10/25/2015fopl: interpretation, models 1 lesson 6 semantics of fopl interpretation, models, semantic...

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  • *FOPL: Interpretation, models*Lesson 6Semantics of FOPLInterpretation, models, semantic tableau

  • FOPL: Interpretation, models**Truth of a formula, interpretation, evaluationWe have seen (in Lesson 4) that the questionIs a formula A true?is reasonable only when we add in the interpretation I for a valuation v of free variables.Interpretation structure is an n-tuple:I = U, R1,...,Rn, F1,...,Fm, where F1,...,Fm are functions over the universe of discourse assigned to the functional symbols occurring in the formula, andR1,...,Rn are relations over the universe of discourse assigned to the predicate symbols occurring in the formula. How to evaluate the truth-value of a formula in an interpretation structure I, or for short in the Interpretation I?

    FOPL: Interpretation, models

  • FOPL: Interpretation, models**Interpretation, evaluation of a formulaWe evaluate bottom up, i.e., from the inside out : First, determine the elements of the universe denoted by terms,then determine the truth-values of atomic formulas, andfinally, determine the truth-value of the (composed) formulaEvaluation of terms: Let v be a valuation that associates each variable x with an element of the universe: v(x) U. By evaluation e of terms induced by v we obtain an element e(x) of the universe U that is defined inductively as follows: e(x) = v(x) e(f(t1, t2,...,tn)) = F(e(t1), e(t2),...,e(tn)), where F is the function assigned by I to the functional symbol f.

    FOPL: Interpretation, models

  • FOPL: Interpretation, models**Interpretation, evaluation of a formulaEvaluation of a formulaAtomic formulas: |=I P(t1,...,tn)[v] the formula is true in the interpretation I for a valuation v iffe(t1), e(t2),...,e(tn) R, where R is the relation assigned to the symbol P (we also say that R is the domain of truth of P)Composed formulas:Propositionally composed A, A B, A B, A B, A B, dtto Propositional LogicQuantified Formulas xA(x), xA(x):|=I xA(x)[v], if for any individual i U holds |=I A[v(x/i)], where v(x/i) is a valuation identical to v up to assigning the individual i to the variable x |=I xA(x)[v], if for at least one individual i U holds |=I A[v(x/i)].

    FOPL: Interpretation, models

  • FOPL: Interpretation, models**QuantifiersIt is obvious from the definition of quantifiers that over a finite universe of discourse U = {a1,,an} the following equivalences hold:x A(x) A(a1) A(an) x A(x) A(a1) A(an) Hence the universal quantifier is a generalization of a conjunction; existential quantifier is a generalization of a disjunction.Therefore, the following obviously holds:x A(x) x A(x), x A(x) x A(x) de Morgan laws

    FOPL: Interpretation, models

  • Satisfiability and validness in interpretationFormula A is satisfiable in interpretation I, if there exists valuation v of variables that |=I A[v]. Formula A is true in interpretation I, |=I A, if for all possible valuations v holds that |=I A[v].Model of formula A is interpretation I, in which is A true (that means for all valuations of free variables).Formula A is satisfiable, if there is interpretation I, in which A is satisfied (i.e., if there is an interpretation I and valuation v such that |=I A[v].)Formula A is a tautology (logically valid), |= A, if A is true in every interpretation (i.e., for all valuations).Formula A is a contradiction, if there is no interpretation I, that would satisfy A, so there is no interpretation and valuation, in which A would be true: |I A[v], for any I and v.

  • Satisfiability and validness in interpretation A: x P(f(x), x) B: x P(f(x), x) C: P(f(x), x) Interpretation I: U=N, f x2, P relation >It is true that: |=I B. Formula B is in N, x2, > true.Formulas A and C are in N, >, x2 satisfied, but not true: for e0(x) = 0, e1(x) = 1 these 0,0, 1,1 are not the elements of >, but for e2(x) = 2, e3(x) = 3, the couples are 4,2, 9,3, the elements of relation >.Formulas A, C are not in N, x2, > true: |I A[e0], |I A[e1], |I C[e0], |I C[e1], only:|=I A[e2], |=I A[e3], |=I C[e2], |=I C[e3],

  • Empty universum?Consider an empty universe U = x P(x): is it true or not? By the definition of quantifiers it is false, because we cant find any individual which would satisfy P, then it is true that x P(x), so x P(x), but this is false as well contradiction. Or it is true, because there is no element of the universe that would not have the property P, but then x P(x) should be true as well, which is false contradiction. Likewise for x P(x) leads to a contradictionSo we always choose a non-empty universe of interpretationLogic of an empty world would not be not reasonable

  • *Existential quantifier + implication?There is somebody such that if he/she is a genius, then everybody is a genius.This sentence cannot be false: |= x (G(x) xG(x))For every interpretation I it holds:If the truth-domain GU of the predicate G is equal to the whole universe (GU = U), then the formula is true in I, because the subformula xG(x) is true; hence G(x) x G(x), and x (G(x) xG(x)) is true in I.If GU is a proper subset of U (GU U), then it suffices to find at least one individual a (assigned by valuation v to x) such that a is not an element of GU. Then G(a) x G(x) is true in I, because the antecedent G(a) is false. Hence x (G(x) xG(x)) is true in I.

  • Existential quantifier + conjunction !Similarly x (P(x) Q(x)) is almost a tautology. It is true in every interpretation I such thatPU U, because then |=I P(x) Q(x)[v] for v(x) PU or QU = U, because then |=I P(x) Q(x) for all valuations So this formula is false only in such an interpretation I where PU = U and QU U.Therefore, sentences of a type Some Ps are Qs are analyzed by x (P(x) Q(x)).

  • Universal quantifier + conjunction? Usually no, but implication!Similarly x [P(x) Q(x)] is almost a contradiction!The formula is false in every interpretation I such that PU U or QU U.So the formula is true only in an interpretation I such that PU = U a QU = U Therefore, sentences of a type All Ps are Qs are analyzed by x [P(x) Q(x)]It holds for all individuals x that if x is a P then x is a Q. (See the definition of the subset relation PU QU)

  • Satisfiability and validness in interpretationFormula A(x) with a free variable x:If A(x) is true in I, then |=I x A(x)If A(x) is satisfied in I, then |=I x A(x).Formulas P(x) Q(x), P(x) Q(x) with the free variable x define the intersection and union, respectively, of truth-domains PU, QU. For every P, Q, PU, QU and an interpretation I it holds: |=I x [P(x) Q(x)]iff PU QU |=I x [P(x) Q(x)] iff PU QU |=I x [P(x) Q(x)] iff PU QU = U|=I x [P(x) Q(x)] iff PU QU

  • Model of a set of formulas, logical entailmentA Model of the set of formulas {A1,,An} is an interpretation I such that each of the formulas A1,...,An is true in I.Formula B logically follows fromA1, , An, denoted A1,,An |= B, iff B is true in every model of {A1,,An}. Thus for every interpretation I in which the formulas A1, , An are true it holds that the formula B is true as well:A1,,An |= B: If |=I A1,, |=I An then |=I B, for all I.Note that the circumstances under which a formula is, or is not, true (see the 1st lesson, Definition 1) are in FOPL modelled by interpretations (of predicates and functional symbols by relations and functions, respectively, over the universe).

  • Logical entailment in FOPLP(x) |= x P(x), but the formula P(x) x P(x) is obviously not a tautology. Therefore, A1,...,An |= Z |= (A1 An Z) holds in FOPL only for closed formulas, so-called sentences.x P(x) P(a) is also not a tautology, and thus the rule x P(x) | P(a) is not truth-preserving; P(a) does not logically follow form x P(x).Example of an interpretation I such that x P(x) is, and P(a) is not true in I: U = N(atural numbers), P even numbers, a 3

  • Semantic verification of an argumentAn argument is valid iff the conclusion is true in every model of the set of the premises. But the set of models can be infinite! And, of course, we cannot examine an infinite number of models; but we can verify the logical form of the argument, and check whether the models of premises do satisfy the conclusion.

  • Semantic verification of an argumentExample:All monkeys (P) like bananas (Q)Judy (a) is monkey Judy likes bananas x [P(x) Q(x)] QU P(a)PU-------------------- a Q(a)

  • RelationsPropositions with unary predicates (expressing properties of individuals) were studied already in the ancient times by Aristotle.Until quite recently Gottlob Frege, the founder of modern logic, developed the system of formal predicate logic with n-ary predicates characterizing relations between individuals, and with quantifiers.Frege, however, used another language than the one of the current FOPL.

  • Aristotle: (384 BC March 7, 322 BC)a Greek philosopher, a student of Plato and teacher of Alexander the Great. He wrote on diverse subjects, including physics, metaphysics, poetry (including theater), biology and zoology, logic, rhetoric, politics, government, and ethics. Along with Socrates and Plato, Aristotle was one of the most influential of the ancient Greek philosophers. They transformed Presocratic Greek philosophy into the foundations of Western philosophy as we know it. Plato and Aristotle have founded two of the most important schools of Ancient philosophy.

  • * Gottlob Frege1848 1925German mathematician, logician and philosopher, taught at the University of Jena.Founder of modern logic.

  • Semantic verification of an argumentMarie likes only winnersKarel is a winner--------------------------------------invalid Marie likes Karel x [R(m,x) V(x)], V(k) R(m,k) ?RU U U: { , , , }VU U: {i1, i2, , Karel,, in}The pair doesnt have to be an elements of RU, it is


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