A%20better%20distinction

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A better distinction

• If a deductive argument is valid, then its
  conclusion follows with equal necessity from its
  premises no matter what else may be the case.
• P All humans are mortal
• Q Socrates is human
• Therefore S Socrates is mortal
• P All humans are mortal
• Q Socrates is human
• R Socrates is ugly
• Therefore S Socrates is mortal

2

• The new argument with an enlarged set of
  premises is valid
• The validity of the original argument It
  is a contradiction that all humans are mortal,
  Socrates is human, and Socrates is not mortal
• The situation where all humans are mortal,
  Socrates is human, Socrates is ugly but, at the
  same time, Socrates is not mortal?
• Still contradiction!!

3

• The conclusion follows strictly from the
  enlarged set of premises because it follows
  strictly from the two original premises initially
  given
• P All humans are mortal
• Q Socrates is human
• R Socrates is not ugly
• Therefore S Socrates is mortal
• P All humans are mortal
• Q Socrates is human
• R Socrates is not human
• Therefore S Socrates is mortal

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Corollary

• Any sentence is deducible from a contradiction
• The following argument is valid whatever Q may
  be
• P
• P
• Therefore, Q
• In general, suppose that Q legitimately follows
  from P1, P2, P3, . . . Then Q legitimately
  follows from R, P1, P2, P3, . ., whatever R may
  be

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Reverse step?

• It is impossible to make a deductively valid
  argument it invalid by adding new premises
• It is possible to make a deductively invalid
  argument valid by adding new premises
• Socrates is human
• Socrates is mortal
• Socrates is human
• All humans are mortal
• Socrates is mortal

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Inductive argument

• Adding new premises to the original argument can
  serve either to weaken or to strengthen the
  result argument
• Augustine is a philosopher and lived a long life
• Aquinas is a philosopher and lived a long life
• Bertrand Russell is a philosopher and lived a
  long life
• Sungho Choi is a philosopher
• Therefore, Sungho Choi will live a long life
• Weakening the support
• Augustine is a philosopher and lived a long life
• Aquinas is a philosopher and lived a long life
• Bertrand Russell is a philosopher and lived a
  long life
• Sungho Choi is a philosopher
• Sungho Choi has a terminal cancer
• Therefore, Sungho Choi will live a long life

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• Strengthening the support
• Augustine is a philosopher and lived a long life
• Aquinas is a philosopher and lived a long life
• Bertrand Russell is a philosopher and lived a
  long life
• Sungho Choi is a philosopher
• Sungho Choi doesnt smoke and exercises on a
  regular basis
• Therefore, Sungho Choi will live a long life
• Deductive argument is a type of argument whose
  conclusion is claimed to follow from its premises
  with absolute necessity, this necessity not being
  a matter of degree

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Truth and validity

• The properties of truth and falsehood are
  predicated of sentences
• Arguments are either valid or invaild
• Meaningless phrases valid sentences, true
  arguments
• To say that an argument is valid is to say that
  the truth of its premises is inconsistent with
  the falsehood of its conclusion
• All humans are mortal
• Socrates is human
• Therefore, Socrates is mortal

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• All non-humans are immortal
• Socrates is non-human
• Therefore, Socrates is immortal
• All humans are mortal
• Socrates is human
• Therefore, Socrates is a philosopher
• All humans are mortal
• Socrates is mortal
• Therefore, Socrates is female
• The truth or falsehood of the conclusion of an
  argument doesnt determine its validity or
  invalidity

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Truth and validity contd

• The only constraint on the truth values of the
  sentences imposed by the validity of an argument
  is that we cannot have true premises and false
  conclusion at the same time
• The truth or falsehood of the conclusion of an
  argument doesnt determine its validity or
  invalidity univocally
• The validity of an argument does not guarantee
  the truth of its conclusion

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Soundness

• An argument is sound if and only if (1) it is
  valid and (2) all of its premises are true
• The conclusion of a sound argument must be true
• The difference between soundness and validity
  the first guarantees but the second does not
  guarantee the truth of the conclusion
• The falsehood of the conclusion
  unsound argument
• either it is invalid or some of its premises are
  false

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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

• Symbolic languages consist of symbolic sentences
• 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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Sentential connectives

• Refer to the phrases like it is not the case
  that and if then
• or and and
• Their main function is to connect sentences to
  one another to form a compound sentence
• Logical connectives

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Michellanies

• the relation between a capital letter and the
  sentence it abbreviates is subject to change
• Sentence letters P, Q, R, . . Z, P1, Q2, R5,
  Z0,

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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
• How to exhaustively characterize the class of
  symbolic sentences
• 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

• Alternative characterization
• Sentences letters are symbolic sentences
• If f is a symbolic sentence, then so is f
• 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 my 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
• Each initial node is a sentence letter
• 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 conditional
  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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• Alternative characterization
• 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 f is a symbolic sentence, then so is f
• If f and ? are symbolic sentences, then so is (f
  ? ?)
• Examples

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Definitions

• Atomic sentences vs. compound sentences
• The main 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

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

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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