Going Formal

1
Going Formal

• Meet the Connectives

2
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

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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

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Sentence Letters

• Translate atomic sentences
• Atomic sentences have no proper parts that are
  themselves sentences
• Examples
• It is raining R
• It is cold C

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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 R C
• If it rains then it pours R ? P

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Parentheses, brackets braces

• Ill go to Amsterdam and Brussels or Calais
• This is ambiguous and we cant tolerate
  ambiguity!

OR
OR
Calais
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Parentheses, brackets braces

• Grouping devices avoid ambiguity (for unique
  readability)
• Ill go to Amsterdam, and then to either Brussels
  or CalaisA (B ? C)
• Ill either go to Amsterdam and Brussels, or else
  to Calais(A B) ? C

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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)

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Plugging into variables
ModusPonens
Substitution Instance of Modus Ponens
?
p ? q p q
(D ? (E ? ? F))
((A ? B) ? C)
((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.

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Plugging into variables
ModusPonens
Substitution Instance of Modus Ponens
?
p ? q p q
((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.

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The Grammar of Propositional Logic

• Constructing WFFs (Well-Formed Formulae)
• Identifying WFFs
• Identifying main connectives

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Rules for WFFs

1. A sentence letter by itself is a WFF A B Z
2. The result of putting ? immediately in front of
   a WFF is a WFF ?A ? B ? ? B ?
   (A ? B) ? (? C ? D)
3. The result of putting ? , ? , ? , or ? between
   two WFFs and surrounding the whole thing with
   parentheses is a WFF (A ? B) (? ? C ? D)
   ((? ? C ? D) ? (E ? (F ? ? G)))
4. Outside parentheses may be dropped A ? B ? ?
   C ? D (? ? C ? D) ? (E ? (F ? ? G))

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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
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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)

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Hints When its not a WFF

• You can't have two WFFs next to one another
  without a two-sided 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 levelBAD! A ? B ? C (? ? C ? D) ?
  (E ? F ? ? G)

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Identifying WFFs Main Connectives
?

• 1 (S ? ? T) ? (? U ? W)
• 2 ? (K ? L) ? (? G ? H)
• 3 (E ? F) ? (W ? X)
• 4 (B ? ? T) ? ? (? C ? U)
• 5 (F ? ? Q) ? (A ? E ? T)

X
X

X
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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)

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Identifying WFFs Main Connectives

• 6 ? D ? ? ( P ? Q) ? (T ? R)
• 7 (D ? ? Q) ? (P ? E) ? A ? ( ? H)
• 8 M (N ? Q) ? (? C ? D)
• 9 ? (F ? ? G) ? (A ? E) ? ? H
• 10 (R ? S ? T) ? ? (? W ? ? X)

?
X
X
?
X
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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)

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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.

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Translation
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Conditionals Biconditionals
If P then Q P ? Q
P, if Q Q ? P
P only if Q 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
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Conditionals
If P then Q P ? Q
P, if Q Q ? P
P only if Q 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
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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!
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Not both and neither/nor

• Not both P and Q
• (P ? Q)

Neither P nor Q ? (P ? Q)
You cant both have your cake and eat it. (H ?
E)
She was neither young nor beautiful. ? (Y ? B)
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Not both and neither/nor

• Not both P and Q
• (P ? Q)

Neither P nor Q ? (P ? Q)
15 Not both Jaguar and Porsche make
motorcycles. (J ? P) 16 Both Jaguar and
Porsche do not make motorcycles. ? J ? P
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Not both and neither/nor

• Not both P and Q
• (P ? Q)

Neither P nor 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
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DeMorgans 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 cant both have your cake
and eat it is equivalent to You either dont
have your cake or you dont eat your cake - NOT
You dont have your cake and you dont eat your
cake.
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So, what do I need for the quiz?

• Identifying WFFs and main connectives
• Translation given an English sentence,which of
  the following symbolizedsentences is the
  correcttranslation?

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The End
WFF