Philosophical Reasoning Introduction to Elementary Logic I' Deduction Induction Distinction

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Philosophical ReasoningIntroduction to
Elementary LogicI. Deduction / Induction
Distinction

• Murali Ramachandran
• University of Sussex

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Definition

• An argument is a collection of propositions, one
  of whichthe conclusionis putatively supported
  (backed-up) by the othersthe premises.

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Definitions

• A deductively valid argument is one where it is
  impossible for the premises to be true and the
  conclusion falsei.e. is one which could not
  have true premises and false conclusion.
• When an argument is valid, we say the premises
  entail the conclusion.

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Argument A

• Singh and Patel went to the party.
• The party was a success if Patel or Jones went.
• 3) Hence, the party was a success.
• Valid

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Argument B

• Vince is a nerd if Brian is.
• Brian isnt a nerd unless he supports United.
• Brian doesnt support United.
• 4) Hence, Vince is a nerd.
• Invalid

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• Propositions p and q are logically equivalent if
  p entails q and q entails p. So e.g. the
  following two statements are logically
  equivalent
• Hillary is in New York or in London.
• If Hillary isnt in NY, she is in London, and if
  she isnt in London, she is in NY.

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Inductive Arguments

• An inductively strong argument is one whose
  premises would provide positive support for the
  conclusion if they were truethe premises render
  the conclusion more likely.
• An invalid argument that is not even inductively
  strong is called inductively weak.

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Argument C

• Kev is an animal-rights activist and Beth is a
  butcher.
• So, if either is a vegetarian, Kev is.
• Invalid but inductively strong

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Argument D

• Malcolm is an accountant.
• Beth was nearly bored to death by some
  accountants at a party once.
• So, Beth will find Malcolm boring too.
• Invalid and inductively weak

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Important point

• An argument may be (deductively) valid or
  inductively strong even if some (or all) of its
  premises are false!
• To say an argment is valid (or inductively
  strong) is to make a claim about how the premises
  are related to the conclusion one is not thereby
  claiming the premises or conclusion to be true.

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• All vegetarians are healthy.
• Babette is a vegetarian.
• So, Babette is healthy.
• Valid argument, but premise (1) is false.

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• Most violinists are vegetarians.
• Yehudi is a violinist.
• So, Yehudi is a vegetarian.
• Inductively strong, but premise (1) false.

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Difference between valid arguments and
inductively strong ones

• Whether an argument is valid or invalid is
  knowable a priori but whether an argument is
  inductively strong or weak often depends on
  background knowledge.
• Consider e.g. Arguments A and C the latters
  strength stems from our knowledge of butchers
  and animal-rights activists.

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Defeasibility of inductive strength

• Inductively strong arguments can be made weaker
  by adding further premises (and vice versa).
  Their strength is defeasible.
• E.g. suppose we added the premise that Beth comes
  from a long line of vegetarian butchers to
  argument C. The conclusion does not seem as
  compelling as before.
• Thus, inductive strength, unlike validity, admits
  of degrees.

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• Given our definition of validity, adding further
  premises to a valid argument cannot make it
  invalid.
• WHY???

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• All vegetarians are healthy.
• Babette is a vegetarian.
• Babette has cancer.
• Therefore, Babette is healthy.
• Question why is this still a valid argument?

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• 90 of children born in South India, have brown
  eyes.
• R was born in South India.
• Hence, it is likely that R has brown eyes.
• Is this a valid argument or merely inductively
  strong?

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• 90 of children born in South India, have brown
  eyes.
• R was born in South India.
• Most children born in Ambattur, S. India, have
  green eyes.
• R was born in Ambattur.
• Hence, it is likely that R has brown eyes.

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• Singh and Patel went to the party. (S and P)
• The party was a success if Patel or Jones
  went.(Q if P or J)
• Hence, the party was a success. (Q)
• Any argument with the same shape will be valid
• S and P, Q if P or J hence, Q

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• So, some valid arguments are valid purely in
  virue of their shape. They are said to have a
  valid logical form.
• Formal logic is the study of logical form and it
  is this we shall be concerned with for the
  remainder of the course, since it provides a
  fundamental and comparatively easy starting point.