Propositional Logic at Kwame Nkrumah University
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Proposition Logic
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SYMBOLIC LOGIC
TUTORIAL
Logic and the English Language
•
Connectives
-
words such as
and, or, if, then
•
Exclusive or
-
one or the other of the given
events can happen, but not both.
•
Inclusive or
-
one or the other or both of the
given events can happen.
Statements and Logical Connectives
•
Statement
- A sentence that can be judged either true or
false.
–
Labeling a statement true or false is called
assigning a truth
value
.
•
Simple Statements -
A sentence that conveys only one idea.
•
Compound Statements -
Sentences that combine two or
more ideas and can be assigned a truth value.
Statements
•
In this section we will study symbolic logic
which was developed in the late 17
th
century.
•
All logical reasoning is based on statements.
•
A
statement
is a sentence that is either true or
false.
Which of the following are
statements?
•
The 2004 Summer Olympic Games were in Athens,
Greece.(Statement - true)
•
Seinfeld
was the best TV comedy of all time. (Not a
Statement – opinion)
•
Did you watch
The Godfather?
(Not a statement – a
question)
•
The Philadelphia Eagles won Super XV. (statement –
false)
•
I am telling a lie. (not a statement – paradox)
Statements
•
Traditionally, symbolic logic uses lower case
letters to denote statements. Usually the
letters
p, q, r, s, t.
•
Statements get labels.
–
p
: It is raining.
Compound Statements
•
A
compound statement
is a statement that
contains one or more simpler statements.
•
Compound statements can be formed by
–
inserting the word
NOT,
–
joining two or more statements with connective
words such as
AND, OR, IF…THEN, ONLY IF, IF AND
ONLY IF.
Logical Connectives
•
The relations between elements that every deductive
argument must employ
•
Helps us focus on internal structure of propositions
and arguments
–
We can
translate
arguments from sentences and
propositions into symbolic logic form
•
“Simple statement”: does not contain any other
statement as a component
–
“Charlie is neat”
•
“Compound statement”: does contain another
statement as a component
–
“Charlie is neat and Charlie is sweet”
Examples
•
Steve did
not
do his homework.
–
This is formed from the simpler statement, Steve did his
homework.
•
Mr. D wrote the MAT114 notes
and
listened to a Pink
Floyd CD.
–
This statement is formed from the simper statements:
Mr. D wrote the MAT114 notes.
Mr. D listened to a Pink Floyd CD.
•
Compound statements are known as negations,
conjunctions, disjunctions, conditionals or
combinations of each.
Alternative Sentential Logic Symbols
•
Negation: -, ¬
•
Conjunction:
∧
,&
•
Disjunction:
∨
(almost always used as in this text)
•
Conditional: →
•
Biconditional: ↔
Compound Statements
•
Statements consisting of two or more simple
statements are called
compound statements.
•
The connectives often used to join two simple
statements are
and, or, if…then…,
and
if and
only if.
Not Statements
The symbol used in logic to show the negation
of a statement is ~. It is read “not”.
NEGATION ~
p
•
The
negation
of a statement is the denial of
that statement. The symbolic representation is
a tilde ~.
•
Negation of a simple statement is formed by
inserting
not
.
•
Example: The senator is a Republican.
The negation is: The senator is not a
Republican.
Quantifiers
•
Negation of a statement -
the opposite
meaning of a statement.
–
The negation of a false statement is always a true
statement.
–
The negation of a true statement is always false.
•
Quantifiers -
words such as
all, none, no,
some, etc…
Example: Write Negations
Write the negation of the statement.
Some candy bars contain nuts.
•
Since some means “at least one” this
statement is true. The negation is “No candy
bars contain nuts,” which is a false
statement.
Example: Write Negations continued
Write the negation of the statement.
All tables are oval.
•
This is a false statement since some tables
are round, rectangular, or other shapes. The
negation could be “Some tables are not
oval.”
Negation
•
“All of Mr. D’s students are Philadelphia Eagles
fans.”
•
The negation is: “Some of Mr. D’s students are
not Philadelphia Eagles fans.”
•
To negate the first statement, we don’t need
to have all the students to be not Eagles fans,
we just need only one student not to be an
Eagles fan. Hence the usage of some.
Negation
•
“No students are math majors.”
•
To deny this statement, we need at least one
instance in which a student does major in
math.
•
“Some students are math majors.”
Negation
•
To summarize negation:
•
All
p
are
q
is negated by Some
p
are not q
•
No
p
are
q
is negated by Some
p
are
q
And Statements
•
is the symbol for a conjunction and is read
“and.”
•
The other words that may be used to express
a conjunction are:
but
,
however
, and
nevertheless
.
Example: Write a Conjunction
Write the conjunction in symbolic form.
The dog is gray, but the dog is not old.
Solution:
Let
p
and
q
represent
the simple statements.
p
: The dog is gray.
q
: The dog is old.
In symbol form, the compound statement is
CONJUNCTION
p ^ q
•
A
conjunction
is a compound statement that consists
of 2 or more statements connected by the word
and.
•
And
is represented by the symbol ^.
•
p ^ q
represents “
p
and
q
”.
•
Example:
p:
Jerry Seinfeld is a comedian.
q
: Jerry Seinfeld is a millionaire.
Express the following in symbolic form:
i. Jerry Seinfeld is a comedian and he is a millionaire.
ii. Jerry Seinfeld is a comedian and he is not a millionaire.
Conjunction
•
Using the symbolic representations
p:
The lyrics are controversial.
q:
The performance is banned.
Express the following in symbolic form:
a. “The lyrics are controversial and the performance is banned.”
b. “The lyrics are not controversial and the performance is not
banned.”
Answers:
a. p ^ q
b. ~p ^ ~q
Conjunction
•
Conjunction of two statements: “…and…”
–
Each statement is called a conjunct
•
“Charlie is neat” (conjunct 1)
•
“Charlie is sweet” (conjunct 2)
•
The symbol for conjunction is a dot •
–
(Can also be “&”)
–
p
• q
•
P and q (2 conjuncts)
Truth Values
•
Truth value
: every statement is either T or F;
the truth value of a true statement is
true;
the
truth value of a false statement is
false
Truth Values of Conjunction
•
Truth value of conjunction of 2 statements is
determined entirely by the truth values of its
two conjuncts
–
A conjunction statement is
truth-functional
compound statement
–
Therefore our symbol
“•” (or “&”) is a truth-
functional connective
Truth Table of
Conjunction •
Given any two statements, p and q
Given any two statements, p and q
A conjunction is true if and only if both conjuncts are true
A conjunction is true if and only if both conjuncts are true
Abbreviation of Statements
•
“Charlie’s neat and Charlie’s sweet.”
–
N
• S
–
Dictionary
: N=“Charlie’s neat” S=“Charlie’s sweet”
•
Can choose any letter to symbolize each conjunct, but it is best to
choose one relating to the content of that conjunct to make your
job easier
•
“Byron was a great poet and a great adventurer.”
–
P
• A
•
“Lewis was a famous explorer and Clark was a
famous explorer.”
–
L
• C
•
“Jones entered the country at New York and went
straight to Chicago.”
–
“and” here does not signify a conjunction
–
Can’t say “Jones went straight to Chicago and entered the
country at New York.”
–
Therefore cannot use the
• here
•
Some other words that can signify conjunction:
–
But
–
Yet
–
Also
–
Still
–
However
–
Moreover
–
Nevertheless
–
(comma)
–
(semicolon)
Or Statements:
•
The disjunction is symbolized by and read
“or.”
Example:
Write the statement in symbolic form.
Carl will not go to the movies or Carl with not
go to the baseball game.
Solution:
If-Then Statements
•
The
conditional
is symbolized by
and is read “if-then.”
–
The
antecedent
is the part of the statement that
comes before the arrow.
–
The
consequent
is the part that follows the arrow.
Example: Write a Conditional
Statement
Let
p
: Nathan goes to the park.
q
: Nathan will swing.
Write the following statements symbolically.
•
If Nathan goes to the park, then he will swing.
•
If Nathan does not go to the park, then he will not swing.
Solutions:
•
a)
b)
DISJUNCTION
p
v
q
•
When you connect statements with the word
or
you
form a
disjunction
.
•
Or
is represented by the symbol
v
.
•
p v q
is read as “
p
or
q
”.
•
Using the
p
and
q
from the last slide, write out in
words
p v q
, and
p v ~q
.
•
p v q
is “the lyrics are controversial or the
performance is banned.”
•
p v ~q
is “the lyrics are controversial or the
performance is not banned.”
Disjunction
•
Disjunction of two statements: “…or…”
•
Symbol is “ v ” (wedge) (i.e. A v B = A or B)
–
Weak (inclusive) sense: can be either case, and possibly
both
•
Ex. “Salad or dessert” (well, you
can
have both)
•
We will treat all disjunctions in this sense (unless a problem
explicitly says otherwise)
–
Strong (exclusive) sense: one and only one
•
Ex. “A or B” (you can have A
or
B,
at least one
but not both
)
–
The two component statements so combined are called
“disjuncts”
Disjunction Truth Table
A (weak) disjunction is false only in the case that both its disjuncts are false
A (weak) disjunction is false only in the case that both its disjuncts are false
Disjunction
•
Translate
:
“You will do poorly on the exam unless you
study.”
–
P=“You will do poorly on the exam.”
–
S=“You study.”
•
P v S
•
“Unless” = v
If-Then Statements
•
The
conditional
is symbolized by
and is read “if-then.”
–
The
antecedent
is the part of the statement that
comes before the arrow.
–
The
consequent
is the part that follows the arrow.
Example: Write a Conditional
Statement
Let
p
: Nathan goes to the park.
q
: Nathan will swing.
Write the following statements symbolically.
•
If Nathan goes to the park, then he will swing.
•
If Nathan does not go to the park, then he will not swing.
Solutions:
•
a)
b)
CONDITIONAL
p
q
•
A
conditional
is of the form “if
p
then
q
”. This is also
known as an implication.
p
is the hypothesis (or
premise), and
q
is the. conclusion.
•
The representation of “if
p
then
q
” is
p
q.
•
Again use the
p
and
q
from the previous 2 slides.
•
“If the lyrics are not controversial, the performance is
not banned.”
•
~p
~q
Sec. 1.2 #28
•
Using the following symbolic representations
•
p:
I am innocent.
•
q:
I have an alibi.
•
express the following in words.
•
A. p ^ q
•
Answer:
“I am innocent and I have an alibi.”
•
B. p
q
•
Answer:
“If I am innocent, then I have an alibi.”
•
C. ~q
~p
•
Answer:
“If I do not have an alibi, then I am not innocent.”
•
D.
q v ~p
•
Answer: “
I have an alibi or I am not innocent.”
Sec. 1.2 #30
•
Using the symbolic representations
•
p: I am innocent.
•
q: I have an alibi.
•
r: I go to jail.
•
Express the following in words.
•
A. (p v q)
~r
•
B. (p ^ ~q)
r
•
C. (~p ^ q) v r
•
D. (p ^ r)
~q
Sec. 1.2 #30
A.
If I am innocent or have an alibi, then I do
not go to jail.
B.
If I am innocent and do not have an alibi,
then I go to jail.
C.
I am not innocent and I have an alibi or I go
to jail.
D.
If I am innocent and go to jail, then I do not
have an alibi.
Sec. 1.2 #23
•
Translate the sentence to symbolic form.
•
If you drink and drive, you are fined or you go
to jail.
•
p: You drink.
•
q: You drive.
•
r: You are fined.
•
s: You are jailed.
•
Answer: (p ^ q)
(r
v
s).
Sec. 1.2 #14
•
Translate into symbolic form.
•
“No whole number is greater than 3 and less
than 4.”
•
p: A whole number.
•
q: A number greater than 3.
•
r: A number less than 4.
•
Answer: ~
p
(q ^ r)
Biconditionals
•
Two sentences are
materially equivalent
when they
have the same truth-value.
•
The symbol “≡” is called the
tribar
and stands for
material equivalence.
•
Compound sentences formed by the tribar are called
material equivalences
, or
biconditionals
.
“Only If” and “Unless”
•
“Only if” sentences indicate necessary conditions,
but not sufficient conditions.
•
“You will pass the class only if you pay attention” can
be symbolized as C
⊃
A.
•
A simple way of symbolizing “unless” sentences is as
‘or” sentences.
If and Only If Statements
•
The
biconditional
is symbolized by and is
read “if and only if.”
•
If and only if is sometimes abbreviated as “iff.”
Example: Write a Statement Using the Biconditional
Let
p
: The dryer is running.
q
: There are clothes in the dryer.
Write the following symbolic statements in words.
a) b)
Solutions:
•
The clothes are in the dryer if and only if the dryer is
running.
•
It is false that the dryer is running if and only if the clothes
are not in the dryer.
Material Biconditionals
•
Two sentences are
materially equivalent
when they
have the same truth-value.
•
The symbol “≡” is called the
tribar
and stands for
material equivalence.
•
Compound sentences formed by the tribar are called
material equivalences
, or
biconditionals
.
“Only If” and “Unless”
•
“Only if” sentences indicate necessary conditions,
but not sufficient conditions.
•
“You will pass the class only if you pay attention” can
be symbolized as C
⊃
A.
•
A simple way of symbolizing “unless” sentences is as
‘or” sentences.
If and Only If Statements
•
The
biconditional
is symbolized by and is
read “if and only if.”
•
If and only if is sometimes abbreviated as “iff.”
Example: Write a Statement Using the Biconditional
Let
p
: The dryer is running.
q
: There are clothes in the dryer.
Write the following symbolic statements in words.
a) b)
Solutions:
•
The clothes are in the dryer if and only if the dryer is
running.
•
It is false that the dryer is running if and only if the clothes
are not in the dryer.
Punctuation
•
As in mathematics, it is important to correctly
punctuate logical parts of an argument
–
Ex. (2x3)+6 = 12 whereas 2x(3+6)= 18
–
Ex. p
• q v r (this is ambiguous)
•
To avoid ambiguity and make meaning clear
•
Make sure to order sets of parentheses when
necessary:
–
Example: { A •
[(B v C)
• (C v D)
] }
• ~E
•
{ [ ( ) ] }Punctuation
•
“Either Fillmore or Harding was the greatest
American president.”
–
F v H
•
To say “Neither Fillmore nor Harding was the
greatest American president.” (the negation of
the first statement)
–
~(F v H) OR (~F)
• (~H)
Punctuation
•
“Jamal and Derek will both not be elected.”
–
~J
• ~D
•
In any formula the negation symbol will be understood
to apply to the smallest statement that the punctuation
permits
•
i.e. above is NOT taken to mean “~[J • (~D)]”
•
“Jamal and Derek both will not be elected.”
–
~(J •D)
Example
•
Rome is the capital of Italy or Rome is the capital of
Spain.
–
I=“Rome is the capital of Italy”
–
S=“Rome is the capital of Spain”
–
I v S
–
Now that we have the logical formula, we can use the
truth tables to figure out the truth value of this statement
•
When doing truth values, do the innermost
conjunctions/disjunctions/negations first, working your way
outwards
Dive into the world of symbolic logic and compound statements with a focus on Propositional Logic at Kwame Nkrumah University in Ghana. Explore the concepts of connectives, simple and compound statements, truth values, and more. Enhance your logical reasoning skills through a tutorial on symbolic logic development.
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Presentation Transcript
Kwame Nkrumah University of Science & Technology, Kumasi, Ghana Title:Proposition Logic Name: Enya Ameza-Xemalordzo Department: Marketing & Corporate Strategy Faculty & College: Business School
SYMBOLIC LOGIC TUTORIAL
Connectives - words such as and, or, if, then Exclusive or - one or the other of the given events can happen, but not both. Inclusive or - one or the other or both of the given events can happen.
Statement - A sentence that can be judged either true or false. Labeling a statement true or false is called assigning a truth value. Simple Statements - A sentence that conveys only one idea. Compound Statements - Sentences that combine two or more ideas and can be assigned a truth value.
Statements In this section we will study symbolic logic which was developed in the late 17th century. All logical reasoning is based on statements. A statement is a sentence that is either true or false.
Which of the following are statements? The 2004 Summer Olympic Games were in Athens, Greece.(Statement - true) Seinfeld was the best TV comedy of all time. (Not a Statement opinion) Did you watch The Godfather? (Not a statement a question) The Philadelphia Eagles won Super XV. (statement false) I am telling a lie. (not a statement paradox)
Statements Traditionally, symbolic logic uses lower case letters to denote statements. Usually the letters p, q, r, s, t. Statements get labels. p: It is raining.
Compound Statements A compound statement is a statement that contains one or more simpler statements. Compound statements can be formed by inserting the word NOT, joining two or more statements with connective words such as AND, OR, IF THEN, ONLY IF, IF AND ONLY IF.
The relations between elements that every deductive argument must employ Helps us focus on internal structure of propositions and arguments We can translate arguments from sentences and propositions into symbolic logic form Simple statement : does not contain any other statement as a component Charlie is neat Compound statement : does contain another statement as a component Charlie is neat and Charlie is sweet
Examples Steve did not do his homework. This is formed from the simpler statement, Steve did his homework. Mr. D wrote the MAT114 notes and listened to a Pink Floyd CD. This statement is formed from the simper statements: Mr. D wrote the MAT114 notes. Mr. D listened to a Pink Floyd CD. Compound statements are known as negations, conjunctions, disjunctions, conditionals or combinations of each.
Negation: -, Conjunction: ,& Disjunction: (almost always used as in this text) Conditional: Biconditional:
Statements consisting of two or more simple statements are called compound statements. The connectives often used to join two simple statements are and, or, if then , and if and only if.
The symbol used in logic to show the negation of a statement is ~. It is read not .
NEGATION ~p The negation of a statement is the denial of that statement. The symbolic representation is a tilde ~. Negation of a simple statement is formed by inserting not. Example: The senator is a Republican. The negation is: The senator is not a Republican.
Negation of a statement - the opposite meaning of a statement. The negation of a false statement is always a true statement. The negation of a true statement is always false. Quantifiers - words such as all, none, no, some, etc
Write the negation of the statement. Some candy bars contain nuts. Since some means at least one this statement is true. The negation is No candy bars contain nuts, which is a false statement.
Write the negation of the statement. All tables are oval. This is a false statement since some tables are round, rectangular, or other shapes. The negation could be Some tables are not oval.
Negation All of Mr. D s students are Philadelphia Eagles fans. The negation is: Some of Mr. D s students are not Philadelphia Eagles fans. To negate the first statement, we don t need to have all the students to be not Eagles fans, we just need only one student not to be an Eagles fan. Hence the usage of some.
Negation No students are math majors. To deny this statement, we need at least one instance in which a student does major in math. Some students are math majors.
Negation To summarize negation: All p are q is negated by Some p are not q No p are q is negated by Some p are q
is the symbol for a conjunction and is read and. The other words that may be used to express a conjunction are: but, however, and nevertheless.
Write the conjunction in symbolic form. The dog is gray, but the dog is not old. Solution: Let p and q represent the simple statements. p: The dog is gray. q: The dog is old. In symbol form, the compound statement is p q .
CONJUNCTION p ^ q A conjunction is a compound statement that consists of 2 or more statements connected by the word and. And is represented by the symbol ^. p ^ qrepresents p and q . Example: p: Jerry Seinfeld is a comedian. q: Jerry Seinfeld is a millionaire. Express the following in symbolic form: i. Jerry Seinfeld is a comedian and he is a millionaire. ii. Jerry Seinfeld is a comedian and he is not a millionaire.
Conjunction Express the following in symbolic form: a. The lyrics are controversial and the performance is banned. b. The lyrics are not controversial and the performance is not banned. Answers: a. p ^ q b. ~p ^ ~q Using the symbolic representations p: The lyrics are controversial. q: The performance is banned.
Conjunction of two statements: and Each statement is called a conjunct Charlie is neat (conjunct 1) Charlie is sweet (conjunct 2) The symbol for conjunction is a dot (Can also be & ) p q P and q (2 conjuncts)
Truth value: every statement is either T or F; the truth value of a true statement is true; the truth value of a false statement is false
Truth value of conjunction of 2 statements is determined entirely by the truth values of its two conjuncts A conjunction statement is truth-functional compound statement Therefore our symbol (or & ) is a truth- functional connective
Truth Table of Conjunction Given any two statements, p and q p T q T p q T T F F F T F F F F A conjunction is true if and only if both conjuncts are true
Charlies neat and Charlies sweet. N S Dictionary: N= Charlie s neat S= Charlie s sweet Can choose any letter to symbolize each conjunct, but it is best to choose one relating to the content of that conjunct to make your job easier Byron was a great poet and a great adventurer. P A Lewis was a famous explorer and Clark was a famous explorer. L C
Jones entered the country at New York and went straight to Chicago. and here does not signify a conjunction Can t say Jones went straight to Chicago and entered the country at New York. Therefore cannot use the here Some other words that can signify conjunction: But Yet Also Still However Moreover Nevertheless (comma) (semicolon)
The disjunction is symbolized by and read or. Example: Write the statement in symbolic form. Carl will not go to the movies or Carl with not go to the baseball game. Solution: p q
The conditional is symbolized by and is read if-then. The antecedent is the part of the statement that comes before the arrow. The consequent is the part that follows the arrow.
Example: Write a Conditional Statement Let p: Nathan goes to the park. q: Nathan will swing. Write the following statements symbolically. If Nathan goes to the park, then he will swing. If Nathan does not go to the park, then he will not swing. Solutions: a)b) p q p q
DISJUNCTION p v q When you connect statements with the word or you form a disjunction. Or is represented by the symbol v. p v q is read as p or q . Using the p and q from the last slide, write out in words p v q, and p v ~q. p v q is the lyrics are controversial or the performance is banned. p v ~q is the lyrics are controversial or the performance is not banned.
Disjunction of two statements: or Symbol is v (wedge) (i.e. A v B = A or B) Weak (inclusive) sense: can be either case, and possibly both Ex. Salad or dessert (well, you can have both) We will treat all disjunctions in this sense (unless a problem explicitly says otherwise) Strong (exclusive) sense: one and only one Ex. A or B (you can have A or B, at least onebut not both) The two component statements so combined are called disjuncts
Disjunction Truth Table p T q T p v q T T F T F T T F F F A (weak) disjunction is false only in the case that both its disjuncts are false
Translate: You will do poorly on the exam unless you study. P= You will do poorly on the exam. S= You study. P v S Unless = v
The conditional is symbolized by and is read if-then. The antecedent is the part of the statement that comes before the arrow. The consequent is the part that follows the arrow.
Example: Write a Conditional Statement Let p: Nathan goes to the park. q: Nathan will swing. Write the following statements symbolically. If Nathan goes to the park, then he will swing. If Nathan does not go to the park, then he will not swing. Solutions: a)b) p q p q
CONDITIONAL p q A conditionalis of the form if p then q . This is also known as an implication. p is the hypothesis (or premise), and q is the. conclusion. The representation of if p then q is p q. Again use the p and q from the previous 2 slides. If the lyrics are not controversial, the performance is not banned. ~p ~q
Sec. 1.2 #28 Using the following symbolic representations p: I am innocent. q: I have an alibi. express the following in words. A. p ^ q Answer: I am innocent and I have an alibi. B. p q Answer: If I am innocent, then I have an alibi. C. ~q ~p Answer: If I do not have an alibi, then I am not innocent. D. q v ~p Answer: I have an alibi or I am not innocent.
Sec. 1.2 #30 Using the symbolic representations p: I am innocent. q: I have an alibi. r: I go to jail. Express the following in words. A. (p v q) ~r B. (p ^ ~q) r C. (~p ^ q) v r D. (p ^ r) ~q
Sec. 1.2 #30 A. If I am innocent or have an alibi, then I do not go to jail. B. If I am innocent and do not have an alibi, then I go to jail. C. I am not innocent and I have an alibi or I go to jail. D. If I am innocent and go to jail, then I do not have an alibi.
Sec. 1.2 #23 Translate the sentence to symbolic form. If you drink and drive, you are fined or you go to jail. p: You drink. q: You drive. r: You are fined. s: You are jailed. Answer: (p ^ q) (r v s).
Sec. 1.2 #14 Translate into symbolic form. No whole number is greater than 3 and less than 4. p: A whole number. q: A number greater than 3. r: A number less than 4. Answer: ~p (q ^ r)
Two sentences are materially equivalent when they have the same truth-value. The symbol is called the tribar and stands for material equivalence. Compound sentences formed by the tribar are called material equivalences, or biconditionals.
Only if sentences indicate necessary conditions, but not sufficient conditions. You will pass the class only if you pay attention can be symbolized as C A. A simple way of symbolizing unless sentences is as or sentences.
The biconditional is symbolized by and is read if and only if. If and only if is sometimes abbreviated as iff.
Let p: The dryer is running. q: There are clothes in the dryer. Write the following symbolic statements in words. ( ) a) b) q p p q Solutions: The clothes are in the dryer if and only if the dryer is running. It is false that the dryer is running if and only if the clothes are not in the dryer.
Two sentences are materially equivalent when they have the same truth-value. The symbol is called the tribar and stands for material equivalence. Compound sentences formed by the tribar are called material equivalences, or biconditionals.
