Logic: With an Emphasis on Deductive Logic and How to Use it.

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Logic With an Emphasis on Deductive Logic and
How to Use it.

• By Aaron Rodriguez
• Crystal Hinojos
• Michelle Barnes

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History of Logic

• For over two thousand years logicians and
  philosophers have been analyzing the character of
  valid arguments and mathematicians have been
  making correct systematic inferences
• Even with this fact, a formal theory of inference
  has been developed only in the last three or four
  decades.
• Since the 4th century B.C., philosophers such as
  Aristotle attempted to explore logic and apply it
  to life in general
• Until around the 18th century, philosophers had
  not yet related logic as the theory of inference
  to the kind of deductive reasoning that are
  continually used in math

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What is Logic

• According to an Intro to Logic book, logic is
  the study of the methods and principles used to
  distinguish correct reasoning from incorrect
  reasoning.
• There are two different kinds of logic Inductive
  Logic and Deductive Logic
• Neither of the two forms of logic are exclusive
  to one specific subject.
• People use logic in everyday conversation and in
  virtually every aspect of their lives.
• Studying logic is not a necessary nor sufficient
  condition for applying the concepts of logic
  efficiently.
• The concern in logic is form and not content.

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Inductive Logic

• Inductive logic is going from one or more
  particular examples and reaching a general
  conclusion from them
• It is commonly referred to as common sense
• An example of inductive logic is In every case
  that the UTEP soccer team has played against
  NMSU, they have one. Therefore, it is likely that
  if the UTEP soccer team plays NMSU this weekend,
  then they will win.
• This kind of logic rests on probability, and the
  more times an instance occurs, the more probable
  it will be that it will occur again.

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Deductive Logic

• Deductive logic is going from true premises to
  reaching a true conclusion.
• In using this form of logic one must ask what
  must be true based on the premises.
• An example of deductive logic is If NMSU does
  not show up to the soccer game, then UTEP will
  win. NMSU will not show up. Therefore, UTEP will
  win.
• Deductive logic is stronger then inductive logic
  in the sense that it relies on certainty rather
  than on probability.

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Difference between Form and Content

• A major pitfall within the subject of deductive
  logic is being concerned with content over form.
• Content is an analysis of the arguments subject
  matter to determine whether it is true or false
• Form is analyzing the arguments validity in the
  way it reaches a conclusion from the given
  premises.
• An example of an argument with a valid form and
  false content is When it rains outside every
  freeway remains completely dry. It is raining
  outside. Therefore every freeway is completely
  dry.
• The statements are actually absurd, but the form
  is completely valid.

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Valid Arguments Modus Ponens

• One form of a valid argument is modus ponens
  (otherwise known as affirming the antecedent).
• Modus Ponens takes the form of
• If A then C A Therefore C.
• This form was seen in an earlier example
  regarding the UTEP soccer team.
• Another example If you commit a crime you will
  be arrested. You committed a crime. Therefore,
  you will be arrested.
• In this case, the subject committing a crime is a
  sufficient condition to conclude that the subject
  will be arrested.

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Valid Arguments Modus Tollens

• Another form of a valid argument is modus tollens
  (otherwise known as denying the consequent).
• Modus Tollens takes the form of
  If A then C not C therefore,
  not A.
• An example of this is If you commit a crime, you
  will be arrested. You were not arrested.
  Therefore you did not commit a crime.
• In this case, not being arrested is a sufficient
  condition to conclude that the subject did not
  commit a crime.

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Invalid Arguments Affirming the Consequent

• An argument that affirms its consequent is an
  invalid argument.
• Affirming the Consequent is when the argument
  takes the form If A then C C therefore A.
• If you commit a crime you will be arrested. You
  will be arrested. Therefore, you committed a
  crime.
• This is an invalid argument because, while the
  argument treats it as such, being arrested is not
  a sufficient condition for committing a crime.

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Invalid Arguments Denying the Antecedent

• An argument that denies its antecedent is an
  invalid argument.
• Affirming the Consequent is when the argument
  takes the form If A then C not A therefore not
  C.
• An example of this is If you commit a crime, you
  will be arrested. You did not commit a crime.
  Therefore, you will not be arrested.
• This is an invalid argument because, while the
  argument treats it as such, not committing a
  crime is not a sufficient condition for not being
  arrested.

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Symbolic Logic

• To make deductive logic easier to use for
  analysis, one can utilize symbols to represent
  the assumptions and the conclusions within the
  arguments.
• One way of doing this is by assigning to
  statements any letter other than x, y, or z (x,
  y, and z are generally reserved for individual
  variables).
• Symbols for Propositional Logic using the letters
  P and Q.
• Conditional (If- Then) If P then Q
• Conjunction (And) P Q
• Disjunction (Or) P v Q
• Biconditional (If and Only If) P Q

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Propositional Logic

• Propositional logic is concerned solely with the
  four types of statements we have learned how to
  symbolize conditional, conjunction, disjunction,
  and biconditional.
• Another symbolization that is important to use in
  propositional logic is the hyphen when meaning to
  say not (i.e. -p for not p).
• In propositional logic, it is possible to
  translate one type of statement into another type
  of statement because there is more then one way
  to interpret a statement.

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Translations

• It can be very helpful to be comfortable in
  translating one type of statement into a
  different type when trying to prove that one
  thing implies another and when seeing if that
  something is implied by something else.
• Examples of translations
• From P Q derive -(-P v-Q) and vice versa
• From P v Q derive -P ? Q and vice versa
• From P Q derive -( P ? -Q) and vice versa
• From (P ? Q and Q ? P) derive P Q and vice
  versa

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Using Translations for Proofs

• When deriving (P v Q -Q v P) from P Q, we
  can use the translations already at our disposal
  to derive this conclusion.
• P Q means (P ? Q Q ? P)
• P? Q means P v Q
• Q ? P means Q v P
• therefore, P Q means (-P v Q -Q v P).
• Similarly we derive (P -Q) (-P ? -Q) from P
  Q
• P Q means (P ? Q Q ? P)
• P ? Q means (P -Q) and Q ? P means P ? -Q
• therefore, P Q means (-P v Q -Q v P).

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Predicate Logic

• Propositional Logic does not account for every
  statement.
• One way in which predicate logic differs from
  propositional logic is while propositional logic
  is the logic of and, or, if, iff and
  not, predicate logic is the logic of all,
  some, none, and related terms.
• Another difference between the two areas of logic
  is that propositional logic can only work by
  analyzing compound statements, while predicate
  logic is capable of analyzing simple statements.

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Symbols for Predicate Logic

• A possible set of symbols that can be used for
  Predicate Logic (taken from Premises and
  Conclusions) is
• individual constants a, b, c, ,v
• individual variables x, y, z
• universal quantifiers (x), (y), (z)
• existential quantifiers ( x), ( y), ( z)
• predicate letters A, B, C, , Z
• Only the meanings assigned to individual
  constants and predicate letters in predicate
  logic are subject to change.

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Functions of Symbols

• Individual constants abbreviate singular terms,
  that is, names (Bill Clinton), pronouns
  (she), and descriptive phrases that refer to a
  single individual (the richest American).
• Individual variables are cross-reference devices.
  They are the subject that the quantifier refers
  to in a statement.
• Predicate letters are placed before an individual
  variable to assert something about that variable.

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Functions of More Symbols

• Existential quantifiers assert that there exist
  at least one of the individual variable being
  predicated.
• Universal quantifiers assert that if there is
  something within the universe that the individual
  variable being quantified was said to be in, then
  that something must necessarily carry with it the
  attributes it was asserted to have by the
  predicate letters within the statement. In other
  words, if your individual variable falls under
  the classification of the universe being
  described, then that individual variable must
  necessarily carry with it the attributes it is
  asserted to have.

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Examples of Predicate Logic

• All mammals are furry.
• Symbolization (x) (Mx ? Fx)
• No mammals are furry.
• Symbolization (x) (Mx ? -Fx)
• Some mammals are furry.
• Symbolization ( x) (Mx Fx)
• Some mammals are not furry.
• Symbolization ( x) (Mx -LX)

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Logically Synonymous Phrases

• Saying that all A is B is the same as saying
  every A is B, each A is B, A are B, and any A is
  B.
• Saying that no A are B is the same as saying A
  are not B, No one is both A and B, There are no A
  Bs, and A are never B.
• Saying that some A are B is the same as saying
  there are A that are B, at least one A is B, A
  that are B exist, and A are sometimes B.
• Saying that, some A are not B, is the same as
  saying there are A that are not B, at least one A
  is not B, not all A are B, and A are not always
  B.

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Sources

• Copi, Irving M., and Cohen Carl, Introduction to
  Logic, Prentice Hall Inc. 2002.
• Pospesel, Howard, and Rodes Robert, Premises and
  Conclusions, Prentice Hall Inc., 1997.
• Quine, W.V. Methods of Logic 4th ed., Harvard
  University Press, 1982.
• Suppes, Patrick, Introduction to Logic, Dover
  Publications Inc., 1957.