Logic Workshop

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Logic Workshop
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Expectations?

• What do you expect to get out of this workshop?
• What do you currently know about logic?

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Schedule

• Day One
• Introduction to logic (what is it? why is it
  important?)
• Historical connections within the context of
  philosophy
• Major philosophers and their contributions to
  logic
• The building blocks of logic
• Day Two
• Applications of logic
• Symbolic logic
• Paradoxes and other interesting philosophical
  questions

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Logic 101

• ?????

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Greek

• ????? logos
• From Classical Greek
• Essentially means thought, idea, argument,
  account, reason, or principle
• A very important term in philosophy (as well as
  in analytic psychology, rhetoric, religion)
• Derives from
• Logic is about just this (logos) its about
  developing an understanding of the methods and
  principles used to distinguish between correct
  and incorrect reasoning
• Traditionally considered a branch of philosophy
• Today it has also spread to other fields such as
  mathematics (e.g. mathematical logic) and
  computer science (e.g. programming, writing code)

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Everyday Use

• We use logic everyday
• Arguments, justifications, reasons, most all
  rational functions
• Listen to just how many logical arguments (with
  premises and conclusions) you hear on TV shows,
  the radio, etc.
• Therefore its important to develop our logical
  knowledge

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because if we dont

• then we might be the victim of Bad Reasoning
• But, as the Stanford Encyclopedia of Philosophy
  notes, logic does not, however, cover good
  reasoning as a whole. That is the job of the
  theory of rationality. Rather it deals with
  inferences whose validity can be traced back to
  the formal features of the representations that
  are involved in that inference, be they
  linguistic, mental, or other representations
• http//www.youtube.com/watch?vG40OEBuIZdM

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Brief Introduction to Logic

• Logic the organized body of knowledge, or
  science, that evaluates arguments
• Our aim then is to develop system of methods to
  use as criteria for evaluating arguments of
  others and for constructing our own to determine
  good arguments from bad arguments

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Logic Evaluates Arguments

• Argument a group of statements, one or more of
  which (premises) are claimed to provide support
  for, or reasons to believe, one of the others
  (conclusions)
• Good argument premises support the conclusion
• Bad argument premises do not support conclusion
  (even if they claim to)
• Syllogism a kind of logical argument in which
  one proposition (the conclusion) is inferred from
  at least two others (the premises) of a certain
  form

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Arguments are Made up of Statements

• Statement a sentence that is either true (T) or
  false (F)
• Melatonin helps relieve jet lag. (T)
• No wives ever cheat on their husbands. (F)
• and is made up of
• 1. Premises
• Statements that set forth the reasons or evidence
• 2. Conclusions
• Statements that the evidence is claimed to
  support or imply (claimed to follow from the
  premises)

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Propositions

• Proposition the meaning or information content
  of a statement (the material of our reasoning)
• PropositionStatement
• Can be simple (making only one assertion) or
  compound (containing two or more simple
  propositions
• Inference a process of linking propositions by
  affirming one proposition on the basis of one or
  more other propositions
• InferenceArgument
• Many sentences, unlike statements, cannot be said
  to be T or F
• Questions (Where is Tom?) Proposals (Lets go to
  a movie.) Suggestions (I suggest you get contact
  lenses.) Commands (Turn off the TV.)
  Exclamations (Wow!)

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Good vs. Bad Arguments

• Good Argument
• 1. All film stars are celebrities. (Premise 1)
• 2. Halle Berry is a film star. (Premise 2)
• C. Therefore, Halle Berry is a celebrity.
  (Conclusion)
• Bad Argument
• 1. Some film stars are men.
• 2. Cameron Diaz is a film star.
• C. Therefore, Cameron Diaz is a man.

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Example

• The space program deserves increased
  expenditures in the years ahead. Not only does
  the national defense depend upon it, but the
  program will more than pay for itself in terms of
  technological spinoffs. Furthermore, at current
  funding levels the program cannot fulfill its
  anticipated potential.

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Argument Reconstruction

• Break up compound statements
• List premises first, then conclusionsP1 The
  national defense is dependent upon the space
  program.P2 The space program will more than
  pay for itself in terms of technological
  spinoffs.P3 At current funding levels the
  space program cannot fulfill its anticipated
  potential.C The space program deserves
  increased expenditures in the years ahead.

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Classes of Arguments

• Inductive Argument
• Claim that the premises support the conclusion
  with a certain degree of probability
• If the premises are true, then its likely the
  conclusion is true
• Ex. This cat is black. That cat is black. A third
  cat is black. Therefore all cats are black.
• Deductive Argument
• Claim that the premises support the conclusion
  conclusively there is no probability involved,
  but logical certainty
• If the premises are true, then its impossible
  the conclusion is false
• Ex. All men are mortal. Joe is a man. Therefore
  Joe is mortal.

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Deductive Arguments

• Look for
• Validity if all premises are true, then the
  conclusion must be true (follows from the
  structure, not the content - no matter how
  ridiculous it may seem - of an argument)
• Soundness if the argument is valid, and the
  premises are true, then the argument is sound
• All sound arguments are valid, but not all valid
  arguments are sound
• Cogency if the argument is valid and sound, is
  it cogent (convincing) to a community of
  listeners
• If all sound arguments are valid, must all cogent
  arguments be sound?

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Belvedere by M.C. Escher

• A structure in which the relations of the base to
  the middle and upper portions are not rational
• A deductive argument rests upon premises that
  serve as its foundation
• To succeed, its parts must be held firmly in
  place by the reasoning that connections those
  premises to all that is built upon them this
  way the argument can stand

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Waterfall by M.C. Escher

• Doesnt make sense the water flows away, and
  going away comes closer
• Flows downward, and going down it comes up,
  returning to the point from which it began
• As perception may be tricked by a clever picture,
  our thinking may be tricked by a clever argument
• In the picture we confront disorder in seeing,
  and then with scrutiny detect its cause
• In logic we confront many bad arguments, and then
  with scrutiny learn what makes them bad

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History

• Logic as a discipline started with the transition
  from the unreflective use of logical methods and
  argument patterns to the reflection on and
  inquiry into these and their elements, including
  the syntax and semantics of sentences
• Mesopotamia (11th century BC)
• Medical diagnostic handbook containing numerous
  axioms and assumptions (i.e. through examination
  and inspection of the symptoms of a patient, it
  is possible to determine the patient's disease,
  its aetiology and future development, and the
  chances of the patient's recovery)
• Babylon (8th 7th centuries BC)
• Babylonian astronomers began employing an
  internal logic within their predictive planetary
  systems, which was an important contribution to
  logic and the philosophy of science.
• Babylonian thought had a considerable influence
  on early Greek thought

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Greece (4th century BC)

• Logic as a fully systematic discipline begins
  with Aristotle (384-322 BC)
• He systematized much of the logical inquiries of
  his predecessors
• Main achievements
• Term Logic
• Fundamental elements of his syllogistic logic are
  terms, which are not true or false in
  themselves, such as human being, animal,
  white (i.e. H, A, and W)
• For Aristotle, a term is simply a thing, a part
  of a proposition and a proposition is just a
  particular kind of sentence, in which the subject
  and predicate are combined so as to assert
  something true or false
• The essential feature of term logic is that, of
  the four terms in the two premises, one must
  occur twice. Thus
• All Greeks are men
• All men are mortal.
• This system became known as the categorical
  syllogistic

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Another System Emerges

• Chrysippus (279-206 BC)
• Was a stoic
• Stoicism teaches the development of
  self-control and fortitude as a means of
  overcoming destructive emotions holds that
  becoming a clear and unbiased thinker allows one
  to understand the universal reason, logos
• Main achievements
• Propositional Logic
• Fundamental elements of logic are whole
  propositions - something over and above the
  terms are true
• Studies ways of joining and/or modifying entire
  propositions, statements, or sentences to form
  more complicated ones
• Also looks at the logical properties derived from
  these methods of combining or altering statements
  (e.g. logical connectives in symbolic logic).
• Does not study those logical properties that
  depend upon parts of statements that are not
  themselves statements on their own, such as the
  subject and predicate of a statement
• This system became known as the hypothetical
  syllogistic

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1300-year Intermission

• 200 BC 1100 AD
• Lack of logical innovations
• Mostly just commentators either preserving
  Aristotelian logic, such as Boethius (480-524
  AD), or criticizing it, such as Islamic
  philosopher Avicenna (980-1037 AD)

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• Peter Abelard (1079-1142 AD)
• Theory of universals in the mind rather than
  natures outside the mind (as Aristotle argued)
• Distinguishes between arguments valid in form and
  arguments valid in content
• Abelard observes that the same propositional
  content can be expressed with different force in
  different contexts
• the content that Socrates is in the house is
  expressed in an assertion in Socrates is in the
  house in a question in Is Socrates in the
  house? in a wish in If only Socrates were in
  the house! and so on.
• Hence Abelard can distinguish the assertive force
  of a sentence from its propositional content, a
  distinction that allows him to point out that the
  component sentences in a conditional statement
  are not asserted, though they have the same
  content they do when asserted
• Ex. "If Socrates is in the kitchen, then Socrates
  is in the house" does not assert that Socrates is
  in the kitchen or that he is in the house
• Likewise, the distinction allows Abelard to
  define negation, and other propositional
  connectives, purely truth-functionally in terms
  of content, so that negation, for instance, is
  treated not-p is false/true if and only if p is
  true/false

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Gottfried Leibniz (1646 1716 AD)

• Leibniz is the most important logician between
  Aristotle and 1847, when George Boole and
  Augustus De Morgan each published books that
  began modern formal logic

• Leibniz enunciated the principal properties of
  what we now call conjunction, disjunction,
  negation, identity, set inclusion, and the empty
  set, essentially building the foundation for
  symbolic logic as we know it today
• Two main principles
• All our ideas are compounded from a very small
  number of simple ideas, which form the alphabet
  of human thought.
• Complex ideas proceed from these simple ideas by
  a uniform and symmetrical combination, much like
  arithmetical multiplication.
• This world is the best of all possible worlds. -
  Leibniz

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Symbolic Logic AP

• Is a technique for analysis of deductive
  arguments
• English (or any) language can make any argument
  appear vague, ambiguous especially with use of
  things like metaphors, idioms, emotional appeals,
  etc.
• We want to avoid these difficulties to move into
  logical heart of argument use symbolic language
• Now can formulate an argument with precision
• Symbols facilitate our thinking about an argument
• These are called logical connectives

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The Logical Connectives

• We can translate arguments from sentences and
  simple or compound propositions/statements into
  symbolic logical form using
• Conjunction (conjunct 1) and (conjunct 2)
• p q p q
• Disjunction (disjunct 1) or (disjunct 2)
• p v q
• Negation It is not the case that
• p
• Conditional If (antecedent), then (consequent)
• p q p ? q

n
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Basic Abbreviation Translation Rules

• Charlies neat and Charlies sweet.
• N S
• Dictionary NCharlies neat SCharlies
  sweet
• Can choose any letter to symbolize each statement
  (in this case, two conjuncts), but it is best to
  choose one relating to the content of that
  conjunct to make it easier to remember

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Punctuation

• As in mathematics, it is important to correctly
  punctuate logical parts of an argument
• Ex. (2x3)6 12 whereas 2x(36) 18
• Ex. p q v r (this is ambiguous)
• To avoid ambiguity and make meaning clear
• Make sure to order sets of parentheses according
  to how the argument reads
• A (B v C) (C v D) E would be a
  different read than (A B) v (C C) v D
  E

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Example

• There were three people involved in the
  accident, and no one was injured.
• (Steps Translate into logic by making a
  dictionary then arrange statements correctly in
  logical form)
• (Hint Note When symbolizing statements, always
  make the statement a positive one. If you have a
  negative statement in the sentence, put its
  positive in the dictionary then when you
  translate, simply negate that sentence)

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Solution

• There were three people involved in the accident,
  and no one was injured.
• T OTThree people were involved in the
  accident.OSomeone was injured.

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One more

• Either you are male or female but not both.

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Solution

• Either you are male or female but not both.
• (M v F) (M F)
• MYou are male.
• FYou are female.

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Inference Rules

• There are many different systems of formal logic,
  each one with its own set of well-formed
  formulas, rules of inference, and even semantics
  (e.g. temporal logic, modal logic, intuitionistic
  logic, quantum logic)
• These rules are like shortcuts for us when doing
  logical proofs
• Take a look at the handout you have on Inference
  Rules

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Nine Basic Inference Rules

• These nine rules of inference correspond to
  elementary argument forms whose validity is
  easily established by truth tables. With their
  use, formal proofs of validity can be constructed
  for a wide range of more complicated arguments.

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Lets put this all together nowFirst, translate
the following argument from English into symbolic
logic

• If Anderson was nominated, the she went to
  Boston.
• If she went to Boston, then she campaigned there.
• If she campaigned there, she met Douglas.
• Anderson did not meet Douglas.
• Either Anderson was nominated or someone more
  eligible was selected.
• C. Therefore someone more eligible was selected.

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Solution (Part 1)

• A B
• B C
• C D
• D
• A v E
• Therefore E

n
n
n
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Now lets try to prove the conclusion from only
the given premises

• This is now where our inference rules may (or may
  not) be of help to us

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Once translated

• Write the premises and the statements that we
  deduce from them in a single column to the
  right of this column, for each statement, its
  justification is written (e.g. the reason why
  we include that statement in the proof)
• Heres what it looks like

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Formal Proof
n

1. A B Premise
2. B C Premise
3. C D Premise
4. D Premise
5. A v E Premise
6. A C 1,2 H.S.
7. A D 6,3 H.S.
8. A 7,4 M.T.
9. E 5,8 D.S.

n
n
The justification for each statement (the right
most column) consists of the numbers of the
preceding statements from which that line is
inferred, together with the abbreviation for the
rule of inference used to get it
n
n
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Proofs

• There are many other examples of simpler (and
  more difficult) proofs, but we dont have time to
  get into the rules associated with proving even
  the most basic of them, so we will skip over that
  for now
• The main point here is to show the application of
  symbolic logic in the context of symbolic
  translations and in constructing formal proofs

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Paradoxes and Questions

• Paradox seemingly sound piece of reasoning
  based on seemingly true assumptions that leads to
  a contradiction or another obviously false
  conclusion
• Zeno's paradoxes are a set of problems devised by
  Zeno of Elea to support Parmenides' doctrine that
  "all is one" and that, contrary to the evidence
  of our senses, the belief in plurality and change
  is mistaken, and in particular that motion is
  nothing but an illusion

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Zenos Paradoxes

• Story of a race between Achilles and the tortoise
• Achilles can run faster so tortoise given big
  head start
• When race starts, Achilless first goal is to get
  to the point where the tortoise started
• By the time he gets there, the tortoise has moved
  again (but only a little) so Achilles must now
  get to that spot
• No matter how many times Achilles reaches the
  tortoises prior location, even if he does it an
  infinite number of times, hell never catch up
  with the tortoise, although hell get awfully
  close
• Tortoise just has to not stop then in order to win

In a race, the quickest runner can never
overtake the slowest, since the pursuer must
first reach the point whence the pursued started,
so that the slower must always hold a lead.
-Aristotle
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• Racetrack paradox
• In order to get to the end of a racetrack, a
  runner must first complete an infinite number of
  journeys
• He must run to the midpoint
• Then he must run to the midpoint of the remaining
  distance
• Then to the midpoint of the still remaining
  distance, etc etc
• Therefore he can never get to the end of the track

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• Arrow paradox
• If everything, when it occupies an equal space,
  is at rest, and if that which is in locomotion is
  always occupying such a space at any moment, the
  flying arrow is therefore motionless. -
  Aristotle
• Imagine an arrow in flight
• Divide up time into a series of indivisible
  moments
• At any given moment, if we look at the arrow it
  has an exact location, so it is not moving
• Yet movement has to happen in the present (it
  can't be that there's no movement in the present
  yet movement in the past or future)
• So throughout all time, the arrow is at rest
• Thus motion cannot happen

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Zen Koans

• A koan is a story, dialogue, question, or
  statement in the history and lore of Zen
  Buddhism, generally containing aspects that are
  inaccessible to rational understanding (yet may
  be accessible to intuition)

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The Gateless Gate

• Joshus Dog
• A monk asked Joshu, a Chinese Zen masterHas a
  dog Buddha-nature or not?
• Joshu answered Mu.
• Mumons comment Enlightenment always comes after
  the road of thinking is blocked. Mu is not
  nothingness, just concentrate your whole energy
  into this Mu, and do not allow any
  discontinuation.

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The Gateless Gate

• Wakuan complained when he saw a picture of
  bearded Bodhidharma Why hasnt that fellow a
  beard?
• Daibai asked Baso What is Buddha? Baso said
  This mind is Buddha.
• A monk asked Baso What is Buddha? Baso said
  This mind is not Buddha.
• Two hands clapping make a sound. What is the
  sound of one hand clapping?

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Thank you!

• Were your expectations met? If not, why?
• What do you now know about logic?