Symbolic Language and Basic Operators

1
Symbolic Language and Basic Operators

• Kareem Khalifa
• Department of Philosophy
• Middlebury College

2
Overview

• Why this matters
• Artificial versus natural languages
• Conjunction
• Negation
• Disjunction
• Punctuation
• Sample Exercises

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Why this matters

• Symbolic language allows us to abstract away the
  complexities of natural languages like English so
  that we can focus exclusively on ascertaining the
  validity of arguments
• Judging the validity of arguments is an important
  skill, so symbolic languages allow us to focus
  and hone this skill.
• Symbolic language encourages precision. This
  precision can be reintroduced into natural
  language.

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More on why this matters

• You are learning the conditions under which a
  whole host of statements are true and false.
• This is crucial for criticizing arguments.
• It is a good critical practice to think of
  conditions whereby a claim would be false.

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Artificial versus natural languages

• Symbolic language (logical syntax) is an
  artificial language
• It was designed to be as unambiguous as possible.
• English (French, Chinese, Russian, etc.) are
  natural languages
• They werent really designed in any strong sense
  at all. They emerge and evolve through very
  organic and (often) unreflective cultural
  processes.
• As a result, they have all sorts of ambiguities.
• The tradeoff is between clarity and expressive
  richness. Both are desirable, but theyre hard to
  combine.

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Propositions as letters

• Logical syntax represents individual propositions
  as letters.
• When we dont care what the proposition actually
  stands for, we represent it with a lowercase
  letter, typically beginning with p.
• When we have a fixed interpretation of a
  proposition, we represent it with a capital
  letter, typically beginning with P.
• Ex. Let P Its raining.
• Sometimes, letters are subscripted. Each
  subscripted letter should be interpreted as a
  different proposition.

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Dispensable translation manuals

• Often, the letters are given an interpretation,
  i.e., they are mapped onto specific sentences in
  English.
• Ex. Let P be It is raining Q be The streets
  are wet, etc.
• However, this is not necessary. The validity of
  an argument doesnt hinge on the interpretation.
• If p then q
• p
• ? q

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Logical connectives some basics

• A logical connective is a piece of logical syntax
  that
• Operates upon propositions and
• Forms a larger (compound) proposition out of the
  propositions it operates upon, such that the
  truth of the compound proposition is a function
  of the truth of its component propositions.
• Today, well look at AND, NOT, and OR.
• Khalifa is cunning and cute.
• Khalifa is not cunning.
• Either Khalifa is cunning or he is foolish.

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Conjunction

• AND-statements
• Middlebury has a philosophy department AND it has
  a neuroscience program.
• Represented either as ? or as
• I recommend , since its just SHIFT7
• p q will be true when p is true and q is
  true false otherwise.

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Truth-tables

• Examine all of the combinations of component
  propositions, and define the truth of the
  compound proposition.

T
F
F
F
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Subtleties in translating English conjunctions
into symbolic notation

• The and does not always appear in between two
  propositions.
• Khalifa is handsome and modest.
• Khalifa and Grasswick teach logic.
• Khalifa teaches logic and plays bass.

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More subtleties

• Sometimes and in English means and
  subsequently.
• The truth-conditions for this are the same as
  , but the meaning of the English expression is
  not fully captured by the formal language.
• Many English words have the same truth-conditions
  as but have additional meanings.
• Ex. but, yet, still, although, however,
  moreover, nevertheless
• General lesson The meaning of a proposition is
  not (easily) identifiable with the
  truth-functions that define it in logical
  notation.

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Negation

• Represented by a
• In English, not, its not the case that,
  its false that, its absurd to think that,
  etc.

F
T
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Disjunction

• Represented in English by or.
• However, there are two senses of or in English
• Inclusive when p AND q are true, p OR q is true
• Exclusive when p AND q is true, p OR q is false
• Logical disjunction (represented as v) is an
  inclusive or.

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Which is inclusive and which is exclusive?

• You can take Intro to Logic in the fall or the
  spring.
• Exclusive. You cant take the same course twice!
• You can take Intro to Logic or Calculus I.
• Inclusive. Youd then be learned in logic and in
  math!

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Truth table for disjunction
T
T
T
F
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Punctuation

• We can daisy-chain logical connectives together.
• Either Polly and Quinn or Rita and Sam will not
  win the game show.
• If we have no way of grouping propositions
  together, it becomes ambiguous
• P Q v R S
• Logic follows the same conventions as math (
  ) , though some logicians prefer to use only (
  ( ( ) ) ).
• (PQ) v (RS)

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A few exceptions

• A negation symbol applies to the smallest
  statement that the punctuation permits.
• Ex. p q is equivalent to (p) q
• It is NOT equivalent to (p q)
• This reduces the number of ( )
• We can also drop the outermost brackets of any
  expression.
• Ex. p (q v r) is equivalent to p (q v r)

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Lessons about punctuation from logic

• Make sure, in English, that you phrase things so
  that there is no ambiguity
• Commas are very useful here
• Be especially sensitive when reading about small
  subtleties about logical structure that would
  change the meaning of a passage.

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Sample Exercise A3 (327)

• London is the capital of England Stockholm is
  the capital of Norway
• T F
• F T
• F

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Sample Exercise A4 (327)

• (Rome is the capital of Spain v Paris the
  capital of France)
• (F v T)
• (T)
• F

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Sample Exercise A9 (327)

• (London is the capital of England v Stockholm is
  the capital of Norway) (Rome is the capital of
  Italy Stockholm is the capital of Norway)
• (T v F) (T F)
• (T) (F T)
• (T) (F)
• F

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Sample Exercise C3 (329)

• Q v X
• Q v F
• Q v T
• T

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Exercise C12 (329)

• (P Q) (P v Q)
• The first conjunct (PQ), can only be true if P
  Q T
• However, this would make the whole conjunction
  false. Heres the proof
• (TT) (T v T)
• (T) (F v F)
• (T) (F)
• F

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Exercise D9 (330)

• It is not the case that Egypts food shortage
  worsens, and Jordan requests more U.S. aid.
• E J