02 Propositional

8/3/2019 02 Propositional

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Computational Applied LogicCSC 503 Fall 2005

Jon Doyle

Department of Computer ScienceNorth Carolina State University

Propositional logic

NC State University 1 / 77 CSC 503 Fall 2005c 2005 by Jon Doyle
http://find/http://goback/


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Propositional logic Statements and their uses

What things can one express?

Sounds/exclamations/marks

Words

Statements

Sets of statements = theories Partial statements

Sets of partial statements

Sequences of statements or sets of statements



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How to do things with sentences

Declarative: facts and descriptions

Interrogative: questions

Imperative: commands and pleas



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What can I express with statements?

Knowledge/facts/opinions/conditions

Ignorance/uncertainty

Goals/desires/intentions

Procedures/methods Propositional attitudes



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Propositional logic Language

Statements

Complete statements = propositions Snow is white

Letters snow-is-white

Four-score-and-seven-years-ago-

our-fathers-brought-forth-

on-this-continent-a-new-nation-

conceived-in-Liberty-

and-dedicated-to-the-proposition-that-all-men-are-created-equal

Ignore spelling, just enumerate A1, A2, . . .


P i i l l i L


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Propositional connectives

Disjunction or Conjunction and Negation not

Conditional implies Biconditional iff = if and only if+ Exclusive or xor| Sheffer stroke nand

n

at least n

And more besides, when we visit description logics


P iti l l i L


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Complex propositions

All combinators use parentheses to provideunique parse tree.

We omit parentheses when parse is clear.

p = A B C p = (((A) B) C)

Depth= depth of parse tree (root has depth 0) Depth(p) = 3

Support= set of letters appearing in tree Support(p) = {A, B, C}


Propositional logic Meaning


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Meaning of propositions

We only consider standard meanings at this time. Standard meanings

True/False (= T/F, 1/0, /)

Multivalued logics Elements of boolean lattices Belnap 4-valued logic {TT, TF, FT, FF}

Probabilistic logics Probability values in [0, 1]

Fuzzy logics Possibility values in [0, 1]




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Logicians are weird

Logical meaning = English (etc.) meaning If 1=2, then Im the Man in the Moon. She is either a lawyer or a professor. Ive won every World Cup game Ive played.

Logical meaning is atemporal 10:00AM Assert 10:01AM Assert Simple inconsistency, or change?




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Truth functionality

Basic connectives are truth functional

Truth of compound statement determined bytruth of the connected substatements

Truth of compound a function of truth ofconstituents

Truth tablesrepresent these functions




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Connective truth tables

( )

T T TT F T

F T T

F F F

( )

T T TT F F

F T F

F F F

()

T FF T




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( )

T T TT F F

F T T

F F T

( )

T T TT F F

F T F

F F T




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p g g


( + )

T T FT F T

F T T

F F F

( | )

T T FT F T

F T T

F F T




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p g g

Truth tables

Complete truth tables One column for each proposition in formation tree

Abbreviated truth tables

Omit one or more intermediate propositions




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Non-truth-functional connectives

Truth values of component propositions do notdetermine truth value of compound proposition.

because

causes necessarily implies

preceded

is a shorter statement than

expresses more information than

is more likely than

Alice believes but Bob claims




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Truth valuations

Truth assignment A : L {T, F} Truth valuation V : L(L) {T, F}

Required to respect truth tables in every connective

Valuations must agree on a propositionwhenever they agree on the propositionssupport




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Logical equivalence

Means and are logically equivalent

Logical equivalence = agreement w.r.t. every

valuation

True just in case the truth table column for contains only Ts

Each row corresponds to a class of valuations Truth table summarizes all valuations restricted to

support of a proposition




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Lattice of meanings

Propositional equivalence classes [] = { | }

2n distinct truth tables over n letters

Thus 2n equivalence classes over n letters

Form a Boolean lattice with respect to , ,

Define lattice order iff [ ] = []




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Metalanguage vs. object language

is part of the logical metalanguage Part of the language we use to talk about logical

statements

Not part of the logical object language in whichpropositions are expressed.

Other metalinguistic notions:

Entailment Satisfiability

Provability




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Adequacy

What can one say with a specific set of connectives?

S is adequate iff every proposition is equivalentto some proposition constructed using onlyconnectives in S

For every truth-functional , there is some over S such that

Claim: {, , } is adequate. Why?

Claim: {, } is adequate. Why?




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Normal forms

Literal = letter or negation of a letter: A, A Clause = disjunction of literals: A1 . . . An Conjunct = conjunction of literals: A1 . . . An

CNF = Conjunctive normal form Conjunction of clauses (A1 . . . An) . . . (B1 . . . Bm)

DNF = Disjunctive normal form Disjunction of conjuncts (A1 . . . An) . . . (B1 . . . Bm)




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Linguistic expressiveness

Choose or change the basis connectives to improve Consision of expression

Cardinality of expression

Complexity of expression Clarity/comprehensibility/convenience of

expression




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Validity

is valid iff every valuation makes it true is a tautology

Taut = set of all tautologies

is nontrivialif neither nor are valid




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Satisfiability

is satisfiable iff some valuation makes it true is possibly true

is unsatisfiable iff no valuation makes it true

is a contradiction

is a tautology




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Consistency

is consistent just in case some valuationmakes every statement in true

For finite , just in case

is satisfiable

is inconsistent if no valuation makes allstatements true

and

are (in)consistent iff{, } is (in)consistent

NC State University 25 / 77CSC 503 Fall 2005

c 2005 by Jon Doyle



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Logical consequence

|= means V() = T whenever V() = T entails

|= means V() = T whenever V() = T for each entails

Cn() is the set of consequencesof Cn() = { | |= }

Taut = Cn()


c 2005 by Jon Doyle



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Properties of consequences

Monotonic: implies Cn() Cn() Supra-tautologous: Taut Cn()

Idempotent: Cn(Cn()) = Cn()

Additive: Cn()


c 2005 by Jon Doyle



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Completeness

A complete theory determines truth values for allpropositions

is complete iff for each p either p Cn(), or

p Cn()

Is {p, p} complete?


c 2005 by Jon Doyle



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Models

Models = interpretations that make true V a model of iff V() = T for each

M() = {V | .V() = T} is the set of allmodels of .

implies M() M()


c 2005 by Jon Doyle

Propositional logic Formalizing theories


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So what good is logic?

Precise concepts for expressing theories Precise concepts for critiquing theories


c 2005 by Jon Doyle


Th f l i l f li i


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The process of logical formalization

Commence with initial formulation Common sense Expertise Informed speculation Wild guesses

Critique the formulation with respect to thedesired qualities

Correct the visible flaws as seems fit

Continue this process until convergence


c 2005 by Jon Doyle


Th NC S K l d Di


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The NC State Knowledge Discovery

Method

Commence to continuously correct the contentvia the critique categories until convergence

Truth

Correctness**Consistency**CompletenessCategoricity

**ContingencyChanceCoverageCourageousness

Goodness

Computability**Complexity**CardinalityCompromises

ConvenienceCharityCompactness

Beauty

ClarityComprehensibilityCleavageCogency

CommonsensicalityContinuity

Perfection

ClosenessCumulativityConvergenceConstancy


c 2005 by Jon Doyle


L i l f li ti h


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Logical formalization as search

Different critiques might suggest incompatiblecorrections; what to desire?

Applied corrections might not work

Confusion or contradiction can suggest retreatto prior formulation; divergence

View this process as a search for the rightformulation

Process state as position in a space of assessment

dimensions Assessment criteria as elements of heuristic

evaluation function Correction methods as possible actions


c 2005 by Jon Doyle

Propositional logic Inference

T bl f


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Tableau proofs

Tableaux = tables Labeled trees, built up from atomic tableaux

Various nice computational properties

We will consider other proof formalisms later


c 2005 by Jon Doyle


At i iti l t bl


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Atomic propositional tableaux

TA FA

T()

F

F()

T


c 2005 by Jon Doyle


Atomic propositional tablea


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T( )

T

T

F( )

T

dd

T

T( )

T

dd

T

F( )

F

F


c 2005 by Jon Doyle




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T( )

F

dd

T

F( )

T

F

T( )

T

dd

F

T F

F( )

T

dd

F

F T


c 2005 by Jon Doyle


Tableaux construction rules


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Tableaux construction rules

Root is proposition under consideration Apply atomic tableau to some proposition in tree

Append atomic tableau at end of branch beneath Head of appended tableau duplicates proposition

being reduced


c 2005 by Jon Doyle


Tableau properties


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Tableau properties

Tableau , path P on , and E an entry on P

E is reduced iff all entries on the atomic tableauwith root E appear on P

P is contradictory iff both T and F appear onP

P is finished iff it is contradictory or every entry

on P is reduced on P is finished iff every path is finished

is contradictory iff every path is contradictory


c 2005 by Jon Doyle


Tableau proof


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Tableau proof

Proof by refutation Tableau proofof = a contradictory tableau with

root F

means is tableau provable

Tableau refutationof = a contradictory tableauwith root T

is tableau refutable iff it has a tableaurefutation


c 2005 by Jon Doyle


Complete systematic tableaux


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Complete systematic tableaux

Construct increasing sequence of tableaux Find highest level with unreduced noncontradictory

entry E Find leftmost path containing such an entry Adjoin atomic tableau with root E to each such path Adjunction means m m+1

Limit (union) of this sequence is the CST


c 2005 by Jon Doyle


Properties of CST


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Properties of CST

Every CST is finished

If a CST is contradictory, it contains a finitecontradictory tableau m

Thus if a CST is a proof, it is a finite tableau.

Every CST is finite


c 2005 by Jon Doyle


Soundness and completeness


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What is the relationship between truth and proof?Between entailment and provability?

Soundnessmeans truth preserving

A logic is sound if p implies |= p

Completenessmeans proof preserving A logic is complete if |= p implies p

Logicians often will use completeness proof tomean a proof of both soundness and completeness


c 2005 by Jon Doyle


Soundness of tableau proof


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Soundness of tableau proof

Each satisfying valuation of a formula mustagree with the labels on some path through thetableau

No valuation can agree with a contradictory path

In a proof, all paths are contradictory


c 2005 by Jon Doyle


Completeness of tableau proof


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Completeness of tableau proof

Each finished but noncontradictory pathprovides a counterexample

Assign T to A if TA appears on the path Assign F to A otherwise


c 2005 by Jon Doyle


Tableau proof from premises


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Tableau proof from premises

Allow a set of premises for use in proofs Add a new atomic tableau Tp for each premise p

Tp can be added to any path that does notcontradict it


c 2005 by Jon Doyle


Complete systematic tableaux from


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Complete systematic tableaux from

premises

Assume an enumeration of the premises

Add premises sequentially to each

noncontradictory finished path


c 2005 by Jon Doyle




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Sound: p implies |= p

Complete: |= p implies p


c 2005 by Jon Doyle


Compactness


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Compactness

Propositional logic is a compact logic |= p iff |= p for some finite subset

One only needs finitely many premises to get

any particular consequence

An infinite set is satisfiable iff every finitesubset of is satisfiable


c 2005 by Jon Doyle


Deductive closure


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Deductive closure

The deductive closureof a set of propositionscontains all the statements deducible from the set

Th() = {p | p}

Soundness and completeness mean Th() = Cn()


c 2005 by Jon Doyle


Deduction theorem


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Deduction theorem

If is finite and

is the conjunction of thesestatements, the following conditions are equivalent:

|=

|=

This shows the desired matching of truth and proof


c 2005 by Jon Doyle


Alternative proof systems


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Alternative proof systems

Proof by intimidation Shut up, he explained. Five-finger argument

Axiomatic proofs

Natural Deduction proofs


c 2005 by Jon Doyle


Axiomatic logics


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g

Axioms Maybe lots Axiom schemata

( ( )) (( ( )) (( ) ( )))

(( ) (( ) ))

Inference rules Usually a small set Modus ponens

p, p q q


c 2005 by Jon Doyle


Axiomatic proofs


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p

Proof = sequence of statements Each statement either An axiom, or A conclusion of an inference rule applied to preceding

statements

Final statement is the theorem


c 2005 by Jon Doyle


Natural deduction proofs


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p

No axioms Lots of inference rules

Rules p correspond to axioms

Introduction and discharge of assumptions

Dependency tracking


c 2005 by Jon Doyle


A sample proof


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p p

In the style of Kalish and Montague

Line Statement Justification Deps.

1. A B Premise {1}

2. B C Premise {2}3. A Hypothesis {3}4. B MP 1,3 {1,3}5. C MP 2,4 {1,2,3}6. A C Discharge 3,5 {1,2}7. A B B C -introduction {1,2}8. (A B B C) (A C) Discharge 7,6 {}


c 2005 by Jon Doyle

Propositional logic Resolution

Resolution


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Language

Inference method

Proof automation

Logic programming


c 2005 by Jon Doyle


Language


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g g

CNF language: Literals Clauses Formulas

Set notation: Clauses as finite sets of literals

Empty clause is always false

Formulae as finite sets of clauses Empty formula {} is always true

Sets mean syntactic irredundance


c 2005 by Jon Doyle


Linguistic models


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Partial truth assignment = consistent set ofliterals Does not contain both A and A Literals in set = what is assigned T

Complete truth assignment contains each letteror its negation

A |= S Means assignment A satisifies formula (set) S

For each C S, C A = S (un)satisfiable iff there is an (no) assignment

that satisfies S


c 2005 by Jon Doyle


Prolog


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Divide clauses into positive and negative literals Interpret each clause as implication

A1 . . . An B1 . . . Bm B1 . . . Bm A1 . . . An

Hornclause: at most one positive literal Programclause: exactly one positive literal

Prolog program = set of program clauses


c 2005 by Jon Doyle


Prolog notation


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Rule: some negative literals Fact or unit clause: no negative literals Goalclause: no positive literals

Rule A B1, . . . , Bm A : B1, . . . , BmFact A A :Goal B1, . . . , Bm : B1, . . . , Bm

Nomenclature for clause parts:

head : bodygoal : subgoals


c 2005 by Jon Doyle


Modus Ponens in Clausal Form


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Modus Ponens:

From and infer

From and infer

Cut rule generalizes Modus Ponens: From and infer

From and infer


c 2005 by Jon Doyle


Resolution rule


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Resolving on literal A:

Clause {A} C1 Clause {A} C2

Infer C1 C2


c 2005 by Jon Doyle


Resolution deduction


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A resolution deductionof C from S consists of

A finite sequence C1, . . . , Cn with Cn = C Each Ci is either

A clause in S or

The resolvent of two preceding clauses in thesequence


c 2005 by Jon Doyle


Resolution refutations


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A resolution refutationof S is a resolution proof of from S

Resolution preserves satisfiability Clauses {A} C1, {A} C2 Resolvent C1 C2

Hence refutation is sound


c 2005 by Jon Doyle


Resolution trees


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A resolution treededuction of C from S:

A labeled binary tree such that

The root is labeled with clause C

The leaves are labeled with the clauses of S

Each nonleaf node is labeled with resolvents ofits children


c 2005 by Jon Doyle


Resolution closure


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The resolution closureR(S) of a set of clauses S isthe closure of S under the operation of takingresolutions

S R(S)

If C1, C2 R(S) and C is a resolvent of C1 andC2, then C R(S)

There is a resolution refutation of S iff R(S)


c 2005 by Jon Doyle


Semantic analysis


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Formula S, literal Literal reductions:

S() = {C R(S) | , / C} If S is unsatisfiable, then so is S()

S = {C {} | C S / C} Formula reduced by assuming is true If S is unsatisfiable, both S and S must be

unsatisifable


c 2005 by Jon Doyle




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S is satisfiable iff either S

or S

is satisfiable The unsatisifiable sentences U are generatedby

If S, then S U If S U and S U, then S U

If S is unsatisfiable, then R(S)


c 2005 by Jon Doyle


Computational complexity


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SAT = set of all satisfiable formulae

Is S satisfiable?

Resolution answer

2-SAT is linear time

SAT is NP-complete

3-SAT is NP-complete

This is good news too, not just bad;more on this later


c 2005 by Jon Doyle


Restricted resolution


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T-resolution: never resolve a tautology

Semanticresolution: one parent is falsified byassignment A

Orderedresolution: order letters, always resolve

on highest-index letter possible Supportrestriction: never resolve two clauses

outside support clauses

These are sound and complete, but others are not


c 2005 by Jon Doyle


Linear resolution


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A linearresolution deduction of C from S is asequence of pairs C0, B0, . . . , Cn, Bn such that

C = Cn C0 S

Each Bi is either in S or is some preceding Cj Each Ci+1 is a resolvent of Ci and Bi.

C is linearly deducible (refutable) from S if there is alinear deduction (refutation) of C from S


c 2005 by Jon Doyle


Nomenclature


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S = inputclauses

C0 = startingclauses

Ci = centerclauses

Bi = sideclauses


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Linear resolution is sound (by restriction) Linear resolution is complete


c 2005 by Jon Doyle


Linear input resolution


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Starts with goal clause All side clauses are input clauses

Incomplete in general

Consider all clauses of two literals Complete when all inputs are program clauses


c 2005 by Jon Doyle


Refinements


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LD-resolution = linear definite resolution Ordered literals = definiteclauses Resolutions maintain ordering within insertions

SLD-resolution = selected linear definite

resolution Resolutions follow syntactic ordering of literals Prolog: always resolve on first goal literal

Both are sound and complete


c 2005 by Jon Doyle


Search and backtracking


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Success and failure on resolution paths Success = find on path Failure = end path with no

Search all paths until success or exhaustion

Depth-first search, breadth-first search, etc.

Pure backtracking DFS can fail!

Intelligent backtracking schemes


c 2005 by Jon Doyle

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02 Propositional