Formal Methods in Computer Science CS1502 FirstOrder Validity and Consequence

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Formal Methods in Computer ScienceCS1502First-O
rder Validity and Consequence

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• To learn about notion of tautology for
  quantified sentences, and its use to capture the
  notion of truth of quantified sentences.
• truth-functional form and the algorithm to
  construct it.
• To learn about the notion of FO validity and
  its use to capture the notion of truth of
  quantified sentences.
• FO validity, non FO validity, and FO
  counterexample
• To learn about the notion of FO consequence and
  its use to capture the notion of logical
  consequence of quantified sentences.
• FO consequence, non FO consequence, and FO
  counterexample

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Different Kinds of Truths
in propositional logic

• Tautologies
• boolean connectives
• Logical truths
• boolean connectives predicates
• Model-specific truths
• boolean connectives predicates restriction of
  the model

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Different Kinds of Truths
in first-order logic

• Tautologies
• boolean connectives
• First-order validity
• boolean connectives quantifiers
• Logical truths
• boolean connectives quantifiers
  predicates
• Model-specific truths
• boolean connectives quantifiers
  predicates restriction of the model.
• Meaning of quantifiers

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Quantified sentences that are tautologies

• Truth-functional form
• Truth-functional form algorithm

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Essence of tautologies

• When we consider if a sentence is a tautology, we
  follow these rules.
• Consider the sentence as a structure consisting
  of basic sentences (basic pieces) connected
  by Boolean connectives.
• Each basic piece has a truth value. Each
  basic piece can be true or false (ignore
  meaning of predicates, quantifiers in the basic
  piece, restriction of the model.)
• Two occurrences of the same basic pieces always
  have the same truth value.
• If all combinations of truth assignments to basic
  pieces (all rows in truth table) yield TRUE for
  the whole sentence, then the sentence is a
  tautology.

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Example (propositional logic)

• Tet(a) \/ (Tet(a) \/ Small(b))
• Basic pieces Tet(a), Small(b)
• Two occurrences have the same truth value
• All rows are true, then its a tautology.

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Truth-Functional Form

• Decomposition of a sentence into basic
  constituent parts that are still sentences (have
  truth value).
• ?x Tet(x) ? ?y (Cube(y) /\ ?z FrontOf(z,y))
• Truth-functional form A ? B

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Truth-Functional Form Algorithm

• Work left to right
• When encounter a quantifier or an atomic sentence
• Begin underline
• If it is a quantifier, underline the quantifier
  and the wff that it applies to.
• If it is an atomic sentence, underline the atomic
  sentence.
• End underline
• Give the underlined constituent a letter (A,B,C,
  )
• If an identical constituent already appears, use
  same letter
• Otherwise, use a new letter
• Repeat step 2 until then end of sentence
• Replace each underlined constituent with a letter
  that labels it.

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Finding the truth-functional form

• Tet(a) /\ ((Tet(b) \/ ?x Cube(x)) ? Tet(b))
• A /\ ((B \/ C) ? B)
• ?x Large(x) /\ ?y(Tet(y) ? ?z LeftOf(y,z))
• A /\ B

B
C
B
A
A
B
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Exercises

• ?x (Tet(x) /\ Small(x))
• ?x Tet(x) /\ ?y Small(y)
• ?x ?y (Tet(x) /\ Small(y))
• (Tet(b) \/ ?x Tet(x)) ? ?x Tet(x)
• ?x (Tet(x) /\ ?y FrontOf(y,x)) ? (Tet(b) /\ ?z
  FrontOf(z,b)

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Tautologies and Quantification

• A quantified sentence is a tautology if and only
  if its truth-functional form is a tautology.

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Which of these sentences are tautologies?

• ?x Tet(x) \/ ?x Tet(x)
• ?x Tet(x) \/ ?y Tet(y)
• ?x (Tet(x) \/ Tet(x))
• ?x Tet(x) \/ ?x Tet(x))
• ?x (Tet(x) \/ Cube(x) \/ Dodec(x))

Tautology
FO validities. The meaning of quantifier is used
Not an FO validity, but a TW-necessity Even
though the meaning of quantifier is used, we
still cannot conclude that it is always true.
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Tautological Consequence

• Given an argument, how to determine that the
  argument is tautologically valid?
• Find the truth-functional form of each sentence
  in the argument.
• Then prove it.

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Which of these are tautologically valid?

• ?x Tet(x) ? ?x Small(x))
• ?x Tet(x)
• ?x Small(x)
• ?x Tet(x)
• ?x Small(x)
• ?x (Tet(x) /\ Small(x))

• ?x Tet(x)
• ?x Small(x)
• ?x Tet(x) /\ ?x Small(x))
• ?x Tet(x)
• ?x Small(x)
• ?x (Tet(x) /\ Small(x))

A tautological consequence
A tautological consequence
Not a tautological consequence An FO consequence
Not valid
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Goals

• To learn about notion of tautology for
  quantified sentences, and its use to capture the
  notion of truth of quantified sentences.
• truth-functional form and the algorithm to
  construct it.
• To learn about the notion of FO validity and
  its use to capture the notion of truth of
  quantified sentences.
• FO validity, non FO validity, and FO
  counterexample
• To learn about the notion of FO consequence and
  its use to capture the notion of logical
  consequence of quantified sentences.
• FO consequence, non FO consequence, and FO
  counterexample

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First-Order validity (FO validity)

• FO validity
• Non FO validity
• FO counterexample

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Different Kinds of Truths
in first-order logic

• Tautologies
• boolean connectives
• First-order validity
• boolean connectives quantifiers
• Logical truths
• boolean connectives quantifiers
  predicates
• Model-specific truths
• boolean connectives quantifiers
  predicates restriction of the model.
• Meaning of quantifiers

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Terminology
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FO validity

• Tautology True solely in virtue of the meaning
  of
• the truth-functional connectives. /\, \/,
• (that is, not include the meanings of predicates)
• (not include the restriction of the model)
• FO validity True solely in virtue of the meaning
  of
• the truth-functional connectives /\, \/,
• the quantifiers ?, ?
• and the identity symbol.
• (that is, not include the meanings of predicates
  besides identity symbols)
• (not include the restriction of the model)

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FO validity

• A sentence is a first-order validity if a
  sentence is true in any world without the need to
  know the meaning of the predicates (other than
  identity).
• A sentence is a first-order validity if a
  sentence is true in any world under any
  interpretation of the predicates (except identity
  which always means identity).
• To help ourselves avoid interpreting the
  predicates in particular ways, replace predicates
  with nonsensical ones.

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Example

• ?x SameShape(x,x)
• Replace the predicate with nonsensical one.
• ?x SameShape(x,x) becomes ?x P(x,x)
• Interpret /\, \/, and ?, ?
• If P(x,y) means x has the same shape as y
• the sentence is true in any circumstance
• If P(x,y) means x is in the same row as y
• the sentence is true in any circumstance
• If P(x,y) means x is larger than y
• the sentence is not true in some circumstance.
• The sentence is false in some world under some
  interpretation of the predicate.
• Not an FO validity.

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Example

• ?x Tet(x) \/ ?x Tet(x)
• Replace the predicate with nonsensical one.
• ?x Tet(x) \/ ?x Tet(x) becomes ?x K(x) \/ ?x
  K(x)
• Interpret /\, \/, and ?, ?
• If K(x) means x is a tetrahedron
• the sentence is true in any circumstance
• If K(x) means x is a great movie
• the sentence is true in any circumstance
• In fact, no matter what K(x) means, the sentence
  ?x K(x) \/ ?x K(x) is always true.
• The sentence is true in any world under any
  interpretation of the predicate.
• (The sentence is true and in any world without
  the need to know the meaning of the predicate ).
• FO validity.

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Which of these are FO validities?

• ?x Small(x) ? Small(b)
• (Tet(a) /\ ad) ? Tet(d)
• (FrontOf(a,b) /\ SameRow(a,c)) ? FrontOf(c,b)

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First-order counterexample

• FO validity
• For any circumstance and any interpretation of
  the predicates, the sentence is true
• Recall Showing non tautology
• Find truth assignment to atomic sentences
• Such that whole sentence is false
• Non FO validity
• For some circumstance and some interpretation of
  the predicates, the sentence is false.

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Example of FO counterexample

• ?x SameShape(x,x)
• Replace the predicate with nonsensical one.
• ?x SameShape(x,x) becomes ?x P(x,x)
• Come up with a interpretation of the predicate
• P(x,y) means x is taller than y.
• Come up with a world (not necessarily Tarskis
  world).
• A world with 3 people John, James, Jack with
  height 4, 5, 6 respectively.
• Verify that the sentence is false under this
  interpretation in this world.
• P(John, John) is false. Thus, ?x P(x,x) is
  false.
• Not an FO validity.

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Exercise

• Come up with an FO counterexample to show that
  the following sentence is not an FO validity.
• ?x (FrontOf(x,a) ? BackOf(a,x))

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Goals

• To learn about notion of tautology for
  quantified sentences, and its use to capture the
  notion of truth of quantified sentences.
• truth-functional form and the algorithm to
  construct it.
• To learn about the notion of FO validity and
  its use to capture the notion of truth of
  quantified sentences.
• FO validity, non FO validity, and FO
  counterexample
• To learn about the notion of FO consequence and
  its use to capture the notion of logical
  consequence of quantified sentences.
• FO consequence, non FO consequence, and FO
  counterexample

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FO consequence

• FO consequence
• Non FO consequence
• FO counterexample

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FO Consequence

• Tautological consequence True solely in virtue
  of the meaning of
• the truth-functional connectives. /\, \/,
• (that is, not include the meanings of predicates)
• (not include the restriction of the worlds)
• FO consequence True solely in virtue of the
  meaning of
• the truth-functional connectives /\, \/,
• the quantifiers ?, ?
• and the identity symbol.
• (that is, not include the meanings of predicates
  besides identity symbols)
• (not include the restriction of the worlds)

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FO consequence

• Q is a first-order consequence of P1, P2, P3 if Q
  always follows from P1, P2, P3 without the need
  to know the meaning of the predicates (other than
  identity).
• Q is a first-order consequence of P1, P2, P3 if Q
  always follows from P1, P2, P3 under any
  interpretation of the predicates (except identity
  which always means identity).
• To help ourselves avoid interpreting the
  predicates in particular ways, replace predicates
  with nonsensical ones.

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• ?x (Tet(x) ? Small(x))
• _ Tet(a)
• Small(a)
• Replace the predicate with nonsensical one.
• ?x (Mida(x) ? Epo(x))
• _ Mida(a)
• Epo(a)
• Interpret /\, \/, and ?, ?
• The conclusion follows from the premises in any
  world under any interpretation of the predicates.
• (The conclusion follows from the premises in any
  world without the need to know the meaning of the
  predicates).
• FO consequence.

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FO consequence?

• ?x Larger(a,x)
• ?x (xb ? Larger(x,b))
• Larger(a,b)

• ?x P(a,x)
• ?x (xb ? P(x,b))
• P(a,b)

?x Larger(x,a) ?x Larger(b,x) Larger(c,d) Larger
(a,b)
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First-order counterexample

• FO consequence
• For any circumstance and any interpretation of
  the predicates, the conclusion follows from the
  premises
• Recall Showing non tautological consequence
• Find truth assignment to atomic sentences
• All premises are true
• Conclusion is false.
• Non FO consequence
• For some circumstance and some interpretation of
  the predicates, the conclusion does not follow
  from the premises

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First-Order Counterexample

• 1. Replace predicates
• 2. Define interpretation
• P(x,y) means x likes y

• 3. Create world

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Exercise

• Create a FO counterexample for this argument.
• ?x (FrontOf(x,a) ? Cube(x))
• Tet(b)
• FrontOf(b,a)

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Goals

• To learn about notion of tautology for
  quantified sentences, and its use to capture the
  notion of truth of quantified sentences.
• truth-functional form and the algorithm to
  construct it.
• To learn about the notion of FO validity and
  its use to capture the notion of truth of
  quantified sentences.
• FO validity, non FO validity, and FO
  counterexample
• To learn about the notion of FO consequence and
  its use to capture the notion of logical
  consequence of quantified sentences.
• FO consequence, non FO consequence, and FO
  counterexample