Statistics of implicational logic

In this paper we investigate the size of the fraction of tautologies of the given length n against the number of all formulas of length n for implicational logic. We are specially interested in asymptotic behavior of this fraction. We demonstrate the relation between a number of premises of implicational formula and asymptotic probability of finding formula with this number of premises. Furthermore we investigate the distribution of this asymptotic probabilities. Distribution for all formulas is contrasted with the same distribution for tautologies only. We prove those distributions to be so different that enable us to estimate likelihood of truth for a given long formula. Despite of the fact that all discussed problems and methods in this paper are solved by mathematical means, the paper may have some philosophical impact on the understanding how much the phenomenon of truth is sporadic or frequent in random logical sentences

Are there Hilbert-style Pure Type Systems?

For many a natural deduction style logic there is a Hilbert-style logic that is equivalent to it in that it has the same theorems (i.e. valid judgements with empty contexts). For intuitionistic logic, the axioms of the equivalent Hilbert-style logic can be propositions which are also known as the types of the combinators I, K and S. Hilbert-style versions of illative combinatory logic have formulations with axioms that are actual type statements for I, K and S. As pure type systems (PTSs)are, in a sense, equivalent to systems of illative combinatory logic, it might be thought that Hilbert-style PTSs (HPTSs) could be based in a similar way. This paper shows that some PTSs have very trivial equivalent HPTSs, with only the axioms as theorems and that for many PTSs no equivalent HPTS can exist. Most commonly used PTSs belong to these two classes. For some PTSs however, including lambda* and the PTS at the basis of the proof assistant Coq, there is a nontrivial equivalent HPTS, with axioms that are type statements for I, K and S.Comment: Accepted in Logical Methods in Computer Scienc

Relative Entailment Among Probabilistic Implications

We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective; the semantics of this sort of implication is defined in terms of a threshold on a conditional probability of the consequent, given the antecedent: we are dealing with what the data analysis community calls confidence of partial implications or association rules. Existing studies of redundancy among these partial implications have characterized so far only entailment from one premise and entailment from two premises, both in the stand-alone case and in the case of presence of additional classical implications (this is what we call "relative entailment"). By exploiting a previously noted alternative view of the entailment in terms of linear programming duality, we characterize exactly the cases of entailment from arbitrary numbers of premises, again both in the stand-alone case and in the case of presence of additional classical implications. As a result, we obtain decision algorithms of better complexity; additionally, for each potential case of entailment, we identify a critical confidence threshold and show that it is, actually, intrinsic to each set of premises and antecedent of the conclusion

Exposing Fake Logic

Exposing Fake Logic by Avi Sion is a collection of essays written after publication of his book A Fortiori Logic, in which he critically responds to derivative work by other authors who claim to know better. This is more than just polemics; but allows further clarifications of a fortiori logic and of general logic. This collection includes essays on: a fortiori argument (in general and in Judaism); Luis Duarte D’Almeida; Mahmoud Zeraatpishe; Michael Avraham (et al.); an anonymous reviewer of BDD (a Bar Ilan University journal); and self-publishing

BCI-Algebras and Related Logics

Kabzinski in [6] first introduced an extension of BCI-logic that is isomorphic to BCI-algebras. Kashima and Komori in [7] gave a Gentzen-style sequent calculus version of this logic as well as another sequent calculus which they proved to be equivalent. They used the second to prove decidability of the word problem for BCI-algebras. The decidability proof relies on cut elimination for the second system, this paper provides a fuller and simpler proof of this. Also supplied is a new decidability proof and proof finding algorithm for their second extension of BCI-logic and so for BCI-algebras

Developmental stages challenging cross-linguistic transfer: L2 acquisition of Norwegian adjectival agreement in attributive and predicative contexts

This study presents cross-sectional data on adjectival agreement in second-language (L2) learners of Norwegian with four different first languages (L1s). The target language has full noun phrase agreement between article, adjective and noun, and the source languages represent different agreement conditions, similar to or different from the target language. Sixteen learners participated in the study, and their oral production of adjective agreement was analysed individually. Two hypotheses were proposed. First, learners will develop adjectival agreement in a piecemeal way and follow the developmental stages predicted by Processability Theory (Pienemann, 1998), with attributive and predicative agreement implicationally ordered. Second, learners with adjective agreement in the L1 will transfer that into the L2, whereas learners without agreement in the L1 will not use agreement. Under the first hypothesis, we expect the learners to be distributed along a developmental scale, with some learners applying agreement in attributive positions only and others applying agreement in both the attributive and predicative positions. Under the second hypothesis, we anticipate a difference between the groups: Learners with agreement in their L1 will mark agreement in all contexts where it occurs in the L1, whereas learners who do not have agreement in their L1 will fail to mark agreement overall. The comparison demonstrates larger differences within the L1 groups than between the L1 groups. This suggests a gradual acquisition of agreement, with the agreement features and positions emerging one by one rather than being transferred from the L1.publishedVersio

The algebraic structure of the densification and the sparsification tasks for CSPs

The tractability of certain CSPs for dense or sparse instances is known from the 90s. Recently, the densification and the sparsification of CSPs were formulated as computational tasks and the systematical study of their computational complexity was initiated. We approach this problem by introducing the densification operator, i.e. the closure operator that, given an instance of a CSP, outputs all constraints that are satisfied by all of its solutions. According to the Galois theory of closure operators, any such operator is related to a certain implicational system (or, a functional dependency) $\Sigma$. We are specifically interested in those classes of fixed-template CSPs, parameterized by constraint languages $\Gamma$, for which the size of an implicational system $\Sigma$ is a polynomial in the number of variables $n$. We show that in the Boolean case, $\Sigma$ is of polynomial size if and only if $\Gamma$ is of bounded width. For such languages, $\Sigma$ can be computed in log-space or in a logarithmic time with a polynomial number of processors. Given an implicational system $\Sigma$, the densification task is equivalent to the computation of the closure of input constraints. The sparsification task is equivalent to the computation of the minimal key. This leads to ${\mathcal O}({\rm poly}(n)\cdot N^2)$-algorithm for the sparsification task where $N$ is the number of non-redundant sparsifications of an original CSP. Finally, we give a complete classification of constraint languages over the Boolean domain for which the densification problem is tractable