The value of special symbols

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The value of special symbols

• Arguments presented in English or any other
  natural languages are often difficult to appraise
• Yesterday there were a number of crickets singing
  in our garden
• Cricket consists of bats and balls
• Therefore, yesterday there were a number of bats
  and balls singing in our garden
• The ambiguity of the term cricket
• Any of various insects that produce a shrill
  chirping sound by rubbing the front wings
  together
• An outdoor game played with bats, a ball, and
  wickets
•  

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• He took off his shoes
• He took off his socks
• Therefore, he took off his socks and (then) shoes
• And in the conclusion has the meaning of
  temporal succession.
• He took off his shoes and (then) socks
• He took off his socks and (then) shoes
• It would be convenient to set up an artificial
  symbolic language

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The use of a special logical notation

• Aristotle
• The special symbols of modern logic helps us to
  exhibit with greater clarity the logical
  structures of sentences and arguments

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One further value of logical notation

• The aid they give in the actual use and
  manipulation of sentences and arguments
• The real superiority of Arabic numerals is
  revealed only in computation
• Multiplying 113 by 9
• Multiplying two numerals in Chinese character
• The drawing of inferences and the appraisal of
  arguments is greatly facilitated by the adoption
  of a special logical notation

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Symbolic language

• The more symbols a symbolic language contains,
  the more representational power it has, and
  therefore, the more accurate account of deductive
  arguments it gives
• it is not the case that ? negation
• if, . . Then ? conditional
• Use the capital letters P through Z to
  symbolize English sentences

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Negation

• It is not the case that Socrates is bald is
  the negation of Socrates is bald
• P Socrates is bald
• P It is not the case that Socrates is bald
• The negation of P
• NB. No parenthesis is required

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Conditional

• P Diogenes is canine
• Q Diogenes is carnivorous
• (P ? Q) If Diogenes is canine, then Diogenes
  is carnivorous
• Conditional formed from P and Q
• P is the antecedent of the conditional and Q
  is the consequent of the conditional
• NB. Conditional symbols are accompanied by
  parentheses

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The most elementary symbolic language

• The only logical symbol is the negation symbol
• P, Q, R, . . Z, P, Q, R, . . Z, P, Q, .
  . Z
• Sentence letters are symbolic sentences
• Negations 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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Inductive definition

• Sentences letters are symbolic sentences
• If f is a symbolic sentence, then so is f
• No Greek letters will be contained in symbolic
  sentences
• The class of natural numbers
• The number 1 is a natural number
• If n is a natural number, n1 is also a natural
  number
• The class of Alfreds ancestors
• Examples of symbolic sentences

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Grammatical Tree

• We can represent this generation of the sentence
  by means of a grammatical tree that displays its
  genealogy
• The top node is the symbolic sentence whose
  genealogy is being displayed

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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

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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 symbolic sign of a compound sentence is
  the sign 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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Outermost parentheses

• 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

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