Truth value analysis

1
Truth value analysis

• The method of constructing a derivation cant be
  employed to test an argument for invalidity
• Even if we fail to construct a derivation for a
  given argument, this doesnt immediately mean
  that the argument is not valid
• There is an automatic method for proving
  invalidity

2
Truth values

• Truth or falsity of a sentence is called its
  truth value
• An assignment is a set of pairs of sentence
  letters and its truth values
• We can determine the truth values of complex
  symbolic sentences on the basis of the truth
  values of component symbolic sentences

3
Conditional sentences

• Normally by a conditional sentence, we assert a
  connection between its antecedent and consequent
• A conditional sentence is true iff its
  antecedent is false or its consequent is true
• No connection whatsoever is required by the
  truth condition of a conditional sentence

4
The truth condition of a conditional

• The clause 3 captures the common part of the
  meaning of all different types of conditional
  sentences
• Under what circumstances should we agree that a
  conditional sentence is false?
• A conditional sentence itself proves false if
  its consequent is false while its antecedent is
  true
• Regard (???) as the central part of the
  meaning of ? ? ?

5
Material conditional

• Most conditional sentences we use in natural
  language have more meaning than (???)
• Causal or logical connection
• Expedient to consider the symbol ? as
  representing an artificial or new kind of
  conditional whose meaning is exhaustively
  captured by the clause 3
• ? ? ? is true iff it is not the case that ? is
  true but ? is false

6

• Material conditional
• No real connection between antecedent and
  consequent is suggested by a material conditional

7
A type of material conditional

• We do sometimes assert a conditional even though
  there is no obvious tie between its antecedent
  and consequent
• If all of you get A grade for this course, I
  will ground myself
• An emphatic or humorous method of denying its
  antecedent
• To affirm such a conditional sentence amounts to
  denying that its antecedent is true

8
General conditional sentence

• If x lt 2 then x lt 4 is true for any number x
  whatsoever
• If 1lt2 then 1lt4 both antecedent and
  consequent are true
• If 3lt2 then 3lt4 the antecedent is false and
  the consequent is true
• If 4lt2 the 4lt4 both antecedent and consequent
  are false
• There is no number which is smaller than 2 but
  not smaller than 4

9
Truth-functionality

• Sentential connectives
• Because
• Truth functionality the truth value of the
  compound sentence is completely determined by the
  truth values of its components
• The truth value of a conjunction is determined
  wholly and entirely by the truth values of its
  conjuncts
• Conjunction, negation, disjunction, material
  conditional, and biconditional are
  truth-functional connectives

10
Because

• Not every sentential connective is
  truth-functional
• The truth value of the because sentence is not
  completely determined by the truth values of its
  components
• To determine the truth value of a because
  sentence, not only we have to take account of the
  truth values of the component sentences but also
  the relationship between the component sentences
• A sentential connective is truth-functional iff
  the truth value of the compound sentence doesnt
  change when a component sentence is replaced by
  another sentence with the same truth value

11
Tautology

• A tautology is a symbolic sentence that is true
  under every possible assignment
• Constructing a truth table

12
Semantic and syntactic validity

• A symbolic argument is syntactically valid iff
  its conclusion is derivable from its premises in
  accordance with the directions for constructing a
  derivation
• A symbolic argument is semantically valid iff it
  is impossible that the premises are true but the
  conclusion is false
• For those symbolic arguments of sentential
  calculus, a symbolic argument is syntactically
  valid iff it is sematically valid

13

• Theorem is to syntactic validity what tautology
  is to semantic validity
• A theorem is a syntactically valid argument that
  has no premises
• A tautology is a semantically valid argument
  that has no premises
• A symbolic argument is a theorem iff it is a
  tautology
• By the equivalence between theorem and
  tautology, the truth value analysis serves as an
  automatic method for determining whether a
  symbolic argument is a theorem or not

14
Reductio procedure

• We first assume that the sentence is false
• Based upon this assumption, we assign truth
  values to sentence letters in accordance with the
  rules for sentential connectives
• If a contradiction is derivable from the
  assumption that the sentence is false
• The assumption is wrong
• There is no consistent assignment of truth
  values that makes the whole sentence false
• The sentence is a tautology

15

• If a contradiction is not derivable from the
  assumption that the sentence is false
• This assumption is not contradictory
• There is a consistent assignment of truth values
  to the sentence letters that makes the whole
  sentence false
• The sentence is not a tautology

16
Separating cases

• No contradiction to the rules has been obtained
• No further assignments are required by the rules
• The reduction procedure cant be completed in a
  single row
• If at least one of the cases leads to an
  assignment with respect to which the given
  sentence is false, then the sentence is not a
  tautology

17

• If all of the cases lead to a contradiction,
  this means the sentence is a tautology
• If we cant obtain a contradiction or a
  consistent assignment of truth values to sentence
  letters, take cases within cases

18
Testing an argument

• The method of truth value analysis for testing
  whether an argument is valid or not
• An argument is semantically valid iff there is
  no assignment of truth values with respect to
  which the premises are true but the conclusion is
  false
• The premises tautologically imply (necessitates,
  entails) its conclusion
• The method of truth value analysis serves to
  determine if an argument is syntactically valid
  or not

19
Quantificational Logic

• All humans are mortal
• Socrates is human
• Therefore, Socrates is mortal
• Within the framework of sentential calculus or
  propositional logic, this argument doesnt have
  any valid symbolization

20
English formulas

• x is greater than 2
• This expression, in and of itself, is neither
  true nor false
• Names are expressions that are intended to
  designate a single thing
• Variables are expressions to be replaced by
  names
• A formula of English is either an English
  sentence or an expression containing variables
  that becomes an English sentence when those
  variables are replaced by names.

21
Satisfaction

• x is greater than 2
• 1 satisfies this formula of English
• 3 does not satisfy this formula of English
• x loves y
• The sequence of names ltRomeo, Julietgt
  satisfies this formula of English
• A sequence of names satisfies an English formula
  if we obtain a true sentence by replacing the
  variables in it by the sequence of names

22
Converting into English sentences

• The first way By replacing variables by names
• The second way By prefixing the phrases for
  each x and there is an object x such that

23

24
Bound and free occurrences of variables

• An occurrence of a variable x is bound when
  either (1) it is part of a quantifier phrase or
  (2) it is contained in a symbolic formula and the
  symbolic formula is located within the scope of a
  quantifier involving x
• An occurrence of a variable x is free iff it is
  not bound
• A variable itself is bound iff at least one of
  its occurrences is bound
• A variable itself is free iff at least one of
  its occurrences is free

25
Symbolic sentences

• Symbolic sentences are symbolic formulas where
  no variables are free
• They take definite truth values
• By prefixing quantifier phrases, we eliminate
  free occurrences of variables and, accordingly
  create bound occurrences of variables

26
Scheme of abbreviation

• Not only English sentences are correlated with
  sentence letters but also English predicates and
  names are correlated with predicate letters and
  name letters
• P John is smart
• A John
• F a is smart
• G a takes a logic course
• H a will smile at the end of the semester

27
Literal and free translation

• Examples (x)Fx, (?x)Fx, (x)Fx, (?x)Fx, (x)(Fx
  ? Gx), (?x)(Fx Gx)
• A general characterization of the procedure for
  literal translation
• An English formula is a free translation of a
  symbolic formula pi when it is a stylistic
  variant of the literal English translation of pi
• Two distinct sentences are stylistic variants of
  each other iff (1) they are synonymous (2) their
  difference concerns phrases of connection

28
Redefining stylistic variants

• We do symbolization and translation not only in
  terms of sentential connectives but also in terms
  of quantifier phrases
• Two distinct sentences are stylistic variants of
  each other iff (1) they are synonymous (2) their
  difference concerns phrases of connection or
  phrases of quantity

29
Examples

• For each x, x is bald
• Everything is bald Each thing is bald All
  things are bald For all x, x is bald
• There is an object x such that x is bald
• Something is bald At least one thing is bald
  There is a bald thing For some x, x is bald
• Nothing is bald

30
Examples

• All who take this course will smile at the end
  of the semester
• For each x, if x takes this course, x will smile
  at the end of the semester
• For each x (x takes this course and x will smile
  at the end of the semester)
• Some who take this course will smile at the end
  of the semester
• There is an object x such that x takes this
  course and x will smile at the end of the
  semester
• There is an object x such that if x takes this
  course, x will smile at the end of the semester

31
Logical form

• The two sentences appear to have almost the same
  logical structure
• The first is a generalization of a conditional,
  whilst the second is a generalization of a
  conjuction
• Similar surface-grammatical structures but quite
  different logical forms
• One of great philosophical turns in the 20th
  century

32
Further examples

• No one who takes this course will smile at the
  end of the semester
• Not all who take this course will smile at the
  end of the semester
• Some who take this course will not smile at the
  end of the semester
• Only those who take this course will smile at
  the end of the semester ? All who smile at the
  end of the semester must have taken this course.
• None but those who take this course will smile .
  . .
• All but those who take this course will smile. .
  .

33

34

35

36

37

38

39

40

41
Truth value analysis

• The method of constructing a derivation cant be
  employed to test an argument for invalidity
• Even if we fail to construct a derivation for a
  given argument, this doesnt immediately mean
  that the argument is not valid
• There is an automatic method for proving
  invalidity