Chapter Two

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Chapter Two

• Symbolizing in Sentential Logic
• This chapter is a preliminary to the project of
  building a model of validity for sentential
  arguments. We need to be able to translate
  English statements into symbols.

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• The syntax of a language shows how to formulate
  correct
• sentences using its vocabulary. The syntax is
  specified in
• formation rules.
• The semantics of a language shows the meaning of
  the
• symbols and under what conditions their
  combinations are
• true and under what conditions they are false.

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1. Atomic and Compound Sentences

• In English longer sentences can be built up of
  shorter sentences using sentence connectives such
  as and and or.
• Sentences built up of shorter sentences by means
  of sentence connectives are compound sentences.
• All other sentences are said to be atomic, or
  simple, sentences.

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2. Truth Functions

• With truth functions we have only two
  truth-values true and false.
• We use symbols to represent common mathematical
  functions. These are called operators.

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Truth Functions, continued

• Our system of logic has five truth-functional
  operators
• (not) takes only one input.
• . (and), v (or), ? (ifthen), and (if
  and only if)
• take two.

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3. Conjunctions

• Compound sentences formed by use of the
  connective and
• are called conjunctions, and the two sentences
  joined by
• and are called conjuncts.

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Conjunctions, continued

• The different truth values of compound sentences
  that are the products of the different truth
  values of their conjuncts can be represented in a
  truth table.
• Sentences can be used to make different
  statements, depending on time, and, in some
  cases, place.

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4. Non-Truth-Functional Connectives

• Many connectives in English are not
  truth-functional, e.g., before.

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5. Variables and Constants

• A statement variable has no truth-value what
  does have truth-value is a statement we
  substitute for it, and the truth-value varies
  according to what statement that happens to be.
• This notion of substitution is analogous to that
  used in algebra.
• It is conventional to use small letters, p, q, r
  as sentence variables, and capital letters, A, B,
  C as sentence constants.

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6. Negations

• Some logical operators generate a new sentences
  out of just one starting sentences.
• Only one operatornegationis used in standard
  sentential logic.
• Negation is symbolized by the tilde symbol, .

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7. Parentheses and Brackets

• By using parentheses we can build up complex
  sentences out of shorter sentences.
• The shorter sentences that are combined to make
  longer sentences are component sentences.
• Parentheses are sued to indicate the scope of
  each logical operator in any sentence the
  sentences over which it operates.

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Parentheses and Brackets, continued

• The main connective of a sentence is the
  truth-functional connective whose scope
  encompasses the entire remainder of the sentence.
• A sentence is well-formed if it is clear which
  operator is the main operator for the sentence
  and for each component sentence contained within
  the sentence.

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Parentheses and Brackets, continued

• Two conventions help eliminate unnecessary
  parentheses
• It is not necessary to place an outermost pair of
  parentheses entirely surrounding a sentence.
• 2) The scope of the operator is always the
  shortest complete sentence that follows it.

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8. Use and Mention

• We must distinguish between using a word, phrase,
  or statement, and talking about that word,
  phrase, or statementthat is, mentioning it.
• The language in which we speak about the logical
  language is the metalanguage.
• The language that we are talking about is the
  object language.

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9. Disjunctions

• Two sentences connected by the word or form a
  compound sentence called a disjunction.
• The two sentences so connected are called
  disjuncts.

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Disjunctions, continued

• There are two different senses of the connective
  or
• Exclusive If the disjunction is true one or
  other of the disjuncts is true, but not both.
• 2) Inclusive If the disjunction is true either
  one of the disjuncts is true, or both are true.

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Disjunctions, continued

• Disjunction is symbolized by the wedge, V,
  which is a
• truth-functional logical connective.

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10. Not Both and Neither Nor

• All it takes to make a not both sentence true
  is for at least one of the two components to be
  false.
• Sentences built around the connective
  neithernor should not be symbolized as
  disjunctions, but as conjunctions with two
  negated conjuncts.

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11. Material Conditionals

• A compound sentence of the form If then is
  called a conditional.
• The sentence between the if and the then is
  called its antecedent.
• The sentence after the then is called its
  consequent.
• The truth functional connective for conditionals
  is the horseshoe, ?.

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Material Conditionals, continued

• A sentence whose main connective is the horseshoe
  is called a material conditional.
• The truth function represented by the horseshoe
  is called material implication.

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12. Material Biconditionals

• Two sentences are materially equivalent when they
  have the same truth-value.
• The symbol is called the tribar and stands
  for material equivalence.
• Compound sentences formed by the tribar are
  called material equivalences, or biconditionals.

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13. Only If and Unless

• Only if sentences indicate necessary
  conditions, but not sufficient conditions.
• You will pass the class only if you pay
  attention can be symbolized as C?A.
• A simple way of symbolizing unless sentences is
  as or sentences.

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14. Symbolizing Complex Sentences

• The first step in symbolizing complex sentences
  is to identify the main connective of the
  sentence.
• The second step is to look for punctuation.
  Parentheses often mirror commas and semicolons.
• Be careful to determine the correct scope of
  negations.

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15. Alternative Sentential Logic Symbols

• Negation -,
• Conjunction?,
• Disjunction ? (almost always used as in this
  text)
• Conditional ?
• Biconditional ?

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Key Terms

• Antecedent
• Atomic sentence
• Biconditional
• Component sentence
• Compound sentence
• Conditional
• Conjunct
• Conjunction

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Key terms, continued

• Consequent
• Disjunct
• Disjunction
• Dot
• Exclusive disjunction
• Exclusive or
• Formation rules
• Horseshoe

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Key terms, continued

• Inclusive disjunction
• Inclusive or
• Main connective
• Material biconditional
• Material conditional
• Material equivalence
• Material impication
• Metalanguage

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Key terms, continued

• Negation
• Object language
• Polish notation
• Scope
• Semantics
• Sentence
• Sentence connective
• Sentence constant

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Key terms, continued

• Sentential logic
• Statement
• Statement variable
• Substitution
• Symbolic logic
• Syntax
• Tilde
• Tribar

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Key terms, continued

• Truth-function
• Truth-functional operator
• Truth table
• Truth-value
• Wedge
• Well-formed