Formal Methods in Computer Science CS1502 The Logic of Boolean Connectives

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Formal Methods in Computer ScienceCS1502The
Logic of Boolean Connectives

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• To understand the concepts of
• Truth, possibility, contradiction, equivalence,
  consequence
• To understand the attempts to capture the notion
  of logical truths using
• Tautologies and model-specific truths.
• To gain skill in classifying sentences using the
  above concepts.

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Truth value

• What is truth value?
• Truth value is a property of a sentence. The
  truth value of a sentence is the state of the
  sentence, of being true or false, with respect to
  a world.
• Notice, the truth value of a sentence depends on
  which world it is evaluated against.

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Truth value

• In other words, a sentence is true in some worlds
  and false in some worlds.
• Ex Cube(b), LeftOf(a,b), Taller(max,claire)
• Is this always the case? Are there sentences
  that are true in all worlds or sentences that are
  false in all worlds?

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Truth value

• Cube(a) \/ Tet(a) \/ Dodec(a)
• S(b) \/ S(b)
• Large(b) /\ Larger(e,b)
• ( P /\ P)
• (A \/ (B /\ C)) /\ ((B \/ C) /\ A)

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Truth-value vs. Truth

• Truth value is a property of a sentence. The
  truth value of a sentence is the state of the
  sentence, of being true or false, with respect to
  a world.
• A sentence is a truth if it is true in all
  worlds.
• (we will further refine this notion later)

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Related Definitions

• Definitions (tentative)
• A sentence is a truth if it is true in all
  worlds.
• A sentence is a contradiction if it is false in
  all worlds.
• A sentence is a possibility if it is true in at
  least one world.
• A sentence is a contingency if it is true in at
  least one world and it is false in at least one
  world.
• (we will further refine these later)

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Truth value

• Cube(a) \/ Tet(a) \/ Dodec(a)
• S(b) \/ S(b)
• Large(b) /\ Larger(e,b)
• ( P /\ P)
• (A \/ (B /\ C)) /\ ((B \/ C) /\ A)

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Natural Questions that Arise

• Given a sentence, is it a truth, a contradiction,
  and/or a possibility?
• Did we have this problem with atomic sentences
  (before Boolean connectives are introduces)?

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Concept Checking Questions

• Can a sentence be both a truth and a
  contradiction?
• If we know that a sentence is a truth, is it
  necessarily a possibility?
• If we know that a sentence is a possibility, is
  it necessarily a truth?
• Can a sentence be neither a truth nor a
  contradiction?
• Can a sentence be neither a possibility nor a
  contradiction?

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Goals

• To understand the concepts of
• Truth, possibility, contradiction, equivalence,
  consequence
• To understand the attempts to capture the notion
  of logical truths using
• Tautologies and model-specific truths.
• To gain skill in classifying sentences using the
  above concepts.

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Problems with current Definition of Truth

• Definitions (tentative)
• A sentence is a truth if it is true in all
  worlds.
• What involves in determining if Cube(b) \/
  Tet(b) \/ Dodec(b) is a truth?
• Which object b refers to.
• How to interpret predicates Cube, Tet, Dodec.
• How to interpret Boolean connectives.
• All worlds. All? What exactly are those?
• Restrictions of worlds as specified in a model.
• e.g. world is an 8x8 board, 3 possible shapes, 3
  possible sizes, etc.
• Which model we are using.
• Many of these have not been specified.
• Thus, our definition is not well-defined, it is
  ambiguous how to determine if a sentence is a
  truth.

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So, is it a truth or not?

• Is Cube(b) \/ Tet(b) \/ Dodec(b) a truth?
• In Tarskis worlds model?
• In the following model?
• An object can be one of the following shapes
• Cube, Tet, Dodec, Sphere
• It depends on the model.

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Model-specific truths and Logical truths

• Definition A sentence is an X-necessity
  (model-specific truth) with respect to a model X
  if
• using the interpretation of predicates according
  to model X,
• the sentence is true in all worlds under model X.
• Description A sentence is a logical truth if
• The sentence is true for all circumstances that
  correspond to legitimate interpretations of the
  predicates.
• Note
• All circumstances mean all possible worlds (in
  all possible models).
• We need to consider all possible legitimate
  interpretations of the predicates, not just one
  specific interpretation.
• Deciding which interpretations are legitimate is
  subjective. Thus, the notion of logical truth is
  not well-defined.

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Is it well-defined?

• Determining if Cube(b) \/ Tet(b) \/ Dodec(b) is a
  TW-necessity.
• Definition A sentence is an X-necessity
  (model-specific truth) with respect to a model X
  if
• using the interpretation of predicates according
  to model X,
• the sentence is true in all worlds under model X.
• Check
• Tarskis worlds model is being used
• The set of worlds to consider (from TW model)
• Names (terms) are determinate
• Predicates are determinate (from TW model)
• Boolean connectives

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Previous Definitions

• Definitions (tentative)
• A sentence is a truth if it is true in all
  worlds.
• A sentence is a contradiction if it is false in
  all worlds.
• A sentence is a possibility if it is true in at
  least one world.
• (we will refine these)

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Model-specific classifications

• Definitions
• A sentence is an X-necessity (model-specific
  truth)with respect to a model X if
• using the interpretation of predicates according
  to model X,
• the sentence is true in all worlds under model X.
• A sentence is an X-contradiction (model-specific
  contradiction) with respect to a model X if
• using the interpretation of predicates according
  to model X,
• the sentence is false in all worlds under model
  X.
• A sentence is an X-possibility (model-specific
  possibility)with respect to a model X if
• using the interpretation of predicates according
  to model X,
• the sentence is true in at least one world under
  model X.

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Classify these sentences

• SameRow(b,c) \/ FrontOf(b,c) \/ BackOf(b,c)
• TW-necessity
• Not a necessity in model with shape Sphere
• Large(b) /\ Larger(e,b)
• TW-contradiction
• Possibility in model with size Huge
• LeftOf(a1,a2) /\ LeftOf(a2,a3) /\ /\
  LeftOf(a7,a8)
• TW-possibility
• Not a possibility in model with board 5x5.

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Logical truth vs Model-specific truth why?

• Intuitively, we want to know what sentences are
  logical truths, logical possibilities, logical
  contradictions.
• Ex Is it possible for object b to be to the left
  of c, and c to be to the left of d, yet c is not
  between b and d?
• However, these notions are vague.
• In our first attempt to capture these notions, we
  made certain assumptions and come up with the
  definitions of model-specific truth, etc.
• Are there other ways to capture these notions?

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Logical truth vs Model-specific truth how?

• The notion of logical truth is vague because
• it is not clear what each predicate means, and
• it is not clear what circumstances need to be
  considered.
• The definition of model-specific truth removes
  these ambiguities by
• clearly defining what worlds are considered, and
• clearly defining what each predicate means (each,
  predicate is determinate)

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Another attempt to capture logical truths

• Ignore interpretation of atomic sentences
  (predicates names)
• Each atomic sentence is considered a contingency.
• A contingency is a sentence that is true in at
  least one circumstance and false in at least one
  circumstance
• Ex aa has 2 possible truth values true and
  false.
• Ex Larger(a,a)
• Ignore any restriction about the worlds
• Interaction between atomic sentences are ignored
• Ex The pair ( Large(a) , Small(a) ) has 4
  possible combinations of truth values (TT, TF,
  FT, FF).
• Ex ( Cube(a), Tet(a), Dodec(a) ) has 8 combos.
• Ex ( Large(b), Larger(e,b) ) has 4 combos.

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Helping ourselves to ignore

• To help ourselves ignore these things, we should
  replace each atomic sentence by a meaningless
  variable.
• Large(b) /\ Larger(e,b) ? A /\ B
• Small(d) \/ Large(d) ? P \/ Q
• (Tet(a) /\ Cube(b)) \/ Tet(a) \/ Cube(b) ?
  (A /\ B) \/ A \/ B

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How many combos of truth values?

• aa /\ bc
• Large(b) /\ Larger(e,b)
• Small(d) \/ Large(d)
• (Cube(d) /\ Cube(d))
• (Tet(a) /\ Cube(b)) \/ Tet(a) \/ Cube(b)
• Is there a general rules to count the number of
  combinations?
• How should we define truth here?

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More examples

• (Large(a) \/ Large(a))
• (LeftOf(a,b) /\ LeftOf(b,c) /\ Adjoins(a,c))
• (Tet(a) /\ LeftOf(a,b)) \/ (Tet(a) /\
  FrontOf(a,b))

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Truth-Table necessity

• Definition A sentence is a tautology if it is
  true in all rows of its truth table.
• Other names Truth-Table necessity,
  TT-necessity, tautological truth.

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Related Definitions

• Definition
• A sentence is a tautology if it is true in all
  rows of its truth table.
• A sentence is a tautological contradiction if it
  is false in all rows of its truth table.
• Other names Truth-Table contradiction,
  TT-contradiction
• A sentence is a tautological possibility if it is
  true in at least one rows of its truth table.
• Other names Truth-Table possibility,
  TT-possibility

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Classify these

• aa /\ bc
• Large(b) /\ Larger(e,b)
• Small(d) \/ Large(d)
• (Cube(d) /\ Cube(d))
• (Tet(a) /\ Cube(b)) \/ Tet(a) \/ Cube(b)

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Checking tautologies

• How easy or how hard it is to check if a sentence
  is a tautology?
• What about model-specific truth and logical
  truth?

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Concept Checking Questions

• If a sentence is a tautology, is it a logical
  truth? The opposite?
• Ex Larger(a,b) \/ Larger(a,b)
• If a sentence is a tautology, is it a
  model-specific truth? The opposite?
• If a sentence is a TT-possibility, is it a
  model-specific truth? The opposite?

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3 interpretations

• Model-specific
• Unambiguous interpretation of predicates and
  worlds
• Logical
• Intuitive interpretation of predicates and worlds
• Truth table
• Doesnt interpret predicates
• Only interpret Boolean connectives

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When should we use which interpretation

• (like on exam)
• A /\ B ? tautological
• Cube(b) \/ Tet(b) ? TW (model-specific)
• Paul is at the library or he is at the movie?
  define a model, state assumptions (a person can
  only be at one place at a time) and use the model.

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Table of Terms
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Exercise

• Create a sentence that is a truth under one
  interpretation, but not a truth under another
  interpretation
• Show them to your neighbors and discuss any
  disagreement.
• possibility
• contradiction

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Goals

• To understand the concepts of
• Truth, possibility, contradiction, equivalence,
  consequence
• To understand the attempts to capture the notion
  of logical truths using
• Tautologies and model-specific truths.
• To gain skill in classifying sentences using the
  above concepts.

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Equivalence

• Description Two sentences P and Q are logically
  equivalent if the truth value of P and Q are the
  same in all circumstances.
• Definition Two sentences P and Q are
  X-equivalentwith respect to a model X if
• using the interpretation of predicates according
  to model X,
• P and Q have the same truth value in all worlds
  under model X.
• Definition Two sentences P and Q are
  tautological equivalent if P and Q have the same
  truth value in all rows of their joint truth
  table.

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TT-equivalence

• Two sentences are tautologically equivalent
  (TT-equivalent) if their truth-values are the
  same in every row of their joint truth table.
• Meaning of predicates are completely ignored.
• (Tet(a) /\ Tet(b)) and (Tet(a) \/ Tet(b)) are
  TT-equivalent
• Tet(a) and Cube(a) \/ Dodec(a) is not
  TT-equivalent

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Logical Equivalence

• Two sentences are logically equivalent if their
  truth values are the same in all circumstances
  that correspond to legitimate interpretations of
  the predicates.
• All circumstances mean all possible worlds in all
  possible models.

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Model-specific Equivalence

• Given a model M, two sentences are M-equivalence
  if their truth values are the same in all worlds
  in model M.
• Two sentences are TW-equivalence their truth
  values are the same in all worlds that can be
  created in the Tarskis model.

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Tautological, logical, and TW-equivalences

• If two sentences are tautologically equivalent,
  then they are logically equivalent but not vice
  versa.
• If two sentences are logically equivalent, then
  they are TW-equivalent but not vice versa.

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Exercise

• Create pairs of sentences that are
• TW-equivalent but are not logically equivalent,
• logically equivalent but are not TT-equivalent,
• not TT-equivalent.
• Show them to a classmate next to you and discuss
  any disagreement.

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Goals

• To understand the concepts of
• Truth, possibility, contradiction, equivalence,
  consequence
• To understand the attempts to capture the notion
  of logical truths using
• Tautologies and model-specific truths.
• To gain skill in classifying sentences using the
  above concepts.

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Consequence

• Description Q is a logical consequence of
  P1,P2,P3 if whenever P1,P2,P3 are all true, Q is
  also true
• Definition Q is an X-consequence of P1,P2,P3
  with respect to a model X if
• using the interpretation of predicates according
  to model X,
• in all worlds under model X that P1,P2,P3 are all
  true, Q is also true.
• Definition Q is a tautological consequence of
  P1,P2,P3 if in all rows of that P1,P2,P3 are all
  true, Q is also true.

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TT-consequence

• Q is a tautological consequence (TT-consequence)
  of P1, P2,,Pn if in every row of the joint truth
  table that P1, P2,,Pn are all true, Q is also
  true.
• Meaning of predicates are completely ignored.
• Tet(a) is a TT-consequence of Tet(a) \/ Cube(a)
  and Cube(a)

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• Tet(a) is a TT-consequence of Tet(a) \/ Cube(a)
  and Cube(a)

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Logical Consequence

• Q is a (logical) consequence of P1, P2,,Pn if in
  all circumstances that correspond to legitimate
  interpretations of the predicates and P1, P2,,Pn
  are all true, Q is also true.
• All circumstances mean all possible worlds in all
  possible models.
• ac is a consequence of ab and bc
• Tet(a) is not a consequence of Cube(a) and
  Dodec(a)

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Model-specific Consequence

• Given a model M, Q is an M-consequence of P1,
  P2,,Pn if in all worlds in model M that P1,
  P2,,Pn are all true, Q is also true.
• Q is a TW-consequence of P1, P2,,Pn if in all
  worlds in the Tarskis model that P1, P2,,Pn are
  all true, Q is also true.
• Tet(a) is a TW-consequence of Cube(a) and
  Dodec(a)

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Tautological, logical, and TW-consequences

• If Q is a tautological consequence of P1, P2,
  ,Pn, then Q is a logical consequence of P1, P2,
  ,Pn, but not vice versa.
• If Q is a logical consequence of P1, P2, ,Pn,
  then Q is a TW-consequence of P1, P2, ,Pn, but
  not vice versa.

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Exercise

• Create sets of sentences that form
• TW-consequence but not logical consequence
• Logical consequence but not TT-consequence,
• not TT-consequence
• Show them to a classmate next to you and discuss
  any disagreement.

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Goals

• To understand the concepts of
• Truth, possibility, contradiction, equivalence,
  consequence
• To understand the attempts to capture the notion
  of logical truths using
• Tautologies and model-specific truths.
• To gain skill in classifying sentences using the
  above concepts.