On a decidable generalized quantifier logic corresponding to a decidable fragment of first-order logic

Question

1. What is the only rule for a normal form of a formula?

2. What is the way to get a node?

3. What is the quantifier depth of a quasivacuous quantifier?

4. What is the weakest logic in the first order?

Accepted Answer

If the only applicable rule is T O , then every formula on the branch at this stage is either atomic, or begins with T<>x or F O x .

Accepted Answer

Given a node A, create first all nodes which can be obtained by propositional rules; then all nodes which can be obtained by the T~ -rule; and then all nodes which can be obtained by the F ~ -rule.

Accepted Answer

If the quantifier depth of all quasivacuous quantifiers in X equals 0, then the procedure described below (taking the subformulas, not having the quantified variable free, out of the scope of the quantifier) brings X in normal form.

Accepted Answer

In Alechina and van Lambalgen (1995b) it was shown that classical first order logic can be reformulated to contain a substitution rule as a structural rule, and logics for several generalized quantifiers can be obtained by weakening this rule; L~in isthe weakest logic considered there.

Accepted Answer

Given a formula ~ with n (free and bound) variables, one can show that if it has a model in which the dependency relation R of all arities less or equal to n satisfies a certain condition on R, then it has a model in which this condition is satisfied for all arities.

Accepted Answer

The authors call a signed formula A a result of decomposing in accordance with the tableau rules a signed formula B (on some branch of a tableau) if(i) either A is obtained by applying one of the tableau rules to B (for example, A = TP(x) is a result of decomposing B = F~P(x) inF- P(x)F-~ TP(x)(ii) or there are signed formulas A1 , . . . ,

Accepted Answer

The quantifier depth of <>~ in this formula equals 0. In <>~(P(x) A Oy(Q(x ,y ) A <>zS(z))) (>x is not quasivacuous, since (>~S(z) is not immediately in the scope of ~ : there is ~ y in between.

Accepted Answer

Roughly speaking, in the history of modal logic modalities and their axioms came first and relational semantics giving them their precise meaning came second, bringing with it among other things the standard translation of modal logic into classical first-order logic.

Accepted Answer

Any result of decomposing a formula in normal form beginning with O~ will contain free a variable which depends on the variable used to instantiate <> x, or this variable itself (Lemma 3).

Accepted Answer

Note that a formula of s ~, is valid (satisfiable) if its translation into first-order logic qo* is valid (satisfiable), with R satisfying, for all permutations 7r,R(a, b) r162 n (a , 7r(b))R(a, blblb) r162 R(a, bib).

Accepted Answer

The authors give an algorithm for finding an equivalent of Oz~, to be called ~1, with the the same free variables, such that in ~1 there are no more subformulas immediately in the scope of ~ x which do not have x free, and the quantifiers in the scope of (>z are still not quasivacuous.

Accepted Answer

Proof Let 2 be the free variables of Ozr The F<> rule with F<>zr 2) as a premise can be repeated only if a new variable d with R(d, 2) is introduced by the T O rule with a formula of the form T<>za(x, 2) as a premise.

Accepted Answer

Otherwise write ~ as a disjunction of conjunctions, where each conjunct is either an atomic formula or its negation, or a formula beginning with a quantifier or its negation.

Accepted Answer

One of the directions for future work is to find a uniform framework for studying logics corresponding to cylindric relativized algebras in the same fashion.

Accepted Answer

A formula X is in normal form if in X every subformula immediately in the scope of a quantifier contains the quantified variable of this quantifier free.

On a decidable generalized quantifier logic corresponding to a decidable fragment of first-order logic

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Most frequently asked questions

1. What is the only rule for a normal form of a formula?

2. What is the way to get a node?

3. What is the quantifier depth of a quasivacuous quantifier?

4. What is the weakest logic in the first order?

5. What is the proof of the cylindric relativized set algebras?

6. What is the meaning of a signed formula?

7. What is the quantifier depth of a subformula?

8. What is the meaning of modal logic?

9. what is the simplest way to decompose a formula?

10. What is the truth condition for a formula of s?

11. What is the simplest way to find a quasivacuous quantifier?

12. What is the only rule that can be used to instantiate a formula?

13. What is the definition of a disjunction of conjunctions?

14. What is the direction for future work?

15. What is the definition of a normal form of a formula?

Citations

Modal languages and bounded fragments of predicate logic

Modal quantification over structured domains

Products of modal logics. Part 2: relativised quantifiers in classical logic

Interpolation for a Sequent Calculus of Generalized Quantifiers

Generalized quantification as substructural logic

References

First-Order Logic

Modal Foundations for Predicate Logic

Natural Deduction for Generalized Quantifiers

First-Order Logic

Correspondence and completeness for generalized quantifiers

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