A new symbolic language

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A new symbolic language

• The only logical symbol is the negation symbol
• P, Q, R, . . Z, (P ? P), (P?Q), (P?R), . .
  (P?Z), (Q?P), (Q?R), (P?(P?P)), ((P?Q) ?Q), .
  .(P? ((P?Z) ?R) ?Z)
• Sentence letters are symbolic sentences.
• Conditionals formed from symbolic sentences are
  symbolic sentences
• Nothing other than sentences letters and
  negations formed from symbolic sentences are
  symbolic sentences

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• Sentences letters are symbolic sentences
• If f and ? are symbolic sentences, then so is (f
  ? ?)
• Examples
• Grammatical tree

3
A more complex symbolic language

• A symbolic language that contains both the
  negation sign and conditional sign
• P, Q, R, . . Z, (P ? P), (P?Q), (P?R),
  . . (P?Z), (Q?P), (Q?R), (P?(P?P))
• Sentences letters are symbolic sentences
• If pi is a symbolic sentence, then so is pi
• If pi and psi are symbolic sentences, then so is
  (pi ? psi)
• Examples

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Definitions

• Atomic sentences vs. compound sentences
• The main logical connective of a compound
  sentence is the connective that is used at the
  last step in building the sentence
• Examples
• Conditional sentence and negation sentence
• Examples

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Grammatical tree

• Each nonbranching node is of the form f and
  it has the symbolic sentence pi as its sole
  immediate ancestor
• Each branching node is of the form (f ? ?),
  having the symbolic sentence f as its immediate
  left ancestor and the symbolic sentence ? as its
  immediate right ancestor
• Any expression that can be generated as the top
  node of a grammatical tree is a symbolic sentence
  

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Parenthesis

• The function of parentheses is just like that of
  punctuation in written language
• The teacher says John is a fool
• P?Q?R
• (P ? Q) vs. (P ? Q)
• If it doesnt rains, I go out without an
  umbrella
• It is not the case that if it rains, I go out
  without an umbrella

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Informal notation

• No confusion will arise if we omit the outermost
  parentheses of a sentence
• When parentheses lie within parentheses, some
  pair may be replaced by pairs of brackets for the
  sake of display and recognition
• In official notation, a symbolic sentence is
  enclosed by a single pair of outermost
  parentheses but in informal notation it is not
• Chapter 1, Section 1 of Terence Parsons article

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Translation

• Translation and symbolization
• Translation into English
• A scheme of abbreviation correlates a sentence
  letter with an English sentence
• Two steps of translation literal vs. free
  translation
• Free translation is a liberal version of literal
  translation

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Literal translation

• Restore any parentheses that may have disappeared
  as a result of informal conventions
• Replace sentence letters by English sentences in
  accordance with the given scheme of abbreviation
• Replace the negation sign with it is not the
  case that
• Replace the conditional sign with if then

10
Free translation

• A free translation or translation simpliciter is
  a sentence we can get from a literal translation
  only by changing its style
• A free translation of f into English is a
  stylistic variant of the literal translation of f
  into English
• How to determine whether a sentence is a
  stylistic variant of the literal translation of f?

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Guideline

• Negation
• It is not the case that John has 4 limbs
• John does not have 4 limbs
• John fails to have 4 limbs
• Conditional
• If John has 4 limbs then John has 2 siblings
• Provided that John has 4 limbs then John has 2
  siblings
• On the condition that John has 4 limbs then
  John has 2 siblings

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• John has 4 limbs only if John has 2 siblings
• To assert that A only if B is to deny that A is
  true but B is false. This is to assert that if A
  then B
• Chapter 1 Section 2 of Terence Parsons article

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Cautionary note

• John owns a car
• 
  Stylistic variants of one another?
• John owns an automobile
• John is an unmarried man
• 
  Stylistic variants of one another?
• John is a bachelor
• John doesnt own a car
• 
  Stylistic variants
• It is not the case that John owns a car

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• If John is old, he can own a car
• 
  Stylistic variants
• In case John is old, he can own a car
• What is the difference?
• In the second case, the expressions at issue are
  phrases of connection but this is not true in the
  first case
• The expressions, car and automobile, are not
  phrases of connection
• Two synonymous sentences are stylistic variants
  of each other only if their difference concerns
  phrases of connection

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Symbolization

• A symbolic sentence f is a symbolization of an
  English sentence ? iff ? is a free English
  translation of f.
• f is a symbolization of an English sentence ?
  iff ? is a stylistic variant of the literal
  English translation of f

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Procedure

• Introduce it is not the case that and if . .
  Then in place of their stylistic variants.
• Replace if . . .then with the conditional sign
• Replace it is not the case that with the
  negation sign
• Replace English sentences by sentence letters in
  accordance with the given scheme of abbreviation
• Omit outermost parentheses according to the
  informal convention

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Grouping together

• If he does not greet, she will be distraught
• If
• She will be distraught if he greets
• only if
• She will be distraught only if he greets

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Ambiguous sentences

• It is not the case that she will be distraught
  if he does not greet
• (P ? Q)
• P ? Q
• if Wilma leaves Xavier stays if Yolando sings
• (Yolando sings) ? ((Wilma leaves) ? (Xavier
  stays))
• (Wilma leaves) ? ((Yolando sings) ? (Xavier
  stays))

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Commas

• A comma indicates that the symbolizations of
  sentences to its left or the symbolization of
  sentences to its right should be combined into a
  single sentence
• If Wilma leaves, Xavier stays if Yolando sings
• Requiring that Xavier stays and Yolando
  sings are grouped together
• If Wilma leaves Xavier stays, if Yolando sings
  
• Wilma leaves and Xavier stays are required
  to be grouped together

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Logical derivation

• A criterion for validity for those arguments
  that are formulated in the symbolic language
  under discussion
• A symbolic argument is an argument whose
  premises and conclusion are symbolic sentences
• A derivation consists of a sequence of steps
  from the premises of a given argument to its
  conclusion
• Each step constitutes an intuitively valid
  argument

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Mathematical derviation

• X 7891011121314
• Therefore, x 84
• X 7891011121314
• X 1591011121314
• X 241011121314
• ..
• Therefore, X 84
• By going through all of these steps, we can get
  from the premise of the original argument to its
  conclusion

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Four inference rules

• Modus Ponens (MP)
• f ? ?
• f,
• Therefore, ?
• Modus Tollens (MT)
• f ? ?
• ?,
• Therefore, f

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• Double Negation (DN)
• f
• Therefore, f
• f
• Therefore, f
• Repetition
• ?
• Therefore, ?

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Three types of derivation

• Direct derivation
• Conditional derivation
• Indirect derivation

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