Logic

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Logic
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• Unsolvability results also imply unprovability in
  logics
• Logics we will look at (all very briefly)
• Aristotelian logic
• Euclidean geometry
• Propositional logic
• First order logic
• Peano axioms
• Zermelo Fraenkel set theory
• Higher order logic

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• This material is presented so as to require a
  minimum of mathematical formalism.

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What is truth?

• Logic can infer the truth of statements in the
  conceptual world of mathematics, or statements
  about the real world.
• The real world is perceived through senses
• We all perceive the same real world
• How is the conceptual world perceived?

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• How do we know we perceive the same conceptual
  world?
• The symbol 1 that we see is just ink on paper
  (or shadow on screen) and not the same as the
  concept of the number one.
• 1 can be written different ways but the concept
  does not change
• The conceptual world is not perceived but
  imagined
• It is a world of ideas rather than objects

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Why study the conceptual world?

• For some reason mathematics is helpful in
  understanding the real world
• Why should this be so?
• A possible argument for the existence of God
• The correspondence between mathematics and
  reality can be seen as an evidence that the
  designer of the world was a thinking being
• Statements may be true or false in the conceptual
  world

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Possible Worlds

• Logic considers not only the world that exists
  but also other potential worlds, the way things
  could be
• Different statements may be true in different
  possible worlds.
• In one world, the statement It is raining may
  be true.
• In another world, this statement may be false.

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• There are many possible worlds
• There are also many possible worlds in the
  conceptual realm of mathematics.
• In a logic a possible world is called an
  interpretation. It assigns meanings to the
  symbols in the logic.
• Thus each interpretation makes some statements
  true and some statements false.
• Then what does it mean to say that a statement in
  mathematics is true?

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What is truth?
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• We believe certain statements are true
• Fermats last theorem
• There are infinitely many primes
• What does it mean that such statements are true?
• The answer is not straightforward
• Primes are not physical objects
• They cant be directly counted
• Even if you could count them you could not count
  infinitely many of them

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• How can one check the truth of a statement in the
  conceptual world?
• In the real world this is easier
• Either its raining or it isnt
• We all perceive the same real world
• The question of meaning is more straightforward
• A mathematical statement is logically true if it
  is true when the symbols in it are given their
  standard mathematical meaning

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• This is called the standard interpretation
• For example, the integers are , -2, -1, 0, 1, 2,
  under the standard interpretation
• Sometimes mathematicians debate about the
  standard interpretation
• Is the axiom of choice true or not?
• Then there can be disagreement about whether a
  logical statement is true or not

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• The purpose of logic is to distinguish correct
  forms of argument from incorrect forms of
  argument
• This is done using only the form of the argument,
  independently of the subject matter

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• A logic consists of a set of statements (syntax),
  an assignment of meaning to the statements
  (semantics), and a method of proving statements.
• A logic L is sound if all statements provable in
  L, are true
• A logic L is effective if the problem of
  determining whether a statement A is provable in
  L, is partially decidable
• We assume all logics are sound and effective

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• A theorem prover for a logic is a Turing machine
  that tests if a statement is provable in the
  logic. If the statement is provable, the Turing
  machine halts. If not, the Turing machine either
  runs forever or halts in the n state.
• Thus if there is a theorem prover for a logic L,
  then it is partially decidable whether a
  statement of L is provable in L.
• All these logics have theorem provers.

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• All these logics also have interpretations of
  formulas.
• An interpretation assigns meanings to the symbols
  in the logic. It is a possible world described
  by the logic.
• A statement in L is valid if it is true in all
  interpretations of L.

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• An interpretation that makes a statement A true
  is called a model of A.
• A nonstandard model has a counterintuitive
  meaning. For example, it may have integers
  larger than infinity.
• We will see that any sufficiently powerful logic
  has nonstandard models.
• If a statement X is valid then it is true in
  standard models so it is true.
• A statement may be true but not valid.

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Aristotelian Logic

• Three part statements called categorical
  syllogisms
• 256 forms of categorical syllogisms in all
• Validity depends only on the form of the syllogism

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• Example of a categorical syllogism
• All P are Q. All Q are R. Thus all P are R.
• An interpretation of this syllogism
• All North Carolinians are Southerners.
• All Southerners are Earthlings.
• Therefore all North Carolinians are Earthlings.
• Another interpretation
• All ducks are sponges.
• All sponges are happy.
• Therefore all ducks are happy.

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• The syllogism is true if
• one of the hypotheses is false or
• the conclusion is true
• Both interpretations make the syllogism true.
• This syllogism is valid. Thus it is true in all
  interpretations.

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• Another syllogism
• All P are Q. All P are R. Thus all Q are R.
• An interpretation
• All North Carolinians are Earthlings.
• All North Carolinians are Southerners.
• Therefore all Earthlings are Southerners.
• Another interpretation
• All students are people.
• All students are mortal.
• Therefore all people are mortal.

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• The first interpretation makes the syllogism
  false.
• The second interpretation makes the syllogism
  true.
• This syllogism is not valid.
• Syllogisms can be conditionally or
  unconditionally valid.

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Aristotelian Logic

• Conditional validity assumes non empty sets P,Q,R
  et cetera
• Unconditional validity has no assumptions
• 15 unconditionally valid syllogisms
• 9 conditionally valid syllogisms
• 24 valid either way

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• Table 9 Valid categorical syllogisms Hurley,
  1985.
• Unconditionally valid
• All M are P. All S are M. Thus All S are P.
• No M are P. All S are M. Thus No S are P.
• All M are P. Some S are M. Thus Some S are P.
• No M are P. Some S are M. Thus Some S are not P.
• No P are M. All S are M. Thus No S are P.
• All P are M. No S are M. Thus No S are P.
• No P are M. Some S are M. Thus Some S are not P.
• All P are M. Some S are not M. Thus Some S are
  not P.
• Some M are P. All M are S. Thus Some S are P.
• All M are P. Some M are S. Thus Some S are P.
• Some M are not P. All M are S. Thus Some S are
  not P.
• No M are P. Some M are S. Thus Some S are not P.
• All P are M. No M are S. Thus No S are P.
• Some P are M. All M are S. Thus Some S are P.
• No P are M. Some M are S. Thus Some S are not P.

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• Conditionally valid
• All M are P. All S are M. Thus Some S are P. (S
  must exist)
• No M are P. All S are M. Thus Some S are not P.
  (S must exist)
• All P are M. No S are M. Thus Some S are not P.
  (S must exist)
• No P are M. All S are M. Thus Some S are not P.
  (S must exist)
• All P are M. No M are S. Thus Some S are not P.
  (S must exist)
• All M are P. All M are S. Thus Some S are P. (M
  must exist)
• No M are P. All M are S. Thus Some S are not P.
  (M must exist)
• No P are M. All M are S. Thus Some S are not P.
  (M must exist)
• All P are M. All M are S. Thus Some S are P. (P
  must exist)

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• Interpretations assign meanings to symbols
• The meaning of S in interpretation I is a set
• This set is called SI.
• Interpretations also assign meanings to
  statements
• Let I be an interpretation of a categorical
  syllogism. Then I is extended to statements as
  follows

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• All S are P means that SI is a subset of PI.
• Some S are P means that SI and PI have nonempty
  intersection.
• No S are P means that SI and PI have empty
  intersection.
• Some S are not P means that SI and the complement
  of PI have nonempty intersection.
• A categorical syllogism is valid if it is true in
  all interpretations.

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• Example All M are P. All S are M. Thus all S are
  P.
• Example I MI is a,b,c, SI is a,b, PI is
  a,b,c,d.
• I is a model of this syllogism if
• (MI ? PI) ? (SI ? MI) ? (SI ? PI).
• This syllogism is true for this particular I.
• Another example I MI is a,b, SI is a,b,c,
  PI is a.
• This syllogism is also true for this I.

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• This syllogism is true for all I, so this
  syllogism is valid.
• Example No M are P. Some M are S. Thus some S
  are not P.
• I is a model of this syllogism if
• (MI ? PI ?) ? (SI ? MI ? ? ) ? (SI - PI ? ?).
• This syllogism is also true for all I so this
  syllogism is also valid.

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• An invalid syllogism
• All S are P. All S are Q. Thus all P are Q.
• An interpretation SI a,b, PI a,b,c,
  QIa,b,c,d.
• This I makes the syllogism true and is thus a
  model of it.
• Another interpretation SI a,b, PI
  a,b,c,d, QIa,b,c.
• This I makes the first two statements true but
  the conclusion false. I is not a model.

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• A categorical syllogism is satisfiable if there
  exists an interpretation I making it true.
• Such an interpretation I is called a model of the
  syllogism.
• It is possible to construct models of valid
  categorical syllogisms.
• It is also possible to construct models of many
  non-valid categorical syllogisms.

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• Venn diagrams can be used to check the validity
  of categorical syllogisms.
• A Turing machine could use the same idea to check
  whether a categorical syllogism is valid.
• A TM could also check that a statement followed
  from a set of statements by a sequence of
  categorical syllogisms.
• Thus there is a theorem prover for this logic.
  In fact the validity problem is decidable.

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• Given the assumptions
• All M are P. All S are M. All P are Q.
• To prove
• All S are Q
• From the first two statements, it follows that
  all S are P.
• From all S are P and all P are Q, it follows
  that all S are Q.
• Thus all S are Q has been proved.

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GeometryEuclid's Axioms and Postulates

• First Axiom Things which are equal to the same
  thing are also equal to one another.
• Second Axiom If equals are added to equals, the
  whole are equal.
• Third Axiom If equals be subtracted from equals,
  the remainders are equal.
• Fourth Axiom Things which coincide with one
  another are equal to one another.
• Fifth Axiom The whole is greater than the part.

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• First Postulate To draw a line from any point to
  any point.
• Second Postulate To produce a finite straight
  line continuously in a straight line.
• Third Postulate To describe a circle with any
  center and distance.
• Fourth Postulate That all right angles are equal
  to one another.
• Fifth Postulate That, if a straight line falling
  on two straight lines make the interior angles on
  the same side less than two right angles, the two
  straight lines, if produced indefinitely, meet on
  that side of which are the angles less than the
  two right angles.

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Hilbert's Axioms of Geometry

• Given below is the axiomatization of geometry by
  David Hilbert (1862-1943) in Foundations of
  Geometry (Grundlagen der Geometrie), 1902 (Open
  Court edition, 1971). This was logically a much
  more rigorous system than in Euclid.
• I. Axioms of Incidence
• For every two points A, B there exists a line a
  that contains each of the points A, B.
• For every two points A, B there exists no more
  than one line that contains each of the points A,
  B.
• There exist at least two points on a line. There
  exist at least three points that do not lie on a
  line.

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• For any three points A, B, C that do not lie on
  the same line there exists a plane alpha that
  contains each of the points A, B, C. For every
  plane there exists a point which it contains.
• For any three points A, B, C that do not lie on
  one and the same line there exists no more than
  one plane that contains each of the three points
  A, B, C.
• If two points A, B of a line a lie in a plane
  alpha, then every point of a lies in the plane
  alpha.
• If two planes alpha, beta have a point A in
  common, then they have at least one more point B
  in common.
• There exist at least four point which do not lie
  in a plane.

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• II. Axioms of Order
• If a point B lies between a point A and a point
  C, then the points A, B, C are three distinct
  points of a line, and B then also lies between C
  and A.
• For two points A and C, there always exists at
  lest one point B on the line AC such that C lies
  between A and B.
• Of any three points on a line there exists no
  more than one that lies between the other two.
• Let A, B, C be three points that do not lie on a
  line and let a be a line in the plane ABC which
  does not meet any of the points A, B, C. If the
  line a passes through a point of the segment AB,
  it also passes through a point of the segment AC,
  or through a point of the segment BC.

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• III. Axioms of Congruence
• 1. If A, B are two points on a line a, and A' is
  a point on the same or on another line a' then it
  is always possible to find a point B' on a given
  side of the line a' through A' such that the
  segment AB is congruent or equal to the segment
  A'B'. In symbols AB A'B'.
• If a segment A'B' and a segment A"B", are
  congruent to the same segment AB, then the
  segment A'B' is also congruent to the segment
  A"B", or briefly, if two segments are congruent
  to a third one they are congruent to each other.
• On the line a let AB and BC be two segments which
  except for B have no point in common.
  Furthermore, on the same or on another line a'
  let A'B' and B'C' be two segments which except
  for B' also have no point in common. In the case,
  if AB A'B' and BC B'C' then AC A'C'.

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• Let angle(h,k) be an angle in a plane alpha and
  a' a line in a plane alpha' and let a definite
  side of a' in alpha' be given. Let h' be a ray
  on the line a' that emanates from the point O'.
  Then there exists in the plane alpha' one and
  only one ray k' such that the angle(h,k) is
  congruent or equal to the angle(h',k') and at the
  same time all interior point of the angle(h',k')
  lie on the given side of a'. Symbolically
  angle(h,k) angle(h',k'). Every angle is
  congruent to itself, i.e., angle(h,k)
  angle(h,k) is always true.
• If for two triangles ABC and A'B'C' the
  congruences AB A'B', AC A'C', angleBAC
  angleB'A'C' hold, then the congruence angleABC
  angleA'B'C' is also satisfied.

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• IV. Axiom of Parallels
• (Euclid's Axiom) Let a be any line and A a point
  not on it. Then there is at most one line in the
  plane, determined by a and A, that passes through
  A and does not intersect a.

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• V. Axioms of Continuity
• (Archimedes' Axiom or Axiom of Measure) If AB and
  CD are any segments, then there exists a number n
  such that n segments CD constructed contiguously
  from A, along the ray from A through B, will pass
  beyond the point B.
• (Axiom of Line Completeness) An extension of a
  set of points on a line with its order and
  congruence relations that would preserve the
  relations existing among the original elements as
  well as the fundamental properties of line order
  and congruence that follow from Axioms I-III, and
  from V,1 is impossible.

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Examples of geometry proofs
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• The diagram is an interpretation of the
  assumptions.
• The two lines in the diagram are the meaning of
  the symbol ?1 in the assumption.
• Other diagrams would be other interpretations of
  these assumptions.

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• The line in the diagram is the meaning of the
  symbol AC in the hypotheses.
• Thus the diagram is an interpretation of the
  hypotheses of the theorem.

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• Given m?1 m?2
• m?3 m?4
• Prove YS ? XZ

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• A Turing machine could generate all possible
  proofs in an attempt to prove a theorem in
  geometry.
• Thus there is a theorem prover for geometry.

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Propositional (Boolean) Logic

• Formulae are composed of Boolean variables p,q,r,
  and Boolean connectives
• ? (conjunction, and)
• ? (disjunction, or)
• ? (negation, not)
• ? (implication, if then)
• ? (equivalence, if and only if)

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• Example formula
• p ? q ? p
• Interpretation
• It is raining and It is Tuesday implies It
  is raining.
• Another interpretation
• All birds are green and All fish are purple
  implies All birds are green.
• Both interpretations make the formula true.
• The formula is valid (true in all interps.)

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• Another example formula
• p ? q ? ? p
• Interpretation
• 22 ? 33 ? 2 ? 2
• Another interpretation
• 22 ? 3 ? 3 ? 2 ? 2
• The first interpretation makes the formula false.
• The second makes it true.
• The formula is not valid.

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• Validity can be determined by truth tables.

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Truth Tables
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• Interpretations assign meanings to symbols.
• In Boolean logic interpretations assign truth
  values (true, false) to the symbols.
• An interpretation in Boolean logic is called a
  valuation.
• Thus a valuation I is an assignment of truth
  values (true or false) to each variable in a
  formula

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• Example Consider the formula (X ? Y) ? X.
• An example of an interpretation of this formula
  assigns true to X and false to Y.
• This interpretation makes the formula true.
• Another example interpretation assigns false to X
  and true to Y.
• This interpretation makes the formula false.

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• If X is a formula then I(X) is the value of X
  with truth values assigned as in I. Thus I(X1 ?
  X2) true iff I(X1)true and I(X2)true, et
  cetera.
• A formula X is satisfiable if for some I, I(X) is
  true. Such an I is called a model of X.
• A formula is valid if for all I, I(X) is true.

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A valid formula
A satisfiable invalid formula
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• An unsatisfiable formula P ? ?P

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• Valid formulas are also called tautologies.
• Unsatisfiable formulas are contradictions.
• One can test validity of a formula with n
  variables by 2n evaluations.
• Thus a Turing machine can test validity of
  propositional formulae. So there is a theorem
  prover for Boolean logic.
• The validity problem for Boolean logic is
  decidable.

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• NP completeness What is the fastest algorithm
  to test satisfiability of Boolean formulae? The
  answer is not known.
• But all known algorithms take exponential time in
  the worst case.

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First Order Logic

• Formulae may contain Boolean connectives and also
  variables x, y, z, , predicates P,Q,R, ,
  function symbols f,g,h, , and quantifiers ? and
  ? meaning for all and there exists.
• Example ?x(P(x) ? ?yQ(f(x),y))

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Individual Constants

• Formulae can also contain constant symbols like
  a,b,c which can be regarded as functions of no
  arguments.
• Example ?x(P(x) ? Q(x,c))

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• Interpretations assign meanings to the symbols in
  a logic.
• First-order formulae have interpretations that
  interpret predicate symbols as predicates,
  function symbols as functions, variables as
  elements of a nonempty set (the domain) and
  individual constants as particular elements of
  the domain. Boolean connectives and quantifiers
  are given the expected interpretations.

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Interpreting first order formulae

• To translate a first order formulae into English,
• choose a set of objects (people, integers for
  example) as the domain
• choose a meaning (interpretation) for the
  predicate and function symbols
• Translate ?xA as for all x, A
• Translate ?xA as there exists x such that A

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• Translate Boolean connectives as follows
• A ? B as A and B
• A ? B as A or B
• A ? B as if A then B
• ?A as not A
• A ? B as A if and only if B
• Translate predicate symbols in English
• P(x,y) as x loves y, x is a child of y, et
  cetera. This assigns a meaning to P.
• f(x) as the age of x, the father of x, et
  cetera. This assigns a meaning to f.

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• If the domain is the set of people and P(x,y) is
  interpreted as x is a child of y then the
  formula ?x?yP(x,y) is translated as for all x
  there exists y such that x is a child of y.
• It can also be translated as for all persons x
  there exists a person y such that person x is a
  child of person y. In other words, everyone is
  a child of someone.
• This formula is true under this interpretation.

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• If the domain is the set of people and P(x,y) is
  interpreted as x is a parent of y then the
  formula ?x?yP(x,y) is translated as for all x
  there exists y such that x is a parent of y. In
  other words, everyone has a child. This formula
  is false under this interpretation.
• Thus this formula is true under at least one
  interpretation but not true in all
  interpretations.

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• A formula X that is true under at least one
  interpretation I is satisfiable. Such an I is
  called a model of X.
• A formula that is true under all interpretations
  is said to be valid.

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• Consider the formula ?y?xP(x,y) ? ?x?yP(x,y).
  Let the domain be the set of people, and let
  P(x,y) be x loves y.
• The formula then is interpreted as if there
  exists y such that for all x, x loves y, then for
  all x, there exists y such that x loves y. In
  other words, if there is someone that everyone
  loves, then everyone loves someone.
• The formula is true under this interpretation.

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• In fact this formula is true under all
  interpretations, and is a valid formula.
• Consider this formula ?x?yP(x,y) ? ?y?xP(x,y).
  Under the same interpretation, this formula
  becomes If for all x, there exists y such that x
  loves y, then there exists y such that for all x,
  x loves y.
• In other words, if everyone loves someone, then
  there is someone that everyone loves.
• This formula is false under this interpretation
  and is not a valid formula.

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• The validity problem for first-order logic has
  the set of first-order formulae as the base set.
  The right answer is yes if the formula is valid
  and no otherwise.
• The validity problem for first-order logic is
  undecidable (unsolvable). But it is partially
  decidable.
• Therefore there is a Turing machine theorem
  prover for first-order logic.

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Peano axioms

• There is a natural number 0.
• Every natural number a has a successor, denoted
  by a 1.
• There is no natural number whose successor is 0.
• Distinct natural numbers have distinct
  successors if a ? b, then a 1 ? b 1.
• If a property is possessed by 0 and also by the
  successor of every natural number it is possessed
  by, then it is possessed by all natural numbers.

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Peano Axioms in Higher Order Logic

• Nat(0)
• ?x(Nat(x) ? Nat(s(x)))
• ?x(Nat(x) ? s(x) ? 0)
• ?x ?y(Nat(x) ? Nat(y) ? x ? y ? s(x) ? s(y))
• ?PP(0) ? ?x(P(x) ? Nat(x) ? P(s(x))) ? ?x(Nat(x)
  ? P(x))

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Induction in Peano Arithmetic

• Using the last axiom, to show that ?x(Nat(x) ?
  P(x)) it suffices to show
• P(0) and
• ?x(P(x) ? Nat(x) ? P(s(x)))
• This is mathematical induction

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• Many proofs about integers can be done in Peano
  arithmetic but not first-order logic.
• The quantification over P is not allowed in
  first-order logic.
• To get an effective logic, properties can be
  restricted to those that are expressible by a
  first-order formula.
• This makes Peano arithmetic into an infinite set
  of first-order formulas, but still much more
  powerful than first-order logic.

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Making Peano axioms first order

• Only the last axiom is the problem
• For all first order formulae A with one free
  (unquantified) variable, have the axiom
• A0 ? ?x(Ax ? Nat(x) ? As(x)) ? ?x(Nat(x) ?
  Ax)
• This gives an infinite set of first-order axioms
  and makes Peano arithmetic effective.
• Some expressivity is lost.

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Instance of last axiom

• Let Ax be ?y(xyyx).
• Then the first-order instance of the last axiom
  is
• ?y(0yy0) ? ?x (?y(xyyx)? Nat(x) ?
  ?y(s(x)yys(x))) ? ?x(Nat(x) ? ?y(xyyx))
• Different formulas A generate different instances
  of this axiom

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Proofs in Peano Arithmetic

• Peano arithmetic permits mathematical induction
• Do some proofs of properties of the integers in
  Peano arithmetic, possibly defining addition and
  showing it is commutative
• Maybe also prove the distributive law
• Such proofs require induction and cannot be done
  in first-order logic

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Nonstandard models of integers

• The compactness theorem a set of first-order
  sentences is satisfiable, i.e., has a model, if
  and only if every finite subset of it is
  satisfiable. Applies to infinite sets of axioms.
• Consequence any theory that has an infinite
  model has models of arbitrary large cardinality.
  So, for instance, there are nonstandard models of
  Peano arithmetic with uncountably many natural
  numbers.

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Zermelo-Fraenkel set theory

• The ten axioms of ZFC are listed below. (Strictly
  speaking, the axioms of ZFC are just strings of
  logical symbols. What follows should therefore be
  viewed only as an attempt to express the intended
  meaning of these axioms in English. Moreover, the
  axiom of separation, along with the axiom of
  replacement, is actually an infinite schema of
  axioms, one for each formula.)
• The axioms of choice and regularity are still
  controversial today among a minority of
  mathematicians.

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• Axiom of extensionality Two sets are the same if
  and only if they have the same elements.
• Axiom of empty set There is a set with no
  elements. We will use to denote this empty
  set.
• Axiom of pairing If x, y are sets, then so is
  x,y, a set containing x and y as its only
  elements.
• Axiom of union For any set x, there is a set y
  such that the elements of y are precisely the
  elements of the elements of x.
• Axiom of infinity There exists a set x such that
  is in x and whenever y is in x, so is the
  union y U y.
• Axiom of separation (or subset axiom) Given any
  set and any proposition P(x), there is a subset
  of the original set containing precisely those
  elements x for which P(x) holds.

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• Axiom of replacement Given any set and any
  mapping, formally defined as a proposition P(x,y)
  where P(x,y) and P(x,z) implies y z, there is a
  set containing precisely the images of the
  original set's elements.
• Axiom of power set Every set has a power set.
  That is, for any set x there exists a set y, such
  that the elements of y are precisely the subsets
  of x.
• Axiom of regularity (or axiom of foundation)
  Every non-empty set x contains some element y
  such that x and y are disjoint sets.
• Axiom of choice (Zermelo's version) Given a set
  x of mutually disjoint nonempty sets, there is a
  set y (a choice set for x) containing exactly one
  element from each member of x.

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First order forms of ZFC axioms

• ? A, ? B, A B ? (? C, C ? A ? C ? B)
  (extensionality)
• ? A, ? B, (B ? A) (empty set)
• ? A, ? B, ? C, ? D, D ? C ? (D A ? D B)
  (pairing)
• ? A, ? B, ? C, C ? B ? (? D, D ? A ? C ? D)
  (union)
• ? ?, ? ? ? (? x, x ? ? ? x ? x ? ?)
  (infinity)
• ? A, ? B, ? C, C ? B ? (C ? A ? P(C))
  (specification)

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• (? X, ?! Y, P(X,Y)) ? ? A, ? B, ? C, C ? B ? (?
  D, D ? A ? P(D,C)) (replacement)
• ? A, ? B, ? C, C ? B ? (? D, D ? C ? D ? A)
  (power set)
• ? S, (S ? ? a, (a ? S ? a n S ))
  (regularity)
• Let X be a collection of non-empty sets. Then we
  can choose a member from each set in that
  collection. That is, there exists a function f
  defined on X such that for each set S in X, f(S)
  is an element of S. (axiom of choice)

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• The specification and replacement axioms are
  axiom schemata, and represent infinitely many
  first-order formulas.
• The elememts of ? in the infinity axiom can be
  regarded as integers.
• ZFC set theory is an infinite set of first-order
  axioms. Thus it also has nonstandard models of
  the integers.
• ZFC can be used to define the real numbers,
  imaginary numbers, continuous functions,
  integration, and differentiation.

90

• We have seen a number of logics Aristotelian
  logic, geometry, propositional calculus, first
  order logic, Peano axioms, and ZFC set theory.
• All these logics have Turing machine theorem
  provers.
• This permits one to show incompleteness of the
  Peano axioms and ZFC using the halting problem.

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• One can also show that nonstandard models exist
  because of the incompleteness of these logics.
• True statements in a logic L are true in all
  standard models
• Provable statements in L are true in all standard
  and nonstandard models.
• Thus if there is a statement that is true but not
  provable, then L has a nonstandard model.