The ubiquity of logic

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The ubiquity of logic

• One common example of reasoning
• If I take an umbrella, I can prevent getting wet
  by rain
• I dont want to get myself wet by rain
• Therefore, I need to come back home to take an
  umbrella
• Logic is the study of the methods and principles
  used to distinguish good from bad reasoning

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Logic as a skill

• Only the student of logic can reason well
• Excellent athletes who know nothing about
  physics and physiology
• A sport science professor who risks his dignity

3
Practicing logic

• A person who has studied logic is more likely to
  reason correctly
• The proper study of logic will approach it as an
  art as well as a science ? Practice makes perfect
• A large part of the study of logic is the
  examination and analysis of fallacies
• The study of logic will give students techniques
  and methods for testing the correctness of
  different kinds of logical reasoning

4
Declarative sentence

• Declarative sentences are either true or false
• Questions to be asked
• Commands to be given
• Exclamations to be uttered

5
Argument

• Inference is a process by which one sentence is
  arrived at and affirmed on the basis of one or
  more other sentences accepted as the starting
  point of the process
• There is an argument that corresponds to every
  possible inference
• Group of sentences of which one is claimed to
  follow from the others
• If I am a superman, I can have laser beam come
  out of my eyes
• I cannot have laser beam come out of my eyes
  Premises
• Therefore, I am not a superman
  
  Conclusion

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• If the Athenians condemned Socrates to death,
  they
• would condemn Plate to death as well
  Premises
• But the Athenians did not condemn Plato to death
• Therefore, the Athenians did not condemn Socrates
  to death Conclusion
• The conclusion of an argument is that
  proposition which is affirmed on the basis of the
  other propositions of the argument
• The premises of an argument are the propositions
  meant to provide evidence or reasons for the
  conclusion

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Deductive and inductive arguments

• A deductive argument involves the claim that its
  premises provide conclusive evidence
• All humans are mortal
• Socrates is human
• Therefore, Socrates is mortal
• All humans are mortal
• Socrates is human
• Therefore, Socrates is a philosopher

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Differences

• What is the difference between good and bad
  arguments
• Given the truth of the premises, the conclusion
  must be true
• Given the truth of the premises, the conclusion
  couldnt be otherwise
• Good arguments are valid but bad arguments are
  invalid

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The validity of deductive argument

• A argument is valid iff (if and only if)
• Given the truth of the premises, the conclusion
  must be true
• The premises provide (conclusive, decisive,
  highest possible, strongest possible) (evidence,
  reason, ground, basis) for the conclusion
• The truth of the premises (necessitates, entails,
  implies) the truth of the conclusion
• It is (contradictory, absolutely impossible,
  logically impossible) that the premises are all
  true but the conclusion is false
• The conclusion (deductively follows, logically
  follows, is deducible, is derivable) from the
  premises5) P provides (conclusive or decisive)
  (evidence, reason, ground, basis) for Q
• We cannot have true premises and false conclusion
  at the same time
• The truth of the premises is (inconsistent,
  incompatible) with the falsehood of the conclusion

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Examples

• In what sense is this argument valid?
• All humans are mortal
• Socrates is human
• Therefore, Socrates is mortal
• In what sense is this argument invalid?
• All humans are mortal
• Socrates is human
• Therefore, Socrates is a philosopher

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Validity and Contradiction

• An argument is valid iff it is (contradictory,
  absolutely impossible, logically impossible) that
  the premises are all true but the conclusion is
  false
• Suppose that the following four sentences are
  jointly contradictory A, B, C, and D
• Then the following arguments are valid
• A, B, C therefore D
• A, B, D therefore, C
• A, C, D therefore, B
• B, C, D therefore, A

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Examples

• The following sentences are jointly
  contradictory
• x is smaller than 2 or x is greater than 2
• x is not smaller than 2
• x is not greater than 2
• The following arguments are valid
• x is smaller than 2 or x is greater than 2
• x is not smaller than 2
• It is not the case that x is not greater than 2
• x is smaller than 2
• x is not greater than 2
• It is not the case that x is not smaller than 2

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The task of deductive logic

• The claim that the premises provide conclusive
  evidence for the truth of the conclusion
• The task of deductive logic is to clarify the
  nature of the relation between premises and
  conclusion in valid arguments, and thus to allow
  us to discriminate valid from invalid arguments

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The task of deductive logic

• The claim that the premises provide conclusive
  evidence for the truth of the conclusion
• The task of deductive logic is to clarify the
  nature of the relation between premises and
  conclusion in valid arguments, and thus to allow
  us to discriminate valid from invalid arguments

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Inductive argument

• An inductive argument involves the claim that
  its premises give inconclusive grounds for the
  truth of its conclusion
• Socrates is human and is mortal
• Plato is human and is mortal
• Aristotle is human and is mortal
• Therefore, all humans are mortal
• Deductively invalid
• It is not the case that given that the premises
  are true, the conclusion must be true
• The truth of the premises doesnt provide
  conclusive ground for the truth of the conclusion

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• Given that the premises are all true, the
  conclusion is more likely to be true than false
• An inductively good argument is an argument
  where the truth of its premises gives some
  support, albeit not decisive support, to the
  truth of the conclusion

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The strength of support

• We can meaningfully speak of the strength of the
  support in terms of the likelihood of the
  conclusion conditional upon the truth of the
  premises
• The strength of the support given by the
  premises to the conclusion varies from one
  inductive argument to another
• Socrates is human and is mortal
• Plato is human and is mortal
• Therefore, all humans are mortal
• The support given to the conclusion by the
  premises in this argument is weaker than the one
  in the original Socrates argument

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• The conclusion of this argument is less likely
  to be true conditional upon its premises than the
  conclusion of the original Socrates argument is
  true conditional upon its premises
• By omitting the third premise of the original
  Socrates argument, we have weakened the resulting
  argument
• Socrates is human and is mortal
• Plato is human and is mortal
• Aristotle is human and is mortal
• Choi is human and is mortal
• Therefore, all humans are mortal
• By adding the fourth premise of the new
  argument, we have strengthened the resulting
  argument

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Evaluating inductive arguments

• Inductive arguments may be evaluated as better
  or worse, according to the strength of the
  support given to their conclusions by their
  premises

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Distinction

• One way to distinguish the two types of
  argument In a deductive argument, the premises
  are claimed to furnish outright support or basis
  for the truth of the conclusion. But in an
  inductive argument, the premises are claimed to
  furnish some inconclusive support or basis for
  the truth of the conclusion.
• Another way?
• One suggestion in a deductive argument we infer
  a particular sentence from universal sentences,
  while in an inductive argument we infer a
  universal sentence from particular sentences
• An intuitive appeal of the suggestion

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Distinction

• Not universally applicable
• It is wrong both about deductive arguments and
  about inductive arguments
• It is not always the case that in deductive
  arguments, particular conclusions are inferred
  from universal premises
• If the Athenians condemned Socrates to death,
  they would condemn Plate to death as well
• But the Athenians did not condemn Plato to death
• Therefore, the Athenians did not condemn Socrates
  to death

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• It is not always the case that, in inductive
  arguments, universal conclusions are inferred
  from particular premises
• All cows are mammals and have lungs
• All horses are mammals and have lungs
• All humans are mammals and have lungs
• Therefore, all mammals have lungs
• Hilter was a dictator and was callous
• Stalin was a dictator and was callous
• Mugabe is a dictator
• Therefore, Mugabe is callous
• The criterion for the distinction between
  deductive and inductive argument in terms of
  generality and particularity of the sentences
  involved are mistaken