An Introduction to Classical Logic Propositional and Predicate Logic

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An Introduction to Classical Logic(Propositional
and Predicate Logic)

• Lecture Notes for ISA 763

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Propositional and Predicate Logic

• Propositional Logic
• The study of statements and their connectivity
  structure.
• Predicate Logic
• The study of individuals and their properties.
• Study syntax and semantics for both.
• Propositional logic more abstract and hence less
  detailed than predicate logic.

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Propositional Logic Syntax

• A collection of atomic propositional symbols.
• Say A ai 0 contradiction
• A collection of logical connectives.
• (and) , (or) v, ( not ) ? , (implies)
• Inductively define propositions as
• If X,Y are propositions, so are
• X Y, X v Y, X Y, ? X.
• Examples
• a1a2, (a a2)v(a3(? a4)) are propositions.

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Propositional Logic Semantics

• A model M of a propositional language consists of
  
• a collection of atoms, say B bi 0 where __ is excluded from B, and
• a partial mapping M from A ai 0 bi 0
• If M(ai) e B, we say that ai is true in M. We
  write ai is true in M as M ai. (Read M
  satisfies ai).
• is referred to as the satisfaction relation.

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Propositional Semantics Continued

• Extend M, and therefore the satisfaction relation
  to all propositions using the following inductive
  definition
• M X Y iff M X and M Y.
• M X v Y iff M X or M Y.
• M X Y if M X then M Y.
• M ? X, if it is not the case that M X.
• Notice usage of truth tables

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Propositional Logic Example

• B a1, a3 where M given as M(a1) a1 and
  M(a2) a2 has the following properties.
• M a1
• M a1 a3
• M ? a2
• M a2 a4
• M does not satisfy the following propositions.
• M a4
• M a1 a4

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Propositional Logic Proofs

• What formulas hold in all models ?
• I.e. can we check if a given proposition is true
  in all models without going through all possible
  models?
• Need proofs to answer this question.
• We use Natural Deduction proofs.
• Recommended Read Ch 2 of Logic and Computation
  by L.C. Paulson.

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Natural Deduction for Prop. Logic

• Proofs are trees of formulae made by applying
  inference rules.
• An inference rule is of the form
• A1 An
• B
• Here A1 .. An are said to be premises (or
  antecedents) of the rule, and B is said to be the
  conclusion (consequent) of the rule.

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Natural Deduction for Prop. Logic

• Hence a proofs is a trees whose
• Root is the theorem to be proved,
• Branches are rules, and
• Leaves are the assumptions (axioms) of the proof.
• Example
• A1 A2 A3 C1 C2 ? Assumptions
• B1 B2 ? Applications of rules
• D ? Theorem being proved
• There are introduction and elimination rules for
  each connective in Natural Deduction proof
  systems.

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Rules for Conjunction

• Introduction
• A B
• A B
• Elimination
• A B A B
• A B

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Rules for Disjunction

• Introduction
• A B
• A v B A v B
• Elimination
• A B
• A v B C C
• C
• X denotes discharged assumption X.

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Rules for Implication

• Introduction
• A
• B
• A B
• Elimination (Modus Ponens)
• A B A
• B

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Rules for Negation

• ? B interpreted as ( B __). Hence we get the
  following rules from those of implication.
• ? Introduction ? Elimination
• B ? B B
• __
  __
• ________
• ? B
• Special Contradiction Rule ? B
• __
• __________
• B

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Propositional Proofs Examples

• Prove ( A B ) (A v B)
• Notice
• The outermost connective is . Hence, the last
  step of the proof must be an implication
  introduction.
• That means, we must assume ( A B ) and prove
  (A v B), and then discharge the assumption by
  using introduction rule.
• In order to prove (A v B) from ( A B ), we must
  use v introduction, and hence must prove either
  A or B from ( A B ).
• This plan forms a skeleton of a proof.

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Prop. Proof Example Continued

• Prove ( A B ) (A v B)
• A B
• A elimination
• A v B v introduction
• ( A B ) (A v B) introduction
• Proofs are analyzed backwards, I.e. start
  unraveling the logical structure of the
  conclusion and work backwards to the assumptions.
  Draw out a plan based on your analysis and write
  down the formal proof.

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Derived Rules

• These are rules derived from other rules.
• Example
• A B
• B A
• Here is the derivation
• A B B A
• B A elimination
• B A introduction

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Soundness and Completeness

• A rule A1 An is said to be sound if for every
• B
• model in which all of A1 An are true, then so
  is B. I.e. if M A1 , , M An, then M
  B.
• A collection of rules are sound if all rules in
  the collection is sound.
• A collection of rules is complete if M A for
  all models M, then A is provable. I.e. there is a
  proof of A using the given set of rules. (Denoted
  R-- A ) where R is the set of rules.

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Predicate Logic

• Language to describe properties of individuals.
• Thus, syntax is able to describe individuals,
  their properties (relationships) and functions.
• These are to be thought of as names of
  individuals, properties (relationships) and
  functions.
• Models are incarnations of these individuals,
  properties (relationships) and functions.
• More detailed than propositional logic.

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Predicate Logic Syntax

• A collection of constants say ci i 0 .
• Constants are names for individuals. E.g. 0, 1.
• Note not all individuals in a model have names.
• A collection of variables say xi i 0 .
• Needed to generically refer to individuals.
• Think of them as standing in place of pronouns
  like it, she.
• A collection of function symbols- say fi i
  0 .
• May be of different arities, and may be typed.
  E.g. (x,y)
• A collection of predicate symbols- say pi i
  0 .
• May be of different arities.
• Encodes properties of individuals. E.g. prime(x).

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Predicate LogicRecursive Definition of Terms

• Every variable is a term.
• Every constant is a term.
• If fi is an n-ary function symbol and t1, .., tn
  are terms, then fi(t1, .., tn) is a term.
• We use ti i
• Examples
• f(x, g(2, y)) is a term, where f, g are function
  symbols and x, y are variables.
• ( x, (3,y)) is a term in arithmetic usually
  written as x (3y)

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Recursive Definition of Formulas

• If pi is an n-ary predicate symbol and t1, .., tn
  are terms, then pi(t1, .., tn ) is an atomic
  formula.
• If A and B are formulas, then so are
• A B, A v B, ? A, A B.
• ? xi A(xi), ? xi A(xi), where xi is a variable.
• ?, ? are referred to as the universal and
  existential quantifier, respectively.
• A formula that does not have either quantifier is
  said to be a quantifier free.

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Free and bound Variables

• In ? x A(x), the variable x is said to be bound
  meaning the name x plays no significant role.
  (compare with he, she, it)
• A variable x occurs bound in a formula if ?x or
  ?x is a part of it. More precisely, x occurs
  bound in
• ?y A(y) or ?y A(y) if x and y are the same
  variable.
• ? A if x occurs bound in A.
• A B, A v B, A B if x occurs bound in either
  A or B.

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Substitutions

• If A is a formula, t is a term and x is a
  variable, then At/x is the formula obtained by
  substituting t for x in A.
• At1/x1, tn/xn is the formula resulting in
  simultaneously substituting x1, xn by t1, tn.
• Note Simultaneous substitution Q(x,y)x/y,y/x
  yields Q(y,x) but iterated substitution
  Q(x,y)x/yy/x yields Q(y,y).

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Substituting Terms for Variables

• In At/x, the free variables of t stand the
  danger of becoming bound in A. Hence, need a
  precise definition.
• If x is y then ?y A(y) x/y is ?y A(y). If not
  let z be a fresh variable (I.e. not in t, x) then
  (?y A(y) )t/x is ?z (A(z/y) t/x).
• Similar definition for ?y A(y).
• Examples
• ?y (y 1) y/y is ?y (y 1). Here x is y and t
  is x.
• ?y (y1 x) 2yx/x is ?z ((z1x)2yx/x I.e.
  ?z (z12yx). Here t is (2yx).

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Substituting Terms Continued

• (? A )t/x is ? (A t/x)
• (A B) t/x is (At/x Bt/x)
• (A v B) t/x is (At/x v Bt/x)
• (A B) t/x is (At/x Bt/x)
• Pi(t1, .. tn) t/x is Pi(t1t/x, .. tnt/x)
  for predicate symbol Pi.

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Predicate Logic Semantics

• A model consists of
• A set (of individuals), say A ai i 0 .
• A set of total functions Fn fni i 0 on
  A.
• I.e. fni(aj) is some ak for every aj.
• A set of predicates Pr pri i 0 over A.
• Do not have to be total.
• Can have many arities.

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Interpreting Syntax

• Mapping from Syntax to Semantics
• A mapping mCons ci I 0 to Aai i
  0.
• Need not be ONTO A. I.e. there could be unnamed
  individuals in the semantic domain.
• A mapping mFun fi I 0 to Fnfni i
  0.
• Need not be onto. I.e. there could be unnamed
  functions in the semantic domain.
• A mapping mPred pi I 0 to Prpri i
  0.
• Need not be onto. I.e. there could be unnamed
  predicates in the semantic domain.

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Interpreting Formulas naming

• We do not interpret formulas with free variables.
  
• In order to interpret quantified formulas, need
  to expand the syntax by adding a constant in the
  syntax for each unnamed individual in the model.
• I.e. for each ai for which there is no cj such
  that Fn(cj ) is ai, add a new constant Cai to the
  syntax.
• Now expand the definition of terms to include
  these new constants. Let newT Nti i 0
  be the collection of new terms so defined.

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Interpreting Formulas

• Let M be a model. We define M F for every
  quantified formula as follows.
• For every n-ary predicate symbol pi , and every
  sequence of new variable free terms Nt1, Ntn
  define M pi(Nt1, Ntn ) if and only if
  mPred(pi)(Nt1, Ntn ).
• I.e. pi(Nt1, Ntn ) is true in M if and only if
  its image under the map mPred holds with
  parameters Nt1, Ntn .

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Interpreting Formulas Continued

• For every formula A , M ?y A(y) if and only if
  M A(Nti) for every Nti e newT.
• For every formula A , M ? y A(y) if and only if
  there is some Nti e newT satisfying M A(Nti).
• M A B if M A and M B .
• M A v B if M A or M B.
• M A B if when M A then M B.
• M ? A if it is not the case that M A.

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Proof Rules for Predicate Logic

• Proof rules of introduction and elimination of ,
  v, , and ?.
• New rules required for introduction and
  elimination of ? and ? quantifiers.

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Proof Rules for ?

• ? Introduction
• A(x) provided x is not free in the
• ?x A(x) assumptions of A
• ? Elimination
• ?x A(x)
• At/x

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Proof Rules for ?

• ? Introduction
• At/x
• ?xA(x)
• ? Elimination
• A provided x is not free
• ?xA(x) B in B nor in the
• B assumptions of B apart from A

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An Example Proof

• Prove ((?x A(x)) B) (?x (A(x) B)) provided
  that x is not free in B.
• Plan
• Since outer connective is , need to use
  introduction at the last step. Hence can use
  (?x A(x)) B as an assumption for the steps
  above.
• Now in order to get ?x (A(x) B) using ?
  introduction, we need to get At/x ) B.
• Can use elimination to (?x A(x)) B and obtain
  B
• Can use ?x elimination to get At/x.

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Example Proof

• ?x A(x) B ?x A(x) B
• ?x A(x) A(t/x) B
• A(t/x) B
• ?x(A(x) B
• The other direction of the proof appears in the
  handout page 32.

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Induction Rule

• A(x)
• A0/x Ax1/x
• A(x)
• Proviso x is not free in the assumptions of
  Ax1/x apart from A(x).

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Equality Reasoning

• Rules for equality
• Reflexivity axiom t t.
• Symmetry rule t u .
• u t
• Transitivity rule s t t u .
• s u
• Congruence laws for each function and predicate
  symbol, or substitution rules.

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Equality Reasoning Continued

• Congruence Law for functions
• t1 u1 . tn un
• f(t1, ., tn) f(u1, .,un)
• Congruence Law for Predicates
• t1 u1 . tn un
• p(t1, ., tn) ? p(u1, .,un)
• Substitution Rule
• t u
• St/x Su/x

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Equality Reasoning An Example

• ?x f(x,x) x
• f(g(z), g(z)) g(z)
• p(f(g(z), g(z)) ? p(g(z))
• ? p(f(g(z), g(z)) ? ? p(g(z))

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Many Sorted Logic

• Sorts can be introduced to first-order logic
• So what we learned is single-sorted first
  order-logic
• ?x Integer ?y Real divides(x,y)
• Integer and Real are two different sorts.
• ?x,y Point ? z Line x ? y isOn(x,y,z)
• Point and Line are sorts

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Translating many sorted Logic to one-sorted logic

• Create a one-ary predicate for each sort and
  write the statement as an implication.
• Many sorted
• ?x Integer ?y Real divides(x,y)
• One Sorted
• ?x ?yisInteger(x) isReal(y)divides(x,y)
• ?x,y Point ? z Line x ? y isOn(x,y,z)
• ?x,y ? z isPoint(x) isPoint(y) x ? y
  isLine(z) isOn(x,y,z)

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Notations from Z

• In Z ?x f(x) is written as ?x .f(x)
• Sometimes types are also used right after the
  quantifier, for example
• ?x opera isComposer(Beethovan,x)
• ?x Integer ?y Real . devides(x,y)

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Typed Logics

• Sorts are expected to be disjoint sets
• That implies
• Integers are NOT a subset of Real Numbers
• Introducing types are more complicated
• Requires that there be type constructors.
• Example Integer ? Real
• Ordered pairs, say ORD
• Example
• ?x ORD, ? z Integer ? z Real x(y,z)
• Here ( , ) is a type constructor.

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Why is this an Issue?

• How to prove (x,y) (a,b) iff xa and yb
• Similar problems occur with sets, lists etc.
• What is the problem?
• The equality theory has to be built.
• Sets create more problems
• Set quantifiers are problematic
• Leads to FULL second order logic
• No complete S1 proof theory

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Using Logic As a Programming Language

• Prolog
• Datalog
• Basic Idea Use them as Rules
• Can specify search
• Raises many issues with semantics

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What Other Forms of Logics are there?

• Can enrich 1st Order Logic by introducing more
  connectives
• Example Modal Logic
• Adding two new constructors to reason about
  possibility and nasality
• ? and ?
• Syntax If f is a formula so are ? f and ? f

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Applications of Modal Logic

• Dynamic Logic
• Temporal Logics
• Theories of Motion
• Theories of Change
• Theories of Knowledge
• Applications in security
• Access control
• Delegation
• See separate transparency on Modal, Dynamic and
  Temporal Logics

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Other Forms to Enhance 1st Order Logic

• Add Richer Quantifier Structures
• Quantify over subsets of structures and/or
  relation symbols
• Add quantifiers such as there exists infinitely
  many, or for infinitely many.
• Example
• ?x ?? y factors(x,y)
• ?? y prime(y)

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Apply existing connectives many more times
infinitary logics

• If f1, fn, .. are formulas so are
• i1 ? fi
• v i1 ? fi
• Different from having infinitary quantifiers

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Restrictions based on Semantics

• Can specify the size of models
• Infinitary but large models
• Such as only those beyond a given cardinal ?1
• Finite model theory
• Many applications in computer science
• Quantifiers take an entirely different meaning

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Those that reject basic assumptions of Classical
Logic

• Intuitionism
• Why is (A ? B) v (B ? A) a tautology ?
• Pigenhole principle in classical truth values
• Similar argument for (A1 ? A2) v (A2 ? A3) . (An
  ? An1) v (An1 ? A1)
• Consequence P v ? P does not have to be true!
• Infinitely many truth values
• Leads to Kripke Semantics with worlds with
  increasing truth values
• Used in the semantics of programming languages
  and type theories