Semantic and syntactic validity

1
Semantic and syntactic validity

• A symbolic argument is valid iff its conclusion
  is derivable from its premises in accordance with
  the directions for constructing a derivation
  presented here
• A symbolic argument is semantically valid iff it
  is impossible for the premises to be true unless
  the conclusion is true as well
• For those symbolic arguments we are concerned
  with, a symbolic argument is syntactically valid
  iff it is sematically valid

2

• The proof that if a symbolic argument is
  syntactically valid, it is semantically valid is
  called soundness proof
• The proof that if a symbolic argument is
  semantically valid, it is syntactically valid is
  called completeness proof
• Kurt Geodels incompleteness theorem
• There is a semantically valid symbolic argument
  that is not syntactical valid

3
Validating English arguments

• To establish the validity of an English argument
• Set up a symbolic argument that corresponds to
  this English argument
• Establish the validity of this symbolic argument
• A symbolization of an English argument S on the
  basis of a scheme of abbreviation A is a symbolic
  argument whose premises and conclusion are
  symbolizations of the premises and conclusion of
  S on the basis of A
• A symbolic argument is a symbolization of an
  English argument S iff there is a scheme of
  abbreviation such that it is a symbolization of S
  on the basis of that scheme of abbreviation

4
Different symbolizations

• There is more than one legitimate symbolization
  of a given English argument
• Different symbolizations for a given English
  argument on the basis of different schemes of
  abbreviationt
• Different schemes of abbreviation give rise to
  different symbolizations

5
Validity of English argument

• An English argument is valid iff there is at
  least one valid symbolic argument that is its
  symbolization
• The longer the symbolization, the more likely it
  is to be valid and among symbolizations of equal
  length, the likelihood of validity increases as
  the number of distinct sentence letters
  decreases.
• The interesting symbolization of an English
  argument
• Reflecting to a maximal degree the logical
  structure of the argument it symbolizes

6

• The interesting symbolization is one of the
  symbolizations of a given English argument
• If the interesting symbolization of an English
  argument is valid, then the English argument is
  valid

7
Interesting symbolization

• Given an English argument, its interesting
  symbolization has the highest probability to be
  valid among its symbolizations
• If the interesting symbolization of a given
  English argument is invalid, its other
  symbolizations are invalid
• If an English argument is valid, its interesting
  symbolization must be valid
• An English argument is valid iff its interesting
  symbolization is valid

8
Validity of English translation

• The English arguments that we can obtain by
  translating it into English
• The symbolic argument that reveals the logical
  structure of the English arguments is valid
• The validity of an argument is completely
  determined by its logical structure

9

• What if a symbolic argument is not valid?
• The English arguments that we can obtain by
  translating it into English
• Even if a symbolic argument is invalid, we can
  obtain a valid English argument by translating it
  into English
• P ? R
• R
• Therefore, Q.

10
Fallacies

• Most restrictions in the directions for
  constructing a derivation are introduced to
  prevent fallacies
• The fallacy of affirming the consequent
• P ? Q
• Q
• Therefore, P
• The fallacy of denying the antecedent
• P ? Q
• P
• Therefore, Q

11
Theorems

• Theorems are valid symbolic arguments with no
  premises
• A theorem is derivable from an empty set of
  premises
• A derivation of a theorem is called its proof
• Transposition (P ? Q) ? (Q ? P)
• Reductio ad absurdum (P ? P) ? P

12
A new symbolic language

• Expressive power
• John is male but wants to be female therefore
  John is male
• We need to add three new logical symbols ?,
  v, ?, which symbolize and, or, and if
  and only if, respectively
• They are used with a pair of parentheses
• Conjunction and conjunct

13

• Disjunction and disjunct
• Biconditional and constituents
• Sentential connectives
• Phrases of connection English phrases
  symbolized by sentential connectives

14
Characterization

• The class of symbolic sentences can be
  exhaustively characterized in this way
• Sentence letters are symbolic sentences
• if f is a symbolic sentence, so is f.
• If f and ? are symbolic sentences, then so are
  (f ? ?), (f ? ?), (f v ?), (f ? ?)

15
Informal notation

• Omit the outermost parentheses of a symbolic
  sentence
• Conditional and biconditional are regarded as
  marking a greater break than conjunction and
  disjunction
• (P v Q) ? R, P v (Q ? R), P ? (Q v R), (P ? Q) v
  R
• Repeated conjunctions or disjunctions may be
  omitted with the understanding that they are
  operated from the left to the right
• P v Q v R, P v Q v R v S, P ? Q ? R, P v Q ? R,
  P ? Q v R

16
Literal translation

• Restore any parentheses that may have
  disappeared as a result of the informal
  conventions
• Replace sentence letters by English sentences in
  accordance with the scheme of abbreviation
• Eliminate sentential connectives in favour of
  the corresponding phrases of connection,
  preserving all parentheses

17
Free translation

• A free translation of a symbolic sentence f on
  the basis of a given scheme of abbreviation is a
  stylistic variant of the literal English
  translation of f based on that scheme
• but though although even though are
  stylistic variants of and
• The book is short, but it is interesting
• The book is short, although it is interesting
• The book is short, even though it is interesting
• 'The book, which is short, is interesting'

18
Literal meaning

• The book is short, and it is interesting
• If the book isn't short, what you have said is
  false, and if it isn't interesting then what you
  have said is false
• If the book is both short and interesting, what
  you have said is true
• If the book is both short and interesting, the
  sentence is misleading.

19
Conversational implicature

• The sentence suggests that, given that the book
  is short, it is improbable that it is interesting
  
• The literal meaning of a given sentence is all
  contained in the information conveyed by it. But
  not all information conveyed by a given sentence
  is its literal meaning.
• Conversational implicature
• Paul Grices famous theory of conversational
  implicature

20
Even if and though

• Even if on the one, and though, although,
  even though, on the other
• Their logical meanings are strikingly different
• Even if the book is short, it is interesting
  is a stylistic variant of If the book is short,
  it is interesting
• The book is short, although it is interesting
  is a stylistic variant of The book is short and
  it is interesting

21
or and iff

• The expression unless and either or are
  stylistic variants of or
• The expression exactly on condition that and
  just in case are stylistic variants of if and
  only if (for short, iff)

22
Symbolization

• A symbolization of an English sentence pi is a
  symbolic argument for which pi is a free English
  translation
• Introduce phrases of connection, accompanied by
  parentheses and occurring canonically in place of
  their stylistic variants
• Replace phrases of connection by the
  corresponding sentential connectives
• Replace English components by sentence letters in
  accordance with the scheme of abbreviation
• Omit parentheses in accordance with the informal
  conventions

23
Ambiguity and commas

• Ambiguity in natural languages
• Commas play the role of grouping items on each
  side
• Exercises what are interesting symbolizations?
• Errors will decrease in the subjects
  performance just in case either motivation is
  absent or learning has not occurred
• Assuming that logic is difficult or the text is
  not readable Alfred will pass only if he
  concentrates

24
The inclusive sense of or

• Two related but distinguishable meanings
• Women with high or increasing blood pressure
  are up to three times more likely to develop
  diabetes
• Weak or inclusive disjunction
• An inclusive disjunction is true in case one or
  the other or both disjuncts are true
• Either, possibly both, and/or

25
The exclusive sense of or

• At least one and at most one
• Salad or dessert on the table dhote menu
• A cookie or a piece of cake
• Strong or exclusive disjunction
• An exclusive disjunction is true iff only one of
  the disjuncts is true
• A or B but not both

26
The two senses of unless

• The ambiguity can also be found in the stylistic
  variants of or
• The picnic will occur unless it rains is a
  stylistic variant of The picnic will occur or it
  rains

27
The inclusive sense of or

• The following inference is semantically valid
  for inclusive disjunction
• A
• Therefore, A or B
• It is impossible that A is true but A or B is
  false
• Given A is true, B is either true or false ? In
  both cases, A or B is true
• It is not valid for an exclusive or as it is
  perfectly possible that A is true but A or B is
  false
• An exclusive or can be defined into other
  logical symbols

28
The ambiguity of conjunction

• The standard sense of and
• p and q is true iff p is true and q is
  true
• The meaning of temporal succession
• John took off his shoes and (then) socks
• he took off his socks and (then) shoes
• We select the standard sense of and

29
Neither . . Nor

• Neither Polk was a president nor Quincy was a
  president.
• It is not the case that Polk was a president and
  it is not the case that Quincy was a president
• Neither P nor Q' is equivalent to Not P and
  not Q

30
Grammatical variants

• Conjunction and disjunction may stand between
  subject or predicate terms
• Byron was a great poet and a great adventurer
  ? Byron was a great poet and Byron was a great
  adventurer
• Byron and Elliot were great poets ? Byron was
  a great poet and Elliot was a great poet.
• Smith is the owner or the manager ? Smith is
  the owner or the Smith is the manager
• Red or Butch is guilty ? Red is guilty or
  Butch is guilty

31
Non-conjuctive simple sentences

• Bill and Hilary are spouses
• Korea lies between China and Japan

32
Negated compound subjects or predicates

• Negation after a compound subject
• Wilma or Veronica didn't sing ? Wilma didn't
  sing or Veronica didn't sing
• Negation before a compound predicate
• Wilma didn't sing or dance ? (Wilma sang or
  danced) ? (Wilma sang or Wilma danced)
• Wilma didn't sing and dance ? (Wilma sang and
  danced) ? (Wilma sang and Wilma danced)

33
Two confusing sentences

• Both John and Michael are not good at logic ?
  the conjunction of two negations
• Not both John and Michael are good at logic ?
  the negation of a conjunction
• Neither John nor Michael are good at logic

34
Grouping

• When connectives occur inside otherwise simple
  sentences, the symbolizations of the sentences
  form a unit
• 'Ruth tap-dances or sings and she plays the
  clarinet
• 'Ruth tap-dances or she sings and plays the
  clarinet
• If neither Wilma nor Sally attends, either
  Robert or Peter will be bored.
• Either Robert or Peter, but not Tom, will be
  bored
• If neither Wilma nor Sally attends, either
  Robert or Peter, but not Tom, will be bored

35
Exercises

• Either Robert or Tom will attend, but not both
• Robert will attend if Sally does, but she won't
  attend if neither Tom nor Wilma attend
• Neither Sally nor Robert will run, but if
  either Tom or Quincy run, Veronica will win
• Given that Sally and Robert won't both run, Tom
  will run exactly if Quincy does

36
New inference rules

• Simplification in two forms
• Adjunction
• Addition, in two forms
• Modus tollendo ponens (MTP), in two forms
• Bicondition-conditional (BC), in two forms
• Conditional-biconditional

37
Cautionary remark

• In connection with the placement of parentheses,
  we need to be careful in applying inference rules
• P ? Q ?R
• Q ? R
• P
• P ? Q ? R

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