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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.”