uses them to propose a general conclusion
inductive argument
inductive argument
from specific to general
uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion
deductive argument
deductive argument
from general to specific
it is customary to place a __ __ between the premises and the conclusion
horizontal line
inductive or deductive
Premises:
I ate candy last week and I had toothache.
I ate candy yesterday and I had toothache.
I ate candy today, and I had toothache.
——————————————————
Conclusion:
Therefore, eating candy gives me toothache.
inductive
inductive or deductive
Premises:
all cats are mammals.
a tiger is a cat.
——————————-
Conclusion:
A tiger is a mammal.
deductive
inductive or deductive
Premises:
all doctors are men.
my mother is a doctor.
———————————
Conclusion:
Therefore, my mother is a man.
deductive
Truth table procedure to determine validity of argument
- write the argument in symbolic form.
- construct truth table that shows truth value of each premise and truth value of conclusion for all combinations of truth values of simple statements.
- if conclusion is true in every row of truth table in which all the premises are true, the argument is valid. if the conclusion is false in any row in which all of the premises are true, the argument is valid.
diagram consisting of various overlapping figures contained within a rectangle
Venn diagram
rectangle of a Venn diagram is called
universe
depict form of “all A are B” or “if A, then B”
two circles, one inside the other
- inner circle = A
- outer circle = B
saying that an argument is valid does not mean that the conclusion is __
true
Standard forms of Four Valid Arguments
- direct reasoning
- contrapositive reasoning
- transitive reasoning
- disjunctive reasoning
direct reasoning symbol
p => q
p
———
∴ q
contrapositive reasoning symbol
p => q
~q
———
∴ ~p
transitive reasoning symbol
p => q
q => r
———
∴ p => r
disjunctive reasoning symbol
p V q
~p
——-
∴ q
or
p V q
~q
——-
∴ p
Determine valid conclusion:
if julia is a lawyer, then she will be able to help us.
julia is not able to help us.
——————————-
∴
Julia is not a lawyer.
Determine valid conclusion:
if they had a good time, they will return.
if they return, we will make more money.
———————————-
∴
If they had a good time, we will make more money.
Determine valid conclusion:
if you can dream it, you can do it.
you can dream it.
————————–
∴
you can do it
Determine valid conclusion:
i bought a car or i bought a motorcycle.
i did not buy a car.
————————-
∴
I bought a motorcycle.
Standard Forms of Two Invalid Arguments
- Fallacy of the converse
- Fallacy of the inverse
Fallacy of the converse or
Fallacy of affirming the conclusion
Fallacy of the inverse or
Fallacy of denying the hypothesis
Fallacy of the converse symbol
p => q
q
——-
∴ p
Fallacy of the inverse symbol
p =>
~p
——-
∴ ~q
