Logic

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Logic
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what is an argument?

• People argue all the time? that is, they have
  arguments.
• It is not often, however, that in the course of
  having an argument people actually give an
  argument.
• Indeed, few of us have ever actually stopped to
  consider what it means to give an argument or
  what an argument is in the first place.
• Yet, there is a whole discipline devoted to just
  this logic.

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general definition

• An argument is a set of statements that includes
• at least one premise which is intended to support
  (i.e. give reason to believe)
• a conclusion.
• So according to this definition is the following
  set of statements an argument?
• Ms. Mayberry left to go to work this morning.
  Whenever she does this, it rains. Therefore,
  the moon is made of blue cheese.

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• Yes. We know this is an argument because of the
  word therefore
• Ms. Mayberry left to go to work this morning.
  Whenever she does this, it rains. Therefore,
  the moon is made of blue cheese.
• This typically indicates that the final sentence
  is intended to follow from (that is, be supported
  by) the preceding sentences.
• Of course, it is easy to see that this is not a
  good argument.
• The premises of the argument seem to be
  irrelevant to (that is, they do not support) the
  conclusion.

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other examples

• Does God exist?
• P1) If there is unnecessary evil in the world,
  then God does not exist.
• P2) There is unnecessary evil in the world.
• C) Therefore, God does not exist.
• Is ethics absolute or relative?
• P1) If there were absolute truth about morality,
  then cultures would not disagree about morality.
• P2) Cultures do disagree about morality.
• C) Therefore, there is no absolute truth about
  morality.

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• The premises in these arguments may be true or
  false.
• Either way, in each of these examples the
  premises support the conclusion.
• That is, if they are true, then they give us
  reason to believe the conclusion.
• But this raises an important question
• What does it mean to say that premises of an
  argument support the conclusion?

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• The move from the premises to the conclusion is
  called an inference.
• The premises support the conclusion only if the
  inference is good.
• Our question, then, concerns what it means for an
  inference to be good.

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validity

• An argument is valid when it contains a good
  deductive inference.
• A deductive inference is good just in case, given
  the truth of the premises, the conclusion must
  also be true that is, if there is no way for
  the premises to be true and the conclusion to be
  false.
• In short, the conclusion must be true, assuming
  the truth of the premises.
• It is extremely important to note that the above
  definition does not say that the premises of the
  argument are true. Rather we assume that the
  premises are true and try to determine whether,
  given this assumption, the conclusion must be
  true as well.

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validity

• P1) If the animal in the barn is a pig, then it
  is a mammal.
• P2) The animal in the barn is a pig.
• Therefore, the animal in the barn is a mammal.
• The truth of the premises guarantees the truth of
  the conclusion.
• There is simply no rational way to accept the
  truth of both premises and still deny the
  conclusion.
• That is why it is valid.

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validity

• P1) If the animal in the barn is a pig, then it
  is a mammal.
• P2) The animal in the barn has feathers and lays
  eggs.
• Therefore, the animal in the barn is a mammal.
• invalid
• The premises do not guarantee the conclusion.

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soundness

• Of course, valid arguments with obviously false
  or highly controversial premises are of little
  real value.
• What we must strive for are arguments with
  obviously true or relatively uncontroversial
  premises.
• That is, what we want are valid arguments with
  true premises i.e., sound arguments.
• A deductive argument is sound just in case the
  argument is valid and, in addition, all of its
  premises are true.

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• P1) If the animal in the barn is a pig, then it
  is a mammal.
• P2) The animal in the barn is a pig.
• C) Therefore, the animal in the barn is a mammal.
• valid
• sound
• P1) If the animal in the barn is a pig, then it
  is purple.
• P2) The animal in the barn is a pig.
• C) Therefore, the animal in the barn is purple.
• valid
• NOT sound

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deductive logical structure

• There is an important difference the form and the
  content of an argument.
• The form is its logical structure.
• The content is its subject matter, or what its
  about.
• Deductive arguments are valid because they
  involve the right sort of logical structure.
• Content is irrelevant for validity
• But not for soundness (why?).

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if-then conditionals

• The if-then conditional plays an important role
  in many deductive arguments
• P1) If Joe is a father, then he is a male.
• P2) Joe is a father.
• C) Therefore, he is a male.
• (P1 is a conditional) conditionals have two
  parts
• If Joe is a father (antecedent),
• then he is a male (consequent).

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• If Joe is a father, then he is a male.
• There are two valid things you can do with a
  conditional
• you can affirm the antecedent,
• or you can deny the consequent.

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modus ponens (affirming the antecedent)

• P1) If the moon is made of blue cheese, then pigs
  fly.
• P2) The moon is made of blue cheese.
• C) Therefore, pigs fly.
• P1) If its raining, then the streets are wet.
• P2) Its raining.
• C) Therefore, the streets are wet.
• These arguments bear an obvious similarity to one
  another. This is because they both have the same
  form. Roughly
• P1) If this, then that. P1) p ? q
• P2) This. P2) p
• C) Therefore, that. C) Therefore, q.

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modus tollens (denying the consequent)

• P1) If the moon is made out of blue cheese, then
  pigs fly.
• P2) Pigs dont fly.
• C) Therefore, the moon is not made out of blue
  cheese.
• P1) If my car can get us to Denver, then it is
  working properly.
• P2) My car is not working properly.
• C) Therefore, my car cannot get us to Denver.
• Once again, these arguments bear an obvious
  similarity to one another, which is the form
• P1) If this, then that. P1) p ? q
• P2) Not that. P2) q
• C) Therefore, not this. C) Therefore, p

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• If Joe is a father, then he is a male.
• There are two invalid things you can do with a
  conditional
• you can deny the antecedent,
• or you can affirm the consequent.

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affirming the consequent

• P1) If the moon is made out of blue cheese, then
  pigs fly.
• P2) Pigs fly.
• C) Therefore, the moon is made out of blue
  cheese.
• P1) If my car can get us to Denver, then it is
  working properly.
• P2) My car is working properly.
• C) Therefore, my car can get us to Denver.
• Once again, these arguments bear an obvious
  similarity to one another, which is the form
• P1) If this, then that. P1) p ? q
• P2) That. P2) q
• C) Therefore, this. C) Therefore, p.

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denying the antecedent

• P1) If the moon is made of blue cheese, then pigs
  fly.
• P2) The moon is not made of blue cheese.
• C) Therefore, pigs dont fly.
• P1) If its raining, then the streets are wet.
• P2) Its not raining.
• C) Therefore, the streets are not wet.
• These arguments bear an obvious similarity to one
  another. This is because they both have the same
  form. Roughly
• P1) If this, then that. P1) p ? q
• P2) Not This. P2) p
• C) Therefore, not that. C) Therefore, q.

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four if-then structures
Valid P1) If p, then q. P2) p. C) Therefore, q. Valid P1) If p, then q. P2) not q. C) Therefore, not p.
Invalid P1) If p, then q. P2) not p. C) Therefore, not q. Invalid P1) If p, then q. P2) q. C) Therefore, p.
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a question

• So, why is it valid to
• affirm the antecedent (modus ponens)
• deny the consequent (modus tollens)
• But not to
• affirm the consequent
• deny the antecedent

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• Consider the following premise
• If Joe is a father, then he is a male.
• If Joe is a father, does it follow that he is a
  male? Yes.
• This is modus ponens.
• If Joe is not a male, does it follow that he is
  not a father? Yes.
• This is modus tollens.

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• Consider the following premise
• If Joe is a father, then he is a male.
• If you know that Joe is a male, then can you
  conclude that he must be a father? No.
• This is affirming the consequent.
• If Joe is not a father, does it follow that he is
  not a male? No.
• This is denying the antecedent.

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formal fallacies

• It is important to realize that the reason that
  these arguments forms are invalid is that their
  logical structures do not guarantee the truth of
  their conclusions.
• Hence, these argument forms are always invalid.
• Because affirming the consequent and denying the
  antecedent are fallacies that arise simply in
  virtue of argument form, they are called formal
  fallacies.

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necessary sufficient conditions

• An if-then conditional is composed of two
  propositions, p and q, related by the connective
  if, then (or ?).
• The first proposition (i.e., the one that follows
  the if) is part of the antecedent of the
  conditional
• The second proposition (i.e., the one that
  follows the then) is part of the consequent.

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sufficient conditions

• Suppose I say, If you give birth to a baby, then
  you are a mother.
• What I am saying is that the antecedent (i.e.,
  giving birth to a baby) is enough (it is all you
  need) to make it true that you are a mother.
• In general, we will say that the antecedent of a
  conditional is a sufficient condition for its
  consequent.
• Thus, giving birth to a baby is a sufficient
  condition for being a mother.
• This is why modus ponens is deductively valid.
• p ? q
• p
• ? q

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necessary conditions

• p ? q means that p is enough (sufficient) for q
• It also means that q is required (necessary) for
  p
• So, if you give birth to a baby, you must be a
  mother (being a mother is required). That is, you
  cant have given birth to a baby, but not be a
  mother.
• In this way, the consequent of the conditional is
  a necessary condition for its antecedent. If the
  antecedent is true, then the consequent must be
  true as well
• Being a mother is a necessary condition for
  giving birth to a baby.
• This is why modus tollens is deductively valid.
• p ? q
• q
• ? p

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• If you give birth to a baby, then youre a
  mother.
• You giving birth is a sufficient condition for
  being a mother.
• So, youve given birth to a baby only if you are
  a mother.
• You being a mother is a necessary condition for
  giving birth.
• So, you are a mother if youve given birth to a
  baby.

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another example

• P1) If Maria is alive, then shes breathing.
• The conditional (1st premise) states that
• Marias being alive is a sufficient condition for
  her breathing.
• So, on the assumption that we have the following
  premise
• P2) Maria is alive.
• It follows that
• C) She is breathing. (modus ponens)
• Conversely, Marias breathing is a necessary
  condition for her being alive.
• P2) Marias not breathing.
• C) Maria is not alive. (modus tollens)

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• If Maria is alive, then Maria is breathing.
• Marias being alive is a sufficient condition for
  her breathing.
• So, more generally, x is alive only if x is
  breathing.
• Marias breathing is a necessary condition for
  her being alive.
• So, more generally, x is breathing if x is alive.

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natural language if-thens

• If p, then q.
• Assuming p, q.
• Whenever p, q.
• Given p, q.
• Provided p, q.
• p only if q. (or Only if q, p.)
• A necessary condition of p is q.

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• p if q.
• This translates as if q, then p.
• So does
• p when q.
• p since q.
• p in case q.
• p so long as q.
• A sufficient condition of p is q.

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counterexamples

• An if-then conditional p ? q is false just in
  case
• p is not sufficient for q, or
• q is not necessary for p, or
• p is true while q is false.
• So, to show that a conditional is false, you must
  show that the antecedent (p) is true but the
  consequent (q) is false.
• Such a situation is called a counterexample.

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lets consider some examples

• Being red is a ? condition for being scarlet.
• Being a horse is a ? condition for being a
  mammal.
• Being a female is a ? condition for being a
  sister.
• Being a father is a ? condition for being a
  male.
• Being tall is a ? condition for being a good BB
  player.

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lets consider some examples

• Being red is a necessary condition for being
  scarlet.
• Being a horse is a sufficient condition for
  being a mammal.
• Being a female is a necessary condition for
  being a sister.
• Being a father is a sufficient condition for
  being a male.
• Being tall is a NEITHER condition for being a
  good BB player.

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• Being red is a necessary condition for being
  scarlet.
• If the ball is scarlet, then it is red.
• If the ball is red, then it is scarlet.
• Being a father is a sufficient condition for
  being a male.
• If he is male, then he is a father.
• If he is a father, then he is male.
• Being a three sided figure is a BOTH for being
  a triangle.
• If it is a 3-sided figure, then it is a triangle.
• If it is a triangle, then it is a 3-sided figure.

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bi-conditionals

• x is a triangle only if x is a three-sided figure
  (necessity)
• x is a triangle if x is a three-sided figure
  (sufficiency)
• x is a triangle if and only if x is a three-sided
  figure
• The last says that being a three sided figure is
  both necessary and sufficient for being a
  triangle.
• Since its the combination of two conditionals,
  this is called a biconditional.
• Conditional 1 x is a triangle ? x is a
  three-sided figure
• Conditional 2 x is a three-sided figure ? x is a
  triangle
• Bi-conditional x is a triangle ?? x is a
  three-sided figure

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analysis

• In certain cases, to give necessary and
  sufficient conditions is to give a definition or
  an analysis.
• To give an analysis of x is to state what x is.
• Analyses or definitions in our sense are not to
  be confused with what you find in dictionaries,
  which often simply list various uses of words
  without stating what it is to be that to which
  the words refer.
• Since one of the primary aims of philosophy is to
  understand the nature of things (to state what
  they are), philosophers are particularly
  interested in such biconditionals.

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• Consider a simple example of an analysis
• x is a bachelor if and only if
• (i) x is an adult,
• (ii) x is male, and
• (iii) x is unmarried.
• The first thing to notice is that the analysis is
  stated as a biconditional (if and only if).
• The second thing to notice is that we want
  analyses to hold necessarily.

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• For example, it turns out that no bachelors are
  over ten feet tall.
• Does this mean that the following is a good
  analysis?
• x is a bachelor if and only if
• (i) x is an adult,
• (ii) x is male,
• (iii) x is unmarried, and
• (iv) x is under ten feet tall.
• This is not a good analysis.
• Why? Because an adult unmarried male over ten
  feet tall would still be a bachelor.
• In other words, being under ten feet tall is not
  essential to being a bachelor its not part of
  what it is to be a bachelor.

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counterexamples, again

• An analysis is false just in case
• there is a possible situation in which one side
  holds while the other does not.
• To show that an analysis is false you simply have
  to find a possible situation in which one side of
  the biconditional is true while the other side is
  false that is, a possible situation in which
  the truth-values of the two sides differ.
• Again, this is called a counterexample.

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a philosophical example

• What is it to know that p (where p is any
  proposition)?
• Consider the view that knowledge is true belief.
• x knows that p iff
• (i) x believes that p, and
• (ii) it is true that p.
• In order to see if this is a good analysis, we
  need to evaluate this biconditional.
• To do this, we must ask
• Is each condition on the right hand side
  necessary for knowledge?
• Are the conditions on the right hand side jointly
  sufficient for knowledge?

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in-class exercise

• Give an analysis of love.
• x loves y iff ?

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extra credit

• Evaluate the following analysis of parent
• x is a (biological) parent of y iff
• (i) x is an ancestor of y, and
• (ii) x is not an ancestor of an ancestor of y.
• Find a counterexample to this analysis. (There is
  at least one.)