Chapter 2: Syntax and Symbolization
PHIL 121: Methods of Reasoning
March 11, 2013
Instructor:Karin Howe
Binghamton University
Syntax = grammar of a logical statement
• Two different categories of basic expressions:
– Atomic formulae
• Sentences that have no logically relevant internal structure
• Examples:
– The cat is in the teapot.
– The cat is all wet.
– Logical connectives
• Serve to connect formulae (atomic or otherwise) to create more
complex formulas.
• Examples:
– Cats are a lot of trouble but they are also a lot of fun.
– If the cat is in the teapot, then it is both mad and wet.
Types of logical operators
• Conjunction
– "and" (&)
• Disjunction
– "or" (∨)
• Conditionals
– "if … then" (→)
• Negation
– "not" (¬)
• Biconditionals
– "if and only if" (↔)
Conjunctions
• Recall that conjunctions usually involve the
word "and"
• However, conjunctions may also be
expressed using any one of a number of
(logical) synonyms for "and"
– but, however, moreover, although, yet, even
though, …
Symbolizing Conjunctions
•
Example: The cat is WET and MAD.
1. Standardize: The cat is WET and the cat is
MAD.
2. Loglish: W and M
3. Symbolize: (W & M)
•
The two parts of a conjunction are called
the right conjunct and the left conjunct
Disjunctions
• Recall that disjunctions usually involve the
word "or"
• However, disjunctions may also be
expressed using any one of a number of
(logical) synonyms for "or"
– either/or, and/or
Disjunctions: Exclusive vs. Inclusive
• You can have either CHERRIES or PICKLES on
your ice cream.
• Two ways you can interpret this:
– Pick one - cherries or pickles
– You can have both! (yuck!)
• We will take "or" in the inclusive sense (thus
and/or is just shorthand for inclusive "or")
• If need be, we can represent exclusive "or" as
follows:
– You can have either CHERRIES or PICKLES on your
ice cream, but not both.
Symbolizing Disjunctions
• Example: Either the cat is WET or MAD.
1.Standardize: Either the cat is WET or the cat is
MAD.
2.Loglish: W or M
3.Symbolize: (W ∨ M)
• The two parts of a disjunction are called the
right disjunct and the left disjunct
Conditionals
• Recall that conditionals usually involve the phrase
"if … then"
• However, conditionals may also be expressed
using any one of a number of (logical) synonyms
for "if … then"
– provided (that), given (that), should, will result in, only
if, is a necessary condition for, is a sufficient condition
for
Symbolizing Conditionals
• Example: If the cat is WET, then it is MAD.
1. Standardize: If the cat is WET, then the cat is MAD.
2. Loglish: if W then M
3. Symbolize: (W → M)
• The two parts of a conditional are called the
antecedent and the consequent
– antecedent appears before the →
– consequent comes after the →
Tricky Conditionals
•
•
•
"only if"
– P only if Q
– Standardized as: If P, then Q
– Symbolized as: (P → Q)
Necessary Conditions
– P is a necessary condition for Q
– Standardized as: If Q, then P
– Symbolized as: (Q → P)
– Mnemonic: neceSSary conditions come second
Sufficient Conditions
– P is a sufficient condition for Q
– Standardized as: If P, then Q
– Symbolized as: (P → Q)
– Mnemonic: suFFicient conditions come first
Tricky Conditionals, con't
 "unless"
– P unless Q
– Standardized as: If not Q, then P
– Symbolized as: (¬Q → P)
– Note: the text considers the possibility that there could be an
inclusive and an exclusive reading of "unless"
• Consider the following: John will pick up Henry at the airport,
unless Mary does it.
• Inclusive reading: (¬M → J)
• Exclusive reading: ((¬M → J) & (M → ¬J))
• Text leaves this question unresolved -- are we using the
"inclusive" interpretation or the "exclusive" interpretation?
– We will use the inclusive interpretation ("unless" will be
interpreted as a single conditional)
Negations
• Recall that negations usually involve the word
"not"
• However, negations may also be expressed using
any one of a number of (logical) synonyms for
"not"
– It is not true that, it is false that, no, never, isn't (won't,
didn't, etc.), it is not the case that, unless (equivalent to
"if not"), without (equivalent to "but not"), neither/nor
(equivalent to "it is false that either/or")
Symbolizing Negations
• Example: The cat is not WET.
1. Standardize: It is not the case that the cat is WET.
2. Loglish: not W
3. Symbolize: ¬W
Biconditionals
• Recall that biconditionals usually involve
the phrase "if and only if"
• However, biconditionals may also be
expressed using any one of a number of
(logical) synonyms for "if and only if"
– just in case …, then and only then, it is a
necessary and sufficient condition for
– Logicians also sometimes abbreviate "if and
only if" as iff
Symbolizing Biconditionals
• Example: You can go to the MOVIES if and only
you CLEAN up your room.
1. Standardize: You can go to the MOVIES if and only
you CLEAN up your room.
2. Loglish: M iff C
3. Symbolize: (M ↔ C)
WFF = well formed formulae (pronounced "woof")
Recursive definition of WFF in sentential logic:
1. Every atomic formula ϕ is a well-formed formula of
sentential logic.
2. If ϕ is a well-formed formula of sentential logic then
so is ¬ϕ.
3. If ϕ and ψ are well-formed formulae of sentential
logic then so are each of the following:
a) (ϕ & ψ)
b) (ϕ ∨ ψ)
c) (ϕ → ψ)
d) (ϕ ↔ ψ)
4. An expression of sentential logic is a well-formed
formula if and only if it can be formed through one or
more applications of rules 1-3.
Building up WFFs using the Recursive Definition
•
•
We can use the recursive definition to show that
a formula is a WFF by showing how the formula
can be "built up" step-by-step using the
recursive definition
Example: ((P & Q) ∨ ¬R)
1.
2.
3.
4.
5.
6.
R
¬R
P
Q
(P & Q)
((P & Q) ∨ ¬R)
Rule 1
Rule 2, line 1
Rule 1
Rule 1
Rule 3a, lines 3, 4
Rule 3b, lines 5, 2
Building up WFFs using the Recursive Definition
•
•
We can also use the recursive definition to show
that a formula is a non-WFF by showing how
we get "stuck" in trying to show how the
formula can be "built up" step-by-step using the
recursive definition
Example: ((P & Q)¬ & R)
1.
2.
3.
4.
5.
R
P
Q
(P & Q)
Stuck!
Rule 1
Rule 1
Rule 1
Rule 3a, lines 2, 3
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
major operator 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
(A & B)
(A & B → C)
¬A
Some comments about punctuation
Why do we care about punctuation and the Two-Chunk Rule??
– WFFs
– Finding the major operator
•
But why do we care about the major operator?
– Truth tables (and truth trees)
– Derivations
WFFs or Not?
•
Show whether the following formulas are
WFFs or non-WFFs using the recursive
definition:
1.
2.
3.
4.
5.
A
(A ¬C)
(¬B → A)
¬(B → A)
(A & B ∨ C)
Practice With Symbolization
• Possession of a hot plate in the dorms is not illegal. (L =
possession of a hot plate in the dorms is legal)
~ student newspaper
• KISS me, and a handsome PRINCE will appear.
~ Wizard of Id
• Marvin's being BUSTED for "pot" possession is a
sufficient condition for his being DROPPED from the
team.
• Nancy's SCORING above 1,000 on the GRE is a necessary
condition for her ADMISSION to graduate school.
• It is illegal to FEED or HARASS alligators.
~ Everglades sign
1. I will either get a PUPPY or a GUPPY, or
possibly both.
2. You may get either a PUPPY or a GUPPY, but
not both.
3. You may get neither a PUPPY nor a GUPPY.
4. Lina is SAD that she cannot get a guppy.
5. Lina thinks that a BETTA would make a better
pet than a guppy anyway.
6. Neither TOM nor LINA nor KARIN are in the
market for a new cat.
7. Lina wants to get a new HAMSETER if Rowan
won't stay out of the TEAPOT, unless she thinks
that a GERBIL would be better behaved.
WFFs or Not?
•
Show whether the following formulas are
WFFs or non-WFFs using the recursive
definition:
1.
2.
3.
4.
5.
A & (B ∨ ¬C)
X & (Y ↔ ¬Z ¬)
¬P ↔ P
A → (B¬ ↔ C)
A → (B ↔ ¬C)
More Practice With WFFs
1.
2.
3.
4.
5.
6.
7.
¬¬A
(A ∨ B) ∨ ¬C
¬(¬A ∨ ~C)
¬A¬ ∨ ¬C
¬¬(A ∨& ¬C)
B & ¬(A & C)
A & ¬(C ∨ ¬D)
8.
9.
10.
11.
12.
13.
14.
(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)
Practice With Symbolizing Conjunctions
• My husband has many fine QUALITIES, but he
has one serious HANGUP.
~ letter to 'Dear Abby"
• War is CRUEL and you cannot REFINE it.
~ General William Sherman
• I May Be FAT, But You're UGLY–And I Can
DIET!
~ bumper sticker
• Santa Claus is ALIVE and WELL and LIVING in
Argentina.
~ bumper sticker
Practice With Symbolizing Disjunctions
• This woman must be either MAD or DRUNK.
~ Plautus dialogue
• Either that man's a FRAUD or he's your
BROTHER.
~ Plautus
• I can either run the COUNTRY or control
ALICE–not both.
~ Theodore Roosevelt
• They'd better lost the ATTITUDE and listen to
their DAD, or they won't get diddly CRAP.
~ newspaper, lottery winner discussing her grandchildren
Practice With Symbolizing Conditionals
• I am extraordinarily PATIENT provided I get my own
WAY in the end.
~ Margaret Thatcher
• If 14-year-olds had the VOTE, I'd be PRESIDENT.
~ Evil Knievel
• If the Grand Jury calls me BACK I will be glad to
COOPERATE fully if my IMMUNITY is extended.
~ CREEP operative
• If the AXIOMS could be so selected that they were
necessarily true, then, if the DEDUCTIONS were valid,
the truth of the THEOREMS would be guaranteed.
~ logician James Carney
Practice With Symbolizing Negations
• [Read my lips], no new TAXES.
~ presidential candidate George Bush (senior)
• I am not a CROOK.
~ Richard Nixon
• Now we shall have duck EGGS, unless it is a
DRAKE.
~ Hans Christian Anderson
• If God didn't WANT them sheared, he wouldn't
have MADE them sheep.
~ Eli Wallach in "The Magnificent Seven"
Practice With Symbolizing Biconditionals
• Our JUSTICE system works if, and only if, witnesses are
willing to come FORWARD.
~ newspaper column
• If and only if it [a motion on racial research] is
APPROVED by a majority of the AAA membership will it
become an official POSITION of the American
Anthropological Association)
~ bulletin
• The assistantship will be offered to MCGRAW if and only
if he does not get a tuition WAIVER; and it will be offered
to SPIEGELMAN if not offered to McGraw.
~ minutes of meeting