Transcript FORMAL LOGIC I: PROPOSITIONAL LOGIC

Propositional Logic
USEM 40a
Spring 2006
James Pustejovsky
Evaluation of Deductive Arguments
• argument A is a deductive argument =df.
A is an argument in which the conclusion is supposed
to follow from the premises with necessity / with
certainty
• deductive argument A is valid =df.
it is not possible for all of A’s premises to be true and
its conclusion false
• deductive argument A is sound =df.
(i) A is valid, and (ii) all of A’s premises are true
(P1) If Grover is dead, then Grover does not
vote.
(P2) Grover is dead.
(C) Therefore, Grover does not vote.
Formal Logic
• with many deductive arguments, validity is a
matter simply of form, of structure
• formal logic studies these cases in which validity
depends solely on form
• not all valid arguments are formally valid:
(P) Grover is a bachelor.
(C) Therefore, Grover does not have a wife.
• argument A is formally valid if, in virtue of A’s
logical form alone, it is impossible for all of A’s
premises to be true and its conclusion false
(P1) All 19th Cent. American presidents are
dead people.
(P2) All dead people are people who do not
vote.
(C) Therefore, all 19th Cent. American
presidents are people who do not vote.
Why study formal logic?
• It gives us a more robust understanding of
validity in general
• It forms the building block for our model of
meaning in language and for reasoning in
general
Introduction to Propositional
(or “Sentential” or “Truth-Functional”) Logic
• deals with propositions – whole statements; meaningful
declarative sentences
• S is a simple proposition =df. S does not contain any
other proposition as a component
Grover is dead.
• S is a compound proposition =df. S contains at least
one simple proposition as a component
Grover is dead and Stevenson is dead.
It is not the case that Grover is beautiful.
 The woman who married Grover is beautiful.
Propositional Forms, Variables, Constants,
and Substitution Instances
• a propositional form is a pattern for a whole class
of propositions
(p & q) v ~p
~&)pq)
• a propositional variable is a lowercase letter (e.g.,
‘p’, ‘q’, ‘r’, ‘s’) for which a proposition may be
substituted
• a propositional constant is a capital letter that
stands for a particular, definite proposition
G = Grover is dead.
S = Stevenson is dead.
• a substitution instance of a propositional form is
the result of uniformly replacing the propositional
variables in that form with propositions
• the same proposition may be replaced with different
variables, but no two different propositions may be
replaced by the same one variable
some examples
• Grover is dead and Stevenson is dead.
G&S
p&q
• Grover and Stevenson are beautiful men.
B&M
p&q
• Grover is dead and Grover is dead.
G&G
p & p or p & q
• Grover and Frances are a couple now.
C
p
Propositional Connectives
(“Logical Operators” or Truth-Functional Connectives”)
• a definition for each connective –
this simply specifies the truth conditions for any
proposition in which the connective occurs
this is a way of giving the meaning of the
connective by specifying its use
• a truth table sets out all of the possible truth value
combinations for the simple component propositions
and shows, for each combination, the value of the
compound proposition
Conjunction
‘and’, ‘but’, ‘also’, ‘as well’,…
p
q
p&q
T
T
T
T
F
F
F
T
F
F
F
F
some examples
Grover and Stevenson are dead.
G&S
 Grover and Frances are a couple now.
C
 All that I have left are photographs and
memories.
A
?? Grover and Frances are in love.
??
Disjunction
‘or’, ‘either… or…’
p
q
pvq
T
T
T
T
F
T
F
T
T
F
F
F
Inclusive Disjunction  “either this or that, and
perhaps both”
Some Examples
• Either Zac wants to avoid you or he’s out of
town.
WvO
• Special consideration is appropriate for elderly
or infirm people.
EvI
 Either Kelly or Kerry is the best singer alive
today.
(B v P) & ~ (B & P)
Exclusive Disjunction  “either this or that, but
not both”
p
q
p vv q
(p v q) & ~ (p & q)
T
T
F
F
T
F
T
T
F
T
T
T
F
F
F
F
Negation
‘not’, ‘it is not the case that...’
p
T
~p
F
F
T
Grover is not alive.
~A
It is not the case that Grover is alive.
~A
 Grover is not very attractive.
~V
 Frances never knew about Grover’s affair. ~ K
The (Material) Conditional
‘if..., then...’
[antecedent]  [consequent]
p
T
q
T
pq
T
T
F
F
F
T
T
F
F
T
• “Why should we count the conditional claim as
true when the antecedent is false and the
consequent true or, especially, when both are
false?”
– If you get an ‘A’ on the final, then you get an
‘A’ for the course.
– If Shane is younger than 31, then Shane is
younger than 33.
• “If p, then q.” =
• “Either q is the case or p is not the case.” =
• “It is not the case that p and not-q.”
pq
is equivalent to
qv~p
is equivalent to
~ (p & ~ q)
p
q
pq
q v ~p
~ (p & ~q)
T
T
T
T
T
T
F
F
F
F
F
T
T
T
T
F
F
T
T
T
• If Grover is decapitated, then Grover is dead.
Some Other Constructions
• ‘unless’ constructions can often be treated as
conditionals
– e.g., Otis remains quiet unless he is spoken to.
~SQ
(also Q v S)
• ‘provided that’, ‘given that’, ‘on condition that’,
and such like phrases
• ‘only if’ constructions are different
– You get to be president only if you are over 34.
PO
Some ‘If’s that Are Not Conditionals
• uncertainty / ‘iffy’
– e.g., Jen is not certain if Jack is competent.
• “Bring a friend – if you have one.”
• “I would appreciate tickets for the second
performance, if there is one.”
Parentheses
(punctuation for propositional logic)
• allow us to specify the scope of an operator
• the truth value of a compound proposition is tied
to the main operator
Mary says John is beautiful. =
“Mary,” says John, “is beautiful.”
or
Mary says, “John is beautiful.”
• there’s a big difference between ‘~ (p v q)’
and ‘~p v q’
Equivalences
pq
is equivalent to
q v ~p
• two compound propositions p and q are
logically equivalent if and only if p and q
always have the same truth value
• two equivalent propositions “have the same
meaning”
an example
• “Neither borrower nor lender be.”
 You should be neither a borrower nor a lender.
 You should not be a borrower and you should not be
a lender.
=
~ (B v L)
=
~B&~L
Propositional Arguments and
Checking for Validity
•
we want a decision procedure for determining
whether a propositional argument is valid:
1. isolate the form of the argument
(“translation”)
2. do the truth table (for the entire argument)
3. determine by inspection whether there are
any cases in which all of the premises are
true but the conclusion is false
• an argument form is a pattern for a whole
bunch of particular arguments
• a substitution instance of an argument form is
the argument that results from uniformly
replacing the propositional variables with
propositions
Checking for Validity:
The Guiding Principles
(GP1) an argument A is valid if A is a
substitution instance of a valid argument form
– an argument can be a substitution instance of a
valid form and of an invalid form at the same time
(P) Grover and Stevenson are dead.
(C) Therefore, Grover is dead.
(GP2) an argument form F is valid if and only if F
has no substitution instances in which all of the
premises are true and the conclusion is false
Some Common Argument Forms:
Conjunction
p
q
PREM
p&q
T
T
T
T
p
T
F
F
T
q
F
T
F
F
F
F
F
F
p&q
therefore, p
therefore, p & q
CONC
p
Disjunctive Syllogism
pvq
~p
therefore, q
P1
P2
CONC
p
q
pvq
~p
q
T
T
T
F
T
T
F
T
F
F
F
T
T
T
T
F
F
F
T
F
Modus Ponens
pq
p
therefore, q
P1
P2
CONC
p
q
pq
p
q
T
T
T
T
T
T
F
F
T
F
F
T
T
F
T
F
F
T
F
F
Modus Tollens
pq
~q
therefore, ~ p
P1
P2
CONC
p
q
pq
~q
~p
T
T
T
F
F
T
F
F
T
F
F
T
T
F
T
F
F
T
T
T
Hypothetical Syllogism
pq
qr
therefore, p  r
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
P1
pq
T
T
F
F
T
T
T
T
P2
qr
T
F
T
T
T
F
T
T
CONC
pr
T
F
T
F
T
T
T
T