Propositional Logic

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Propositional Logic

• USEM 40a
• Spring 2006
• James Pustejovsky

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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 As premises to be
  true and its conclusion false
• deductive argument A is sound df.
• (i) A is valid, and (ii) all of As premises are
  true

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• (P1) If Grover is dead, then Grover does not
• vote.
• (P2) Grover is dead.
• (C) Therefore, Grover does not vote.

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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.

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• argument A is formally valid if, in virtue of As
  logical form alone, it is impossible for all of
  As 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.

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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

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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.

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Propositional Forms, Variables, Constants, and
Substitution Instances

• a propositional form is a pattern for a whole
  class of propositions
• (p q) v p ) p q )
• a propositional variable is a lowercase letter
  (e.g., p, q, r, s) for which a
  proposition may be substituted

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• 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

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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

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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

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Conjunctionand, but, also, as well,
p q p q
T T T
T F F
F T F
F F F
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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. ??

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Disjunctionor, either or
p q p v q
T T T
T F T
F T T
F F F
Inclusive Disjunction ? either this or that,
and perhaps both
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Some Examples

• Either Zac wants to avoid you or hes out of
  town. W v O
• Special consideration is appropriate for elderly
  or infirm people. E v I
• ? Either Kelly or Kerry is the best singer alive
  today. (B v P) (B P)

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• 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
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Negationnot, it is not the case that...
p p
T 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
Grovers affair. K
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The (Material) Conditionalif..., then...

• antecedent ? consequent

p q p ? q
T T T
T F F
F T T
F F T
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• 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.

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• p ? q is equivalent to
• q v 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.

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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

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Some Ifs 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.

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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.
• theres a big difference between (p v q)
• and p v q

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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

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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

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Propositional Arguments and Checking for Validity

• we want a decision procedure for determining
  whether a propositional argument is valid
• isolate the form of the argument (translation)
• do the truth table (for the entire argument)
• determine by inspection whether there are any
  cases in which all of the premises are true but
  the conclusion is false

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• 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

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Checking for ValidityThe 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

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Some Common Argument FormsConjunction
PREM CONC
p q p q p
T T T T
T F F T
F T F F
F F F F
p q
therefore, p

p
q
therefore, p q
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Disjunctive Syllogism
p v q
p
therefore, q
P1 P2 P2 CONC
p q p v q p q
T T T F T
T F T F F
F T T T T
F F F T F
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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
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Modus Tollens
p ? q
q
therefore, p
P1 P1 P2 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
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Hypothetical Syllogism
p ? q
q ? r
therefore, p ? r
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P1 P2 P2 CONC
p q r p ? q q ? r p ? r
T T T T T T
T T F T F F
T F T F T T
T F F F T F
F T T T T T
F T F T F T
F F T T T T
F F F T T T