Counterexample Arguments
All valid arguments are deductive
Some valid arguments are not sound
Some deductive arguments are not sound
All dogs are reptiles.
Some dogs are not animals.
Some reptiles are not animals
All dogs are animals
Some animals are not mammals
Some mammals are not dogs
All mammals are animals.
Some animals are not cats.
Some cats are not mammals
Going Formal
Meet the Connectives
The Language of Propositional Logic
• Syntax (grammar, internal structure of the
language)
– Vocabulary: grammatical categories
– Identifying Well-Formed Formulae (“WFFs”)
• Semantics (pertaining to meaning and truth value)
– Translation
– Truth functions
– Truth tables for the connectives
The Vocabulary of Propositional Logic
• Sentence Letters: A, B, … Z
• Connectives (“Sentence-Forming Operators”)
~
negation
“not,” “it is not the case that”
⋅
conjunction
“and”
∨
disjunction
“or” (inclusive)
⊃
conditional
“if – then,” “implies”
≣
biconditional
“if and only if,” “iff”
• “Parentheses”: (, ), [, ], {, and }
Sentence Letters
• Translate “atomic” sentences
• Atomic sentences have no proper parts that
are themselves sentences
• Examples:
– It is raining
R
– It is cold
C
Sentential Connectives
• Connect to sentences to make new sentences
• Negation attaches to one sentence
– It is not raining
∼R
• Conjunction, disjunction, conditional and
biconditional attach two sentences together
– It is raining and it is cold
– If it rains then it pours
R∙C
R⊃P
Parentheses, brackets & braces
• I’ll go to Amsterdam and Brussels or Calais
• This is ambiguous and we can’t tolerate ambiguity!
Brussels
Amsterdam
AND
OR
Calais
Amsterdam
OR
Calais
AND
Brussels
Parentheses, brackets & braces
•
Grouping devices avoid ambiguity (for “unique readability”):
– I’ll go to Amsterdam, and then to either Brussels or Calais
A ∙ (B ∨ C)
Amsterdam
Brussels
AND
OR
Calais
– I’ll either go to Amsterdam and Brussels, or else to Calais
(A ∙ B) ∨ C
Amsterdam
OR
Calais
AND
Brussels
Variables: p, q, …
• Sometimes we want to talk about all sentences
of a given form, e.g.
A  (B  C)
F  (M  X)
(K  M)  [(N   O)  P]
• So we use variables as place-holders
• Each of the above sentences is of the form:
p  (q  r)
Plugging into variables
Modus
Ponens
pq
p
q
Substitution Instance of Modus Ponens
((A  B)  C)

(D  (E   F))
((A  B)  C)
(D  (E   F))
• Variables are like expandable boxes
• To do proofs in logic you have to see how sentences
plug into those boxes.
Plugging into variables
Modus
Ponens
pq
p
q
Substitution Instance of Modus Ponens
((A  B)  C)

(D  (E   F))
((A  B)  C)
(D  (E   F))
• Variables are like expandable boxes
• To do proofs in logic you have to see how sentences
plug into those boxes.
The Grammar of Propositional Logic
• Constructing WFFs (Well-Formed Formulae)
• Identifying WFFs
• Identifying main connectives
Rules for WFFs
1.
A sentence letter by itself is a WFF
A
2.
Z
The result of putting  immediately in front of a WFF is a WFF
A
3.
B
B
B
 (A  B)
 ( C  D)
The result of putting  ,  ,  , or  between two WFFs and
surrounding the whole thing with parentheses is a WFF
(A  B)
4.
(  C  D)
((  C  D)  (E  (F   G)))
Outside parentheses may be dropped
AB
CD
(  C  D)  (E  (F   G))
WFFs
• A sentence that can be constructed by
applying the rules for constructing
WFFs one at a time is a WFF
• A sentence which can't be so
constructed is not a WFF
• No exceptions!!!
woof
Main Connective
• In constructing a WFF, the connective that goes
in last, which has the whole rest of the sentence
in its scope, is the main connective.
• This is the connective which is the “furthest
out.”
• Examples
(  C  D)  (E  (F   G))
 ( C  D)
Hints: When it’s not a WFF
• You can't have two WFFs next to one another without a twosided connective between them.
BAD!
AB
CD
(E  F)G
• Two-sided connectives have to have WFFs attached to both
sides.
BAD!
A
(B  C)  ( D  E)
GH
• You can't have more than one two-sided connective at the same
level
BAD!
ABC
(  C  D)  (E  F   G)
Identifying WFFs & Main Connectives
∨ 1
(S   T)  ( U  W)
X2
 (K  L)  ( G  H)
X 3
(E  F)  (W  X)
≡ 4
(B   T)   ( C  U)
X 5
(F   Q)  (A  E  T)
Identifying WFFs & Main Connectives
1
(S   T)  ( U  W)
X 2
 (K  L)  ( G  H)
X 3
(E  F)  (W  X)
 4
(B   T)   ( C  U)
X 5
(F   Q)  (A  E  T)
Identifying WFFs & Main Connectives
∨
X
X
⊃
X
Identifying WFFs & Main Connectives
 6  D   [ ( P  Q)  (T  R) ]
X 7 [ (D   Q)  (P  E) ]  [A  (  H) ]
X 8 M (N  Q)  ( C  D)
 9 (F   G)  [ (A  E)   H]
X 10 (R  S  T)   ( W   X)
Why should we care about this?
• Because in formal logic we determine whether
arguments are valid or not by reference to their
form.
• And that assumes we can identify the form of
sentences, i.e. that we can identify main
connectives.
• In doing formal derivations in particular, we
have be able to immediately see what the forms
of sentences are in order to formulate
strategies.
Translation
Conditionals & Biconditionals
If P then Q
P, if Q
P only if Q
PQ
QP
PQ
P if and only if Q
PQ
Note: A biconditional is a “conditional going both ways”:
so P  Q is the conjunction of P  Q and Q  P
Conditionals
If P then Q
P, if Q
P only if Q
PQ
QP
PQ
5 If Chanel has a rosewood fragrance then so does Lanvin.
CL
6 Chanel has a rosewood fragrance if Lanvin does.
LC
8 Reece Witherspoon wins best actress only if Martin
Scorsese wins best director.
WS
Biconditionals
P if and only if Q
PQ
7 Maureen Dowd writes incisive editorials if and
only if Paul Krugman does.
DK
A biconditional is a “conditional going both
ways”: so P  Q is the conjunction of P  Q and
Q  P. “Only if” is only half of “if and only if.” Be
careful!
Not both and & neither/nor
Not both P and Q
Neither P nor Q
~ (P  Q)
 (P  Q)
You can’t both have your cake and eat it.
~ (H  E)
She was neither young nor beautiful.
 (Y  B)
Not both and & neither/nor
Not both P and Q
Neither P nor Q
~ (P  Q)
 (P  Q)
15 Not both Jaguar and Porsche make
motorcycles.
~ (J  P)
16 Both Jaguar and Porsche do not make
motorcycles.
J~P
Not both and & neither/nor
Not both P and Q
Neither P nor Q
~ (P  Q)
 (P  Q)
18 Not either Ferrari or Maserati makes economy cars.
19 Neither Ferrari nor Maserati makes economy cars.
 (F  M)
20 Either Ferrari or Maserati does not make
motorcycles.
F~M
DeMorgan’s Laws
~ (P  Q) is equivalent to  P   Q
 (P  Q) is equivalent to  P   Q
“She was neither young nor
beautiful” is equivalent to “She was
old and ugly” - NOT “She was old
or ugly.”
“You can’t both have your cake and
eat it” is equivalent to “You either
don’t have your cake or you don’t
eat your cake” - NOT “You don’t
have your cake and you don’t eat
your cake.”
So, what do I need for the quiz?
• Identifying WFFs and main connectives
• Translation: given an English sentence, which
of the following symbolized sentences is the
correct translation?
The End
WFF