Three Dogmas, a Puzzle and its Solution
Elnaserledinellah Mahmood Abdelwahab
Modern Logics, as formulated notably by Frege, Russell and Tarski involved
basic assumptions about Natural Languages in general and Indo-European
Languages in particular, which are contested by Linguists. Based upon those
assumptions, formal Languages were designed to overcome what Logicians claimed
to be 'defects' of Natural Language. In this paper we show that those
assumptions contradict basic principles of Arabic. More specifically: The
Logicians ideas, that within Natural Language words refer to objects,
'ToBe'-constructions represent identity statements, Indefinite Descriptions
must be replaced by existential quantifiers to form meaningful Sentences and
Symbols can have no interpretation-independent meanings, are all falsified
using undisputed principles of Arabic. The here presented falsification serves
two purposes. First, it is used as a factual basis for the rejection of
approaches adopting Semantic axioms of Mathematical Logics as Models for
meaning of Arabic Syntax. Second, it shows a way to approach the important
computational problem: Satisfiability (SAT). The described way is based upon
the realization that parsing Arabic utilizes the existence of
'meaning-particles' within Syntax to efficiently recognize words, phrases and
Sentences. Similar meaning-particles are shown to exist in 3CNF formulas,
which, when properly handled within the machinery of 3SAT-Solvers, enable
structural conditions to be imposed on formulas, sufficient alone to guarantee
the efficient production of non-exponentially sized Free Binary Decision
Diagrams (FBDDs). We show, why known exponential Lower Bounds on sizes of FBDDs
do not contradict our results and reveal practical evidence, obtained for
multiplication circuits, supporting our claims.