Logic

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Philosophy and Logic
The Process of Correct Reasoning
October 23, 2006
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Logic is the art science ofcorrect reasoning
Analysis Clarification Evaluation
Words Statements Arguments
of
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What is a word???
According to Websters New World Dictionary (2nd
College Edition), a word is (a) a speech sound,
or series of them, serving to communicate meaning
and consisting of at least one base morpheme with
or without prefixes or suffixes but with a
superfix a unit of language between the
morpheme and the complete utterance (b) a
letter or group of letters representing such a
unit of language, written or printed usually in
solid or hyphenated form . . . . italics added
OK, whats a morpheme, a base morpheme, a prefix,
a suffix, a superfix and what is meant by the
complete utterance? To find out, do some
research on this.)
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Statements anOFFICIAL definition
A statement (also known as a proposition) is a
verbal expression that is either true or false
and that may therefore be either affirmed or
denied.
A verbal expression is an expression in words,
either spoken or written.
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Problem Are statements (propositions) really
verbal expressions?

• Cant I believe, for example, that it is
  raining without saying or writing it?

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In other words, can a propositional mental
state (i.e., a belief) be considered a
statement (in a sense)???
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Another problem

• There may be various verbal expressions of the
  same statement or proposition.

It is raining. Es regnet. Il pleut. Esta
lloviendo.
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Still another problem

• Must a statement or proposition be either true or
  false?

• What about The present King of France is bald
  This sentence is false???

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The present King of France is bald.

• If there is no present King of France, how can it
  be true that he is bald?
• But is it false that he is bald? In that case,
  it would be true that he is haired. But how
  can that be since there is no King of France at
  present?

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Louie the Bald

• The present King of France is bald is a
  sentence and therefore a verbal expression.
• Is it a statement?
• If so, then some statements are (APPARENTLY)
  neither true nor false.

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However, . . .
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the great 20th century philosopher, Bertrand
Russell (1872-1970), thought that The present
King of France is bald and The present King of
France is not bald should be interpreted as
follows
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1. There is a present King of France, and he
is bald. 2. There is a present King
of France, and he is not bald.
According to Russell, both of these statements
are false.
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What makes conjunctions true as opposed to false?
P Q T T T T F F F F T F F F
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Is it possible that The present King of France
is bald is NOT a statement?

• What do you think and why?

And how about . . . .
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This sentence is false ? ? ?
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If This sentence is false is true . . . .

• then it is false (because it truly states that it
  is false) and
• if it is false, then its true (again because it
  truly states that it is false).
• Is the sentence both true and false at the same
  time??? How can that be???

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Very Hard Question

• Is the set of all sets that are not members of
  themselves
• a member of itself
• or not???

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What follows if it is?

• And what follows if it isnt?

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Sets that are and sets that arent members of
themselves

• Most sets are NOT members of them-selves.
• E.g., the set of all cats is not a cat the set
  of all tables is not a table the set of all
  human beings is not a human being so on.

• But there ARE sets that ARE members of
  themselves.
• E.g., the set of all countable things is a
  countable thing the set of all conceivable
  things is a conceive-able thing so on.

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But, once again,

• is the set of all sets that are not members of
  themselves a member of itself or not???

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Well, anyway . . . .
lets move on.
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For most or all PRACTICAL purposes, we can assume
that a STATEMENT is a verbal expression that is
EITHER TRUE OR FALSE and that may therefore be
either affirmed or denied.
Now,
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having discussed words and statements, lets talk
about ARGUMENTS.
The OFFICIAL DEFINITION of an ARGUMENT is as
follows
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An ARGUMENT is

• a group, series, or set of STATEMENTS
• in which one of the statements, known as the
  CONCLUSION,
• is claimed by the arguer to follow logically (by
  way of INFERENCE) from the other statements in
  the argument,
• which are known as PREMISES
• (and which the arguer claims to be TRUE).

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All arguments have the same basic structure or
format
1. Premise 2. Premise 3. Premise 4. Conclusion
Factual Claim (premises are true)
Inference
Inferential Claim - that the truth of the
conclusion follows logically (by way of
inference) from the ASSUMED truth of the premises
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Factual Claim Inferential Claim

• The factual claim in an argument is the claim,
  made by the arguer, that all of the premises in
  the argument are true (as opposed to false or
  unconvinc-ing).

• The inferential claim in an argument is the
  claim, made by the arguer, that the conclusion of
  the argument follows logically from its premises,
  assuming that the premises are true.

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How to (1) analyze and (2) evaluate an argument
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First, we need to find an argument to analyze and
evaluate.
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Suppose someone were to argue something really
silly, like
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All bats have two heads because all bats are
kangaroos, and all kangaroos have two heads.
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The argument must be subjected to a 6-step
analysis evaluation.
Step 1. Identify the conclusion. Step 2.
Identify the premises. Step 3. Set the argument
up in standard form.
These three steps constitute an argument
analysis. An argument evaluation consists of
the next three steps, which are
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Step 4. Evaluate the factual claim. Are
the premises true, false, or
unconvincing? Step 5. Evaluate the inferential
claim. Does the conclusion follow
logically from the premises (assuming that
they are true)? Step 6. Evaluate the argument as
a whole. Is it sound or unsound?
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Lets apply the six-step method to our sample
argument about kangaroos and bats.
(1) All bats have two heads because (2) all bats
are kangaroos and (3) all kangaroos have two
heads.
Step 1. Whats the conclusion?
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Step 1 Can you see that the conclusion of the
argument is All bats have two heads
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and that the premises are Step 2 All bats
are kangaroos and All kangaroos have two
heads?
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Step 3

• Thus, the logical (or standard) form of the
  argument is

1. All kangaroos have two heads. 2. All bats
are kangaroos. 3. All bats have two heads.
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Or to put it more abstractly,
1. All K is T. 2. All B is K. 3. All B is T.
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and even more abstractly,
Two- headed things
Kangaroos
Bats
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We have now

• (1) identified the conclusion of the argument,
• (2) identified the premises of the argument,
• and
• (3) represented the argument in STANDARD FORM.

That is what is meant by an ARGUMENT ANALYSIS.
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Now we need an ARGUMENT EVALUATION.
Is the argument successful (sound)?
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For the argument to be sound,

• the premises of the argument must be true (as
  opposed to false or unconvincing)
• and
• the conclusion of the argument must follow
  logically from the premises (assuming that they
  are true).

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Step 4 Are the premises of the argument true,
or false, or unconvincing?
Premise 1 Is it true or are you convinced that
all kangaroos have two heads? Premise 2 Is it
true or are you convinced that all bats are
kangaroos?
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This step is easy (in this case). It is obvious
to anyone in her (or his) right mind that both
premises in this argument are FALSE.
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Another point about Step 4 We need to explain
WHY we think the premises are true, false, or
unconvincing.
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Step 5 But what about the INFERENCE (or
INFERENTIAL CLAIM) in this argument?

• Does the conclusion follow logically from the
  premises (assuming that they are true)?
• In other words,
• IF all kangaroos were two-headed, and
• IF all bats were kangaroos,
• would it follow logically that all bats have two
  heads?

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It would, wouldnt it?
The inference (reasoning) in the argument is
good. The conclusion does follow logically
from the premises (on the assumption that the
premises are true, which is an assumption we
always make at Step 5).
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Step 6 Is the argument as a whole sound?

• Well, at Step 5 we saw that the inference
  (reasoning) in the argument is good,
• but at Step 4
• we found that both premises in the argument are
  false.

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• For an argument to be sound,
• all of its premises must be true (i.e., the
  factual claim in the argument must be
  justified) (Step 4), and
• the inference in the argument must be good (i.e.,
  the inferential claim in the argument must be
  justified) (Step 5).

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The argument we have been considering is UNSOUND
because, although it contains good reasoning, at
least one of its premises is not true.
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For an argument to be sound as opposed to unsound,

• both the factual claim and the inferential claim
  in the argument must be justified.
• If the factual claim is not justified (i.e., if
  at least one premise is false or unconvincing),
  then the argument is unsound.
• If the inferential claim is not justified (i.e.,
  if the conclusion does not follow logically from
  the premises, assuming that they are true), then
  the argument is unsound.

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And, of course,

• if NEITHER the factual claim NOR the inferential
  claim is justified (i.e., if the argument fails
  on both counts),
• then the argument is unsound.

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Lets now apply the six-step method of argument
analysis and evaluation to a few simple (and
unrealistic) arguments, beginning with this one
All cats are animals, and all tigers are cats.
Therefore, all tigers must be animals.
The argument contains three statements, right?
Which one of them is the conclusion (Step 1)?
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All cats are animals, and all tigers are cats.
Therefore, all tigers must be animals.
Step 2 What are the premises of this
argument? Step 3 What is the logical
(standard) form of the argument? (See next
slide)
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This is it, right?
1. All cats are animals. 2. All tigers are
cats. 3. All tigers must be (are) animals.
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Argument Evaluation
1. All cats are animals. 2. All tigers are
cats. 3. All tigers are animals.
Step 4 Is the factual claim justified? That
is, are both premises true (as opposed to false
or unconvincing)?
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Step 5 Does the conclusion follow logically
from the premises?
1. All cats are animals. 2. All tigers are
cats. 3. All tigers are animals.
That is, if all cats are animals, and if all
tigers are cats, does it follow that all tigers
are animals?
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If all cats are animals,
Animals
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and if all tigers are cats,
Animals
it looks like all tigers must be animals, right?
Cats
Tigers
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Step 6 Is the argument as a whole sound or
unsound?
That is, are the factual claim and the
inferential claim both justified? Are the
premises true (Step 4), and does the conclusion
follow logically from the premises (assuming that
they are true) (Step 5)?
1. All cats are animals. 2. All tigers are
cats. 3. All tigers are animals.
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What about the following argument?
Tigers must be cats because all cats are animals
and all tigers are also animals.
Class Participation Exercise Write a six-step
analysis evaluation of this argument.
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Heres the STANDARD FORM of the argument
1. All cats are animals. 2. All tigers are
animals. 3. All tigers are cats.
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A possible misconceptionat Step 6

• To prove that an argument is unsound is not to
  prove that its conclusion is false.

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Enthymemes, i.e., incompletely expressed arguments
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Deductive vs. nondeductive (inductive) arguments

• A deductive argument is one that contains a
  deductive inferential claim.
• A nondeductive (inductive) argument is one that
  contains a nondeductive inferential claim.

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Deductive vs. nondeductive inferential claims

• A deductive inferential claim is the claim, made
  by the arguer, that the truth of the conclusion
  follows with the force of absolute logical
  necessity from the assumed truth of the premises.
• A nondeductive inferential claim . . . .

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. . . is the claim, made by the arguer, that the
truth of the conclusion follows with some
significant degree of probability from the
assumed truth of the premises.
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Deductive inferential claims are either valid
or invalid.

• Nondeductive inferential claims are either
  strong or weak.

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A deductive inferential claim (or argument) is
valid

• when the truth of its conclusion follows
  necessarily from the assumed truth of its
  premises.

1. If Polly is a cat, then Polly is an
animal. 2. Polly is a cat. 3. Polly is an
animal.
is valid.
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A deductive inferential claim (or argument) is
invalid

• when the truth of its conclusion DOES NOT follow
  necessarily from the assumed truth of its
  premises.

1. If Polly is a cat, then Polly is an
animal. 2. Polly is an animal. 3. Polly is a
cat.
is invalid.
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A nondeductive inferential claim (or argument) is
strong

• when the truth of its conclusion follows with
  some significant degree of probability from the
  assumed truth of its premises.

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1. Millions of crows have been observed. 2. All
of them have been black. 3. All crows are black
(probably).
is strong.
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A nondeductive inferential claim (or argument) is
weak

• when the truth of its conclusion DOES NOT follow
  with any significant degree of probability from
  the assumed truth of its premises.

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1. The great majority of college professors are
politically liberal. 2. Patricia Quinn is a
college professor. 3. Patricia Quinn is
(probably) politically liberal.
is weak.
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Necessary Contingent Statements
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Statements (e.g., premises in an argument) are
either true or false.

• However, some statements are necessarily true or
  false,
• while others are contingently true or false.
• Thus, there is a distinction between necessary
  and contingent statements (or propositions).

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Necessary statements

• Necessarily true -- formal and informal
  tautologies
• Necessarily false -- formal and informal
  contradictions (re the law of non-contradiction)
• A priori verification falsification (i.e.,
  verification or falsification by logical analysis
  alone (no empirical appeal)

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Tautologies contradictions

• The negation of a tautology is a contradiction,
  and the negation of a contradiction is a
  tautology.

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Some tautologies their negations

• Either angels exist, or they dont.
• All triangles have three sides.
• Every effect has a cause.
• If God would not permit the existence of pain and
  pain exists, then God does not exist.

• Angels both exist and do not exist.
• Some triangles do not have three sides.
• Some effects are uncaused.
• If God would not permit the existence of pain and
  pain exists, then the existence of God is still
  possible.

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Some contradictions their negations

• It is raining, and it is not raining.
• Polly is a cat, but she is not an animal.
• Johns siblings are all males, but Mary is Johns
  sister.
• A perfectly good being is partly evil

• Either it is raining, or it is not raining (not
  both).
• All cats are animals
• If Johns siblings are all males, then John has
  no sisters.
• A perfectly good being is not evil at all.

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Contingent statements

• Neither necessarily true (tautology) nor
  necessarily false (contradiction)
• True under some conditions false under others
• A posteriori verification falsification (i.e.,
  verification or falsification on empirical
  grounds -- not by logical analysis alone --
  verification/falsification not always possible)

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Some Contingent Statements

• There are rocks (or rocks exist).
• Washington DC is the capital of the US.
• Oranges are not grown in Antarctica.
• Unicorns exist.
• Abraham Lincoln is now President of the US.
• There are palm trees growing on the moon.

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Tautologies, contradictions, contingent
statements

• The negation of a tautology is a contradiction
  the negation of a contradiction is a tautology
  and the negation of a contingent statement is
  neither a tautology nor a contradiction, but
  another contingent statement.

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A Final Point

• Necessary statements can be proved (tautologies)
  or disproved (contradictions) via logical (a
  priori) analysis of their form or meaning.
• Many contingent statements can be proved or
  disproved a posteriori (empirically), but some
  are (at least at present) beyond verification and
  falsification because the needed evidence is
  unavailable. In some cases, the evidence will
  become available in time in other cases, the
  evidence will never be available.

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To Be

• Continued . . . .