Unit 2: The Structure of
Sentential Logic
Friday, September 9, 2011
TA: Karin Howe
Last week…
• Unit 1
– Propositions
• These are also known as "declarative sentences" (Klenk) or
statements (all of these terms can be used more or less
interchangeably in the context of this course)
– Argument (and Statement) Forms, and Argument
Instances
– Inductive vs. Deductive Reasoning
– Validity, Invalidity and Soundness
– Counter-examples (for proving invalidity)
This week …
• Unit 2
– More on declarative sentences
(statements/propositions)
• Simple vs. compound sentences
– Sentential operators
– Finding the major operator (and why we care)
Sentential Logic
•
•
•
This is also known as "propositional
logic."
In sentential logic, we will learn how to
analyze arguments into simple
sentences/propositions and operators.
The two basic units:
1) The simple sentence
2) The sentential operator
Simple and Compound Sentences
• A compound sentence is a (declarative) sentence that
contains another complete declarative sentence as a
component. (Klenk, pp. 23, 30)
• When is one sentence a component of another?
– A sentence is a component of another if, whenever
the first sentence is replaced by a third declarative
sentence, the resulting whole is still a grammatical
sentence. (Klenk, pp. 23, 31)
• A simple sentence is a sentence that is not compound: it
does not contain another complete declarative sentence
as a component. (Klenk, pp. 23, 30)
Examples: Simple Sentences
1.
2.
3.
4.
5.
6.
The kitten frolicked in the teapot.
The kitten knocked over the teapot.
I bought a new teapot yesterday.
Death likes cats.
Cats are nice.
Kangaroos can fly.
Examples: Compound Sentences
• Example: Either the kitten knocked over the teapot or it is
raining tea in here.
• Recall that: A sentence is a component of another if,
whenever the first sentence is replaced by a third declarative
sentence, the resulting whole is still a grammatical sentence.
(Klenk, pp. 23, 31)
• Try it out!
– Either ____________ or it is raining tea in here.
– Either the kitten knocked over the teapot or _________.
• Note that it is not necessary for the new sentence to make
sense. 
– Either the kitten knocked over the teapot or I like bananas.
How many components do
compound sentences have?
• It varies -- some have two components,
some have more than two, and some have
only one component!
• Examples of compound sentences with only
one component:
– I don't like bananas.
– I believe that kangaroos can fly.
– I am 86.0989% sure that kangaroos can fly.
Compound or simple?
1. Jacques went to the store, and the store sells
baguettes.
2. Libya is experiencing political instability.
3. Libya and Egypt are in Africa.
4. Obama loves peanut brittle and roasted puppy.
5. Rosa and Steve are hooking up.
6. Carlos and John are gay and married (to one
another).
7. Canada is not defenseless.
8. Dinosaurs go roar.
Sentential Operators
• Recall that 'Obama loves peanut brittle and roasted puppy'
has the structure: ___________ and ____________.
• The 'and' here is playing the role of an operator. It operates
on the simple sentences: here, it brings two of them
together into a conjunction. (It conjoins them.)
– To put it roughly, it asserts that both are the case.
• Klenk defines an sentential operator as: "an expression
containing blanks such that when the blanks are filled with
complete sentences, the result is a sentence." (Klenk, pp.
26, 31)
• Recall some earlier examples:
– I don't like bananas.
• It is false that I like bananas.
– I believe that kangaroos can fly.
• I believe that kangaroos can fly.
– I am 86.0989% sure that kangaroos can fly.
• I am 86.0989% sure that kangaroos can fly.
• Claim: There are an infinite number of
operators.
• However, we will only be concerned with
five of them in sentential logic. (phew!)
Your Five New Best Friends
• I like to eat apples and bananas.
______ and _______ (conjunction)
• I like to eat apples or bananas.
_________ or ________ (disjunction)
• If I like to eat apples, then I like to eat bananas.
If ________ then ________ (conditional)
• I like to eat apples if and only if I like to eat bananas.
________ if and only if _________ (biconditional)
• I don't like to eat bananas.
It is false that _______ (negation)
Symbolizing Statements
• We will symbolize simple statements using capital letters
(A, B, C, etc.)
 Example:
A = I like aardvarks
• We symbolize the sentential operators as follows:
 "____ and _____" using the dot (•): ____ • ____
 "____ or ____" using the wedge (): ____  ____
 "if ____ the ____" using the horseshoe ():
____  ____
 "____ if and only if ____" using the triple bar ():
____  ____
 "It is false that ____" using the tilde (~): ~ ____
Symbolizing Statements: Examples
• Dictionary:
– A = I like to eat apples
– B = I like to eat bananas
•
•
•
•
I like to eat apples and I like to eat bananas.
A•B
I like to eat apples or I like to eat bananas.
AB
If I like to eat apples, then I like to eat bananas. A  B
I like to eat apples if and only if I like to eat bananas.
A
B
• It is false that I like to eat bananas.
~B
A word about negations
• The negation symbol (~) does not join two components, like our
other operators do.
• The negation symbol operates on only one component. (Recall
the form of our negation operator: It is false that _____)
• Thus, when we symbolize negations, we do not need extra
punctuation: "It is false that p" will be symbolized as ~p.
• Note: the component that is negated can be a compound
component!
– Example: It is false that I like apples and bananas.
• Thus, the following two statements are different statements:
~(A • B)
I don't like apples and bananas.
(~A • B)
I don't like apples but I do like bananas.
Practice Makes Perfect: Simple or
1.
Compound?
Cats enjoy getting into teapots if and only if they have warm
tea in them.
2. The gerbil likes to get into the teapot whether or not it has
warm tea in it.
3. Kangaroos do not like to get into teapots.
4. The kitten's mother tries to keep the kitten out of the teapot.
5. Neither the cat nor the gerbil are fond of baths.
6. The cat and the gerbil will need a bath if they keep playing in
teapots.
7. Life on Earth is doomed if gerbils do not stay out of our
teapots.
8. The gerbil lay down in the teapot and took a nap.
9. The cat likes to play in the teapot with the gerbil.
10. The gerbil will have to either sink or swim if there is a lot of
tea in the teapot.
Finding the Major Operator
• The major operator of a formula is the one
that determines the overall form of the
sentence.
• Find the major operator in the following
formulas:
– (A  B)
– [(A • B)  (C • D)]
– {[A  (B • C)]  [A  (C • D)]}
The Two Chunk Rule
• The Two Chunk Rule says: "Once more than one
logical connective symbol is necessary to translate a
statement, there must be punctuation that identifies the
main connective of a symbolic statement. In addition,
there cannot be any part of a statement in symbols that
contains more than two statements, or chunks of
statements, without punctuation."
• Examples:
– (A  B  C)
– [(A  B)  C]
– [A  (B  C)]
(wrong!)
(correct)
(correct)
WFFs (Well-Formed Formulas)
•
•
WFF (pronounced "woof") stands for wellformed formulas
A well-formed formula in sentential logic is
defined recursively as follows:
1) (Base clause) Any statement constant is a WFF
2) (Recursion clause) If  and  are any WFFs, then all
the following are also WFFs: (a) (  ), (b) (  ),
(c) (  ), (d) (  ), and (e) ~.
3) (Closure clause) Nothing will count as a WFF unless
it can be constructed according to clauses (1) and (2).
Why do we care about the major operator??
•
•
We care about the main connective (major operator)
because it tells us what kind of logical statement we are
looking at. (e.g., if we see that the main connective is a
"," then we know that the statement is a disjunction)
This is important in two different contexts (preview of
coming attractions):
1. Working With Truth Tables
o
It lets us know what the truth values of the statement are at
each line of the truth table. We look at the main connective
to tell us the truth value of that statement as a whole (on each
line of the truth table).
2. Working With Derivation Rules
o
It tells us what derivation rules we can use on that statement
- either to take it apart or to build it up (or to convert it into
another logically equivalent statement).
Practice Makes Perfect: Finding the major operator
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
A
(B  C)
(A  ~C)
(~B  A)
~(B  A)
~(~B  A)
[A  (B  C)]
[(A  B)  ~C]
~(~A  ~C)
(~~A  ~C)
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
~~(A  ~C)
[B  ~(A  C)]
[A  ~(C  ~D)]
[~(B  C)  ~D]
[B  ~(~B  A)]
~[A  ~(B  ~D)]
[(~C  B)  (A  ~C)]
[~(C  B)  (A  ~C)]
~[(C  B)  (A  ~C)]
[~(C  B)  ~(A  ~C)]