Philosophy 115 Lecture 3 Formalizing an argument

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Philosophy 115Lecture 3Formalizing an argument

• By David Kelsey

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Evaluating an argument

• When one comes across an argument her task is to
  evaluate it.
• To evaluate an argument is to critique it. It is
  to inspect the argument for flaws?
• You are out to determine if there is anything
  wrong with the argument.
• So you look at every premise to see if it might
  be false.
• And you look at the conclusion to see if it might
  be false.
• And you look at the pattern of reasoning which
  the argument gives leading from the premises to
  the conclusion, to see if that pattern is valid
  or strong or neither.
• Before one can evaluate an argument she must
  understand the argument.
• She must understand what is meant by each
  premise. And she must understand what is meant by
  the conclusion.
• And she must understand just how it is that the
  premises are supposed to support the conclusion.
• She must understand the argument as its author
  does.
• In order to fully understand an argument one must
  formalize the argument.

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Formalizing an argument

• To formalize an argument is to break that
  argument down into its most simplified form.
• In formalizing an argument one writes down the
  premises and the conclusion of the argument in
  the form of sentences.
• Her aim in writing these sentences down is to
  simplify them as much as is possible.
• She is then to arrange the sentences in an order
  that fits the structure of the argument.
• Once she has the order right she can then
  evaluate it. She can look to see if it has any
  flaws, for instance an invalid pattern of
  reasoning or a false premise.

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The principle of charity

• In formalizing an argument one must make sure
  that when she writes down the premises and
  conclusion of the argument, she writes what the
  arguments author intended.
• One must make sure to abide by the principle of
  charity.
• The guiding principle in formalizing an argument
  is simply this give the author the benefit of
  the doubt.
• If the argument appears to be flawed as it stands
  then consider what premise might make the
  argument valid and give the author the benefit of
  the doubt. He probably intended that the premise
  be included in the argument but just left it out
  for one reason or another.
• If I gave the argument about bloodhounds you
  wouldnt say it was flawed because I didnt
  include in it the implicit premise that all
  bloodhounds have a keen sense of smell. Instead,
  you would just assume that this premise is part
  of my argument and then evaluate the argument as
  such.

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Simplify and number

• In formalizing an argument one must make sure to
  simplify each premise into as few words as is
  possible.
• Once we have simplified our premises and
  conclusion we number them.
• In numbering them we must make sure that any
  inference the argument makes follows what it is
  inferred from.
• Thus, in formalizing an argument, its conclusion
  must follow each of the premises of the argument.

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The structureof an argument

• As was just stated, once we have simplified our
  premises and conclusion we must number them.
• In numbering we must make sure that any inference
  the argument makes follows what it is inferred
  from.
• So we need some means of making clear the
  inferences an argument makes. We need some means
  of clarifying an arguments structure.

• An arguments structure is its pattern of
  reasoning from the first premise to the
  conclusion.
• An arguments structure is the path it follows
  from its first premise to its conclusion.
• The structure of an argument is the way in which
  its premises lead to its conclusion.

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Clarifying an arguments structure

• As was stated earlier, in formalizing an argument
  we pick the propositions of the argument out of
  the text, then simplify those propositions, then
  number them.
• Start out by numbering the propositions of the
  argument according to the order in which they
  fall in the text itself.
• Thus, at this point we number the premises and
  conclusion irrespective of having to make sure
  any inference the argument makes follows what it
  is inferred from.
• Thus, at this point some of our numbers will be
  out of order.

• Once we have all of the propositions of the
  argument numbered, we can then use these numbers
  along with the help of some other symbols to
  clarify the structure of the argument.

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Symbols

• We will use a downward pointing arrow ? to
  stand for any inference the argument makes.
• So when we have one proposition Q inferred from
  another P we write
• P
• ?
• Q
• So ? stands for thus, hence, therefore,
  is intended as a reason for, is intended as a
  premise for, etc.

• When you have two or more propositions, PQ, that
  dependently support some other proposition of the
  argument, R, we put a sign between P and Q
  and then underline all of it, I.e. PQ. From
  the middle of the underline we then place a
  downward pointing arrow (?).
• Like this
• PQ
• ?
• R

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Symbols 2

• When we have two or more propositions, P and Q,
  that independently support some third proposition
  of the argument, R, we make sure that each
  premise is isolated with an arrow pointing down
  to the supported proposition.
• Like this
• P Q
• ??
• R

• And when we have a proposition, P, that supports
  more than one proposition of the argument, Q and
  R, we write P downward pointing arrow Q, and P
  downward pointing arrow R in the shape of a
  triangle like this
• P
• ??
• Q R

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Symbols 3

• When we two propositions, P and Q, that
  dependently support another, S, and we also have
  a fourth proposition, R, that independently
  supports S we write it all out like this
• S PQ
• ? ?
• R

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Counter-arguments

• When we have a proposition, P, that is a reason
  against some other proposition of the argument,
  Q, we write P down arrow Q. We then put a slash
  mark through the arrow like this
• P
• ?
• ?
• Q
• (Please Note that we should put the slash all the
  way through an uninterrupted arrow but I couldnt
  find the symbol for this!)
• Notice that P in this case is a reason given in
  favor of the conclusion that some proposition
  given in the argument is false.
• So P might be the claim that the conclusion of
  the argument is false or it might be the claim
  that some premise of the argument is false.

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Counter-arguments 2

• When one runs into a claim that runs counter to
  her own argument she wants to try and defend her
  argument, A, against this counter claim.
• In defending A, one must consider the argument
  given in favor of the counter claim, the counter
  argument or C.
• If one can show that C is unsound she has
  defended A.
• To show that C is unsound one must first
  formalize C.
• One must then evaluate C with the aim of showing
  that either some premise of C is false or C is
  invalid.
• One can then try to make an argument to the
  conclusion that C is unsound.
• If C is an inductive argument, then she can
  defend A against C if she can show that C isnt
  cogent.
• To show that C isnt cogent she must formalize C.
• She must then evaluate C with the aim of showing
  that either some premise of C is false or that C
  is a weak argument.
• She can then make an argument to the conclusion
  that C is not cogent.

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An example of a formalization

• The passage
• I dont think we should get Carlos is own car.
  As a matter of fact, he is not responsible
  because he doesnt care for his things. And
  anyway, we dont have enough money for a car,
  since even now we have trouble making ends meet.
  Last week you yourself complained about our
  financial situation, and you never complain
  without really good reason.
• Our first task in formalizing this argument is to
  find the sentences in which the premises and the
  conclusion are contained.
• What we are looking for here are the propositions
  of the argument, both the premises and the
  conclusion.
• Looking for premise indicators will help us to
  find these propositions.
• We see, for instance, the premise indicators
  because and since in the second and third
  sentences.

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The first example 2

• Let us now just compose a list of the
  propositions of the argument.
• We will compose this list in the order in which
  the propositions come in the original passage.
• The List
• I dont think we should get Carlos his own car.
• As a matter of fact, he is not responsible.
• He doesnt care for his things
• And anyway, we dont have enough money for a car.
• Since even now we have trouble making ends meet.
• Last week you yourself complained about our
  financial situation.
• You never complain without really good reason.

• Let us now simplify the propositions in our list
  and then number them in the order we have them
  listed
• 1) We shouldnt get Carlos his own car.
• 2) Carlos is not responsible.
• 3) Carlos doesnt care for his things
• 4) We dont have enough money for a car.
• 5) We have trouble making ends meet.
• 6) Last week you complained about our financial
  situation.
• 7) You never complain without really good reason.

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The first example 3

• Now that we have numbered the propositions of our
  argument let us clarify the structure of the
  argument. We can do so using the numbers just
  assigned plus our assortment of logical symbols.
• We know that because is a premise indicator so
  it appears that 3 is in support of 2. Thus, we
  can write
• 3
• ?
• 2
• But we also know that 2 is offered as support for
  1. Thus, we can write
• 2
• ?
• 1

• We also know that since is a premise indicator
  so it appears that 5 is in support of 4. Thus,
  we write
• 5
• ?
• 4
• Looking closely at 6 and 7 though, we can see
  that they are both dependently in support of 4.
  Thus, we write
• 67
• ?
• 4

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The first example 4

• So far we have the following
• 67 and we have 5
• ? ?
• 4 4
• We can now combine all of this symbolization,
  making the shape of a triangle, like this
• 67 5
• ? ?
• 4
• But in looking at 4 though we can see that it
  supports 1. (Of course the claim we dont have
  enough money for a car is in support of the claim
  We shouldnt get Carlos his own car.) Thus, we
  write
• 4
• ?
• 1

• We already saw from before that 2 also supports
  1. We see now that it does so independently of
  4s support for 1. And so we can combine all of
  the symbolization into this
• 3 5 67
• ? ? ?
• 2 4
• ? ?
• 1

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Formalizing Example 1

• Once you have clarified the structure of an
  argument you can finish your formalization of it.
• Since you initially numbered the propositions of
  the argument as they came in the passage, you
  must now renumber the propositions of the
  argument to map onto its structure.
• In renumbering, make sure that any inference the
  argument makes follows what it is inferred from.

• Here is what the argument looks like after
  renumbering the propositions
• 1) Carlos doesnt care for his things.
• Thus, 2) Carlos isnt responsible. (from 1)
• 3) Last week you complained about our financial
  situation.
• 4) You never complain without really good reason.
• 5) We have trouble making ends meet now.
• Thus, 6) We dont have enough money for a car.
  (from 34 and 5.)
• Thus, 7) We shouldnt get Carlos his own car.
  (from 2 and 6.)
• So we have our finished formalization at last!
  We can now set about evaluating it.
• For now though, lets push on to a more difficult
  example so we can evaluate it.

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A Second Example

• Let us now look at a more difficult example of a
  formalization.
• The following passage is taken from The
  Ontological Argument by Saint Anselm.
• So Lord--you who reward faith with
  understanding--let me understand, insofar as you
  see fit, whether you are as we believe and
  whether you are what we believe you to be. We
  believe you to be something than which nothing
  greater can be conceived. The question, then, is
  whether something with this nature exists, since
  the fool has said in his heart that there is no
  God Ps. 141, 531. But surely, when the fool
  hears the words something than which nothing
  greater can be conceived, he understands what he
  hears, and what he understands exists in his
  understanding--even if he doesnt think that it
  exists. For it is one thing for an object to
  exist in someones understanding, and another for
  him to think that it exists.

• And the passage continues
• This should convince even the fool that something
  than which nothing greater can be conceived
  exists, if only in the understanding--since the
  fool understands the phrase that than which
  nothing greater can be conceived when he hears
  it and whatever a person understands exists in
  his understanding. And surely that than which a
  greater cannot be conceived cannot exist just in
  the understanding. If it were to exist just in
  the understanding, we could conceive it to exist
  in reality too, in which case it would be
  greater. Therefore, if that than which a greater
  cannot be conceived exists just in the
  understanding, the very thing than which nothing
  greater can be conceived is something than which
  a greater can be conceived. But surely this
  cannot be. Without doubt, then, something than
  which a greater cant be conceived does
  exist--both in the understanding and in reality.

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Premise and Conclusion indicators

• Now that we have the passage lets look for
  premise and conclusion indicators
• But surely in the sentence But, surely, when
  the fool hears the words something than which
  nothing greater can be conceived. is a premise
  indicator.
• And This should convince in the sentence This
  should convince even the fool that something than
  which nothing greater can be conceived exists
  is a premise indicator.

• Therefore in the sentence Therefore, if that
  than which a greater cannot be conceived exists
  just in the understanding is a conclusion
  indicator as well.
• This merely indicates an inference Anselm has
  made in order to get to his conclusion.
• Without doubt, then in the sentence Without
  doubt, then, something than which a greater cant
  be conceived does exist indicates the
  conclusion of Anselms argument.

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The Argument

• Let us now write the premises and conclusion of
  the Anselm argument.
• In doing so we must make sure to write down both
  explicit and implicit premises.
• And we must make sure to write what the author
  intended.

• The argument
• When one hears the words something than which
  nothing greater can be conceived, he understands
  what he hears.
• Whatever a person understands exists in his
  understanding.
• Something than which nothing greater can be
  conceived exists in the understanding.
• If something than which nothing greater can be
  conceived were to exist just in the
  understanding, we could conceive it to exist in
  reality too, in which case it would be greater.
• Therefore, if that than which a greater cannot be
  conceived exists just in the understanding, the
  very thing than which a greater cannot be
  conceived is something than which a greater can
  be conceived.
• Without doubt, then, something than which a
  greater cant be conceived does exist.

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Simplify and number

• Taking our formalization to the next step, lets
  simplify each premise of the argument into as few
  words as is possible.
• When one hears the words something than which
  nothing greater can be conceived, (or SOMETHING)
  he understands what he hears.
• Whatever a person understands exists in his
  understanding.
• Thus, SOMETHING exists in the understanding.
• If SOMETHING were to exist just in the
  understanding, we could conceive it to exist in
  reality too.
• If SOMETHING were to exist in reality too, it
  would be greater.
• Thus, if SOMETHING exists just in the
  understanding, it is something than which a
  greater can be conceived.
• Thus, SOMETHING does exist in reality.

• Once we have simplified our premises and
  conclusion we number them, making sure that any
  inference the argument makes follows what it is
  inferred from.
• 1) When one hears the words something than
  which nothing greater can be conceived, (or
  SOMETHING) he understands what he hears.
• 2) Whatever a person understands exists in his
  understanding.
• 3) Thus, SOMETHING exists in the understanding.
• 4) If SOMETHING were to exist just in the
  understanding, we could conceive it to exist in
  reality too.
• 5) If SOMETHING were to exist in reality too, it
  would be greater.
• 6) Thus, if SOMETHING exists just in the
  understanding, it is something than which a
  greater can be conceived.
• 7) Thus, SOMETHING does exist in reality.

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Evaluating the formalization

• Once an argument is formalized you must then
  evaluate it. You must determine if it is a good
  argument.
• If the argument is deductive then you can first
  determine if it is valid.
• If the argument is inductive then you can first
  determine if it is strong
• If you find a conclusion given only weak support,
  then the argument has made a weak inference.
• When an argument has made a weak inference the
  argument can be challenged.

• Consider this argument
• 1) All of the 10 ravens I have ever seen
  were black.
• Thus, 2) All ravens are black.
• We might challenge the move from the
  premise to the conclusion just because the
  support given in the premise for the conclusion
  is weak and thus the inference made from the
  premise to the conclusion is weak.

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Evaluating the argument

• Your second step in evaluating an argument is to
  examine its premises.
• You are to examine each premise individually
  determining whether you think it is true, false
  or dubious.
• If you think any of the premises are false or
  dubious, then, of course, the argument can also
  be challenged.

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Evaluating Anselms argument

• What about the Anselm argument.
• Does premise 1 seem dubious to anyone?
• Do you understand the words Something than which
  nothing greater can be conceived when you hear
  them?
• Can one really understand something so great?
• I have a hard time understanding how the engine
  on my truck works and how my computer works and
  these things arent anywhere near as great as
  something than which nothing greater can be
  conceived.
• What about premise 4?
• Even if we could understand something so great
  can we understand what it means for this thing to
  exist in reality? What is it for God to exist in
  reality? Just where might he exist?
• Another Galaxy?
• Here on Earth?
• Everywhere? How does that work? Is God in me
  right now? That would surely be weird wouldnt
  it!