1st-order Predicate Logic (FOL)

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1st-order Predicate Logic (FOL)

• Now a real logic Think and concentrate !

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Simple arguments, where propositional logic does
not suffice

• All monkeys like bananas.
• Judy is a monkey.
• ? Judy likes bananas.
• From the viewpoint of Propositional Logic (PL)
  the above are simple (atomic) sentences
• p, q, r, and p, q does not entail r.
• All students are clever
• Charles is not clever
• ? Charles is not a student
• What are the valid schemata of these arguments?

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Logical form (scheme) of an argument

• The schemata of the arguments above remind
  valid schemata of PL
• p ? q, p q (modus ponens) or
• p ? q, ?q ?p (modus tollens)
• But, in PL we cannot refine the analyses of
  simple sentences. Let us reformulate them
• Every individual, if it is a Monkey, then it
  likes Bananas
• Judy is an individual with the property of being
  a Monkey
• ? Judy is an individual that likes Bananas
• ?x M(x) ? B(x), M(J) B(J), where x is an
  individual variable, M, B are predicate symbols,
  J is a functional symbol
• It is again a schema For M, B, J we can
  substitute other properties and individual,
  respectively. For instance, man for M, mortal for
  B and Charles for J.
• M, B, J are here only symbols which stand for
  properties and individuals

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Formal language of FOL (first-order predicate
logic) Alphabet

• Logical symbols
• individual variables x, y, z, ...
• Symbols for truth-connectives ?, ?, ?, ?, ?
• Symbols for quantifiers ?, ?
• Special symbols
• Predicate Pn, Qn, ... n arity number
  of arguments
• Functional fn, gn, hn, ... -- --
• Auxiliary symbols (, ), , , , , ...

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Formal language of FOLGrammar

• terms
• each symbol for a variable x, y, ... is a term
• if t1,,tn (n ? 0) are terms and if f is an n-ary
  functional symbol, then the expression f(t1,,tn)
  is a term If n 0, then we talk about
  individual constant (denoted a, b, c, )
• only expressions due to i. and ii. are terms

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Formal language of FOLGrammar

• atomic formulas
• If P is an n-ary predicate symbol and if t1,,tn
  are terms, then P(t1,,tn) is an atomic formula
• (composed) formulas
• each atomic formula is a formula
• if A is a formula, then ?A is a formula
• if A and B are formulas, then (A ? B), (A ? B),
  (A ? B), (A ? B) are formulas
• if x is a variable and A a formula, then ?x A
  and ?x A are formulas

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Formal language of FOL1st-order

• We can quantify only over individual variables
• We cannot quantify over properties or functions
• Example Leibnizs definition of identity
• If two individuals have all the properties
  identical, then it is one and the same individual
• ?P P(x) P(y) ? (x y) here we need a
  2nd-order language, because we quantify over
  properties

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Example the language of arithmetic

• We need special functional symbols
• 0-ary symbol 0 (the constant zero)
• constant is a 0-ary functional symbol
• unary symbol s (the successor function)
• binary symbols and ? (functions of adding and
  multiplying)
• Examples of terms (using infix notation for and
  ?)
• 0, s(x), s(s(x)), (x y) ? s(s(0)), etc.
• Formulas are, e.g. ( is here a special
  predicate symbol)
• s(0) (0 ? x) s(0), ?x (y x ? z), ?x (x
  y) ? ?y (x s(y))

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Transforming natural language into the language
of FOL

• all, every, none, nobody, any, ... ? ?
• somebody, something, some, there is, ...
  ? ?
• A sentence often needs to be reformulated (in an
  equivalent way)
• No student is retired (For any student it holds
  that he is not retired)
• ?x S(x) ? ?R(x)
• But Not all students are retired (It is not true
  that any student is retired)
• ??x S(x) ? R(x) ? ?x S(x) ? ?R(x)

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Transforming natural language into the language
of FOL

• An auxiliary rule ? ?, ? ? (almost
  always)
• ??x P(x) ? Q(x) ? ?x P(x) ? ?Q(x)
• It is not true that all Ps are Qs ? Some
  Ps are not Qs
• ??x P(x) ? Q(x) ? ?x P(x) ? ?Q(x)
• It is not true that some Ps are Qs ? No P
  is a Q
• de Morgan laws in FOL

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Transforming natural language into the language
of FOL

• The lift is used only by employees
• ?x L(x) ? E(x)
• All employees use the lift
• ?x E(x) ? L(x)
• Mary likes only the winners
• Hence, for all individuals it holds that if Mary
  likes him then he must be a winner
• ?x L(m, x) ? W(x),
• to like is a binary relation, not a property !!!

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Transforming natural language into the language
of FOL

• Everybody loves somebody sometimes
• ?x ?y ?t L(x, y, t)
• Everybody loves somebody sometimes but Hitler
  doesnt like anybody
• ?x ?y ?t L(x, y, t) ? ??z L(h, z)
• Everybody loves nobody ambiguous
• Nobody loves anybody ambiguous
• Everybody dislikes anybody ?x ?y ?L(x, y) ? ?x
  ??y L(x, y)

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Free, bound variables

• ?x ?y P(x, y, t) ? ??x Q(y, x)
• bound, free free, bound
• Formula with clear variables each variable has
  only free occurrences, or only bound occurrences
  each quantifier has its own variables.
• For instance, the above formula does not have
  clear variables x in the second conjunct is
  another variable than the x in the first
  conjunct, similarly for y. Clear formula
• ?x ?y P(x, y, t) ? ??z Q(u, z)

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Substitution of terms for variables

• A?x/t? ? arises from A by a correct (i.e.,
  collisionless) substitution of a term t for the
  variable x.
• There are two rules for a correct substitution
• We can substitute a term t only for free
  occurrences of a variable x in a formula A, and
  we have to substitute for all the free
  occurrences.
• No individual variable that occurrs in the term t
  can become bound in A(in such a case the term t
  is not substitutable for x in the formula A).

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Substitution, example

• A(x) P(x) ? ?y Q(x, y), term t f(y)
• After executing the substitution A(x/f(y)), we
  obtain
• P(f(y)) ? ?y Q(f(y), y).
• The term f(y) is not substitutable for x in A
• Wed change the sense of the formula

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Semantics of FOL !!!

• P(x) ? ?y Q(x, y) is this formula true?
• A non-reasonable question
• For, we do not know what the symbols P, Q mean,
  what they stand for. They are only symbols which
  can stand for any predicate (property).
• P(x) ? P(x) is this formula true?
• YES, it is and it is always so, in all the
  circumstances. It is necessarily true.

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Semantics of FOL !!!

• ?x P(x, f(x)) we have to specify first,
• ?x P(x , f(x)) how to understand these formulas
• What do they talk about we have to choose the
  universe of discourse any non-empty set U ? ?
• What does the symbol P denote it is binary, with
  two arguments it has to denote a binary relation
  R ? U ? U
• What does the symbol f denote it is an unary,
  one-argument symbol it has to denote a function
  F ? U ? U, denoted F U ? U

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Semantics of FOL !!!

• A ?x P(x, f(x)) we have to specify
• B ?x P(x , f(x)) how to understand these
  formulas
• Let U N (the set of natural numbers)
• let P denote the relation lt (i.e., the set of
  pairs, where the first element is strictly less
  than the second one ?0,1?, ?0,2?, ,?1,2?, )
• Let f denote the function second power x2, i.e.,
  the set of pairs where the second element is the
  power of the first one ?0,0?, ?1,1?, ?2,4?,
  ,?5,25?,
• Now we can evaluate the truth values of the
  formulas A, B

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Semantics of FOL !!!

• A ?x P(x, f(x))
• B ?x P(x , f(x)) We evaluate from the inside
  
• First evaluate the term f(x). Each term denotes
  an element of the universe. Which one? It depends
  on the valuation e of the variable x. Let e(x)
  0, then f(x) x2 0.
• Let e(x) 1, then f(x) x2 1, Let e(x)
  2, then f(x) x2 4, etc.
• Now by evaluating P(x , f(x)) we have to obtain a
  truth value e(x) 0, 0 is not lt 0 False e(x)
  1, 1 is not lt 1 False,
• e(x) 2, 2 is lt 4 True.

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Semantics of FOL !!!

• A ?x P(x, f(x))
• B ?x P(x , f(x))
• The formula P(x , f(x)) is in the given
  interpretation True for some valuations of the
  variable x, and False for other valuations.
• The meaning of ?x (?x) the formula is true for
  all (some) valuations of x
• Formula A False in our interpretation I ?I A
• Formula B True in the interpretation I I B

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Model of a formula, interpretation

• A ?x P(x, f(x))
• B ?x P(x , f(x))
• We have found an interpretation I in which the
  formula B is true. The Interpretation structure
  ?N, lt, x2? satisfies the formula B it is a model
  of the formula B.
• How to adjust the interpretation in order it were
  a model of the formula A? There are infinitely
  many possibilities, infinitely many models.
• For instance ?N, lt, x1?, ?N/0,1, lt, x2?, ?N,
  ?, x2?,
• All the models of the formula A are also models
  of the formula B (what holds for all, it holds
  also for some)

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Model of a formula, interpretation

• C ?x P(x, f(y)) what are the models of this
  formula (with a free variable y)?
• Let us again
• choose a Universe U N
• to the symbol P assign a relation ?
• to the symbol f assign a function x2
• Is the structure IS ?N, ?, power? a model of
  the formula C? In order it were so, the formula
  C would have to be true in IS for all the
  valuations of the variable y. Hence the formula
  P(x, f(y)) would have to be true for all
  valuations of x and y.
• But it is not so, for instance, if e(x) 5, e(y)
  2, then 5 is not ? 22

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Model of a formula, interpretation

• C ?x P(x, f(y)) what are the models of this
  formula (with a free variable y)?
• The structure ?N, ?, x2? is not a model of
  formula C.
• A (trivial) model is, e.g., ?N, N ? N, x2?. The
  whole Cartesian product N ? N, i.e. the set of
  all the pairs of natural numbers, is also a
  relation over N.
• Or, the structure ?N, ?, F?, where F is the
  function, mapping N ? N, such that F associates
  all the natural numbers with the number 0.