![]() (Co-authors: John Corcoran and Hassan Masoud) The mathematical results in this paper are not difficult they could have been discovered and proved as early as the 1920s. As documented below, these results correct false or misleading passages even in respected logic texts.ĬORCORAN-MASOUD ON MATHEMATICS OF FIRST-ORDER-LOGIC EXISTENTIAL IMPORT Existential-import mathematics. Existential import is not an isolated phenomenon. Existential-import implications hold as widely as they fail. How abundant or widespread is existential import? How abundant or widespread are existential-import predicates in themselves or in comparison to import-free predicates? We show that existential-import predicates are quite abundant, and no less so than import-free predicates. An existential-import predicate Q(x) is one whose existentialization, ∃x Q(x), is logically true otherwise, Q(x) is existential-import-free or simply import-free. A predicate is an open formula having only x free. whether it implies its corresponding existentialized conjunction. The antecedent S(x) of the universalized conditional alone determines whether the universalized conditional has existential import, i.e. We characterize the proexamples by proving the Existential-Import Equivalence: ∀x implies ∃x iff ∃x S(x)is logically true. Lewis, 1st-order logic, and Existential import Contrary to common misconceptions, today's logic is not devoid of existential import: the universalized conditional ∀x implies its corresponding existentialized conjunction ∃x, not in all cases, but in some. Research Interests: Logic, History of Logic, Philosophy of Logic, Mathematical Logic, C. We prove the Existential-Import Equivalence: In any first-order logic, for a universalized conditional to imply the corresponding existentialized conjunction it is necessary and sufficient for the existentialization of the antecedent predicate to be tautological. Central to our campaign is the fact that first-order logic has limited existential import: the universalized conditional implies its corresponding existentialized conjunction in some but not all cases. Many useful examples presented in usable form. All terminology is not only explained but discussed. Ranked sixth on the “Most-read list” at History and Philosophy of Logic, this demanding but self-contained and widely accessible paper refutes over a century of mistakes about existential import. Existential import today: New metatheorems historical, philosophical, and pedagogical misconceptions. We here discuss this sort of matter in detail, and hope to get to the bottom of the issue, and perhaps to a universal solution.Ībstract: John Corcoran and Hassan Masoud. However, when we transfer to CL, in having the antecedent false, and the consequent false as well, we have that the implication, which some call material, is true. If x is in that interval, x+2 should not be 5, unless we are talking about moduli of vectors. ![]() As another point, our discussions in Entailment led us to choose the sense has as a consequence for entailment, so that it is not really acceptable that we have, as a consequence, using normal language, of x being inside of the real interval (7,10) that if x +2=5, then x=3. The second instance of entailment does not seem to be justifiable if our intuition is consulted: Even though we could say that absurdity implies anything in CL, entailment should be a concept that belongs to the meta-logic, not to the logic of the system, so that we should not be inside of the Classical Logic World by the time we assess things in what regards entailment. Notwithstanding, we would also have that if P: x belongs to the interval (7,10), and Q: x +2=5 => x=3, P |= Q. Basically, the mathematical notion of entailment seems to be connected to the inferential rules from Classical Logic, so that if we have P: x belongs to the reals, and Q: x+2=5 => x=3, P |= Q. We here propose a solution to the problem we have raised in (Pinheiro, 2016).
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