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**11 - 16**of**16**### Logical Equations in Monadic Logic

, 2007

"... A logical formula F (X, P) can be treated as an equation to be satisfied by the solutions X0(P) with the expressions P as parameters for the expressions X (if there are such solutions). J. McCarthy [7] considers the parameterization of the models of formulas, gives the general solution in the case o ..."

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A logical formula F (X, P) can be treated as an equation to be satisfied by the solutions X0(P) with the expressions P as parameters for the expressions X (if there are such solutions). J. McCarthy [7] considers the parameterization of the models of formulas, gives the general solution in the case of propositional logic and states the problem for other logics. In the present paper we find the general solution for the formulas in the first-order language with monadic predicates and equality. The solutions are obtained via quantifier elimination and parametrized by ɛ-terms. 1 Formulas as Equations A logical formula F (X, P) can be treated as an logical equation to be satisfied by the solutions X0(P) with the expressions P as parameters for the unknowns X = X1,..., Xn, if there are such solutions. For instance, the equation ∀x(P x → Xx) is solved by substitution of X by P x ∨ Y x with an arbitrary predicate Y. For

### INSTANTIATION AND DECISION PROCEDURES FOR CERTAIN CLASSES OF QUANTIFIED SET-THEORETIC FORMULAE

, 1978

"... Set-theoretically oriented mechanical theorem provers can exploit the fact that all notions in set theory can be derined in terms of the two primitives equality and set membership. However, since set theory permits very general constructions, such provers find it hard to decide, even in apparentlytr ..."

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Set-theoretically oriented mechanical theorem provers can exploit the fact that all notions in set theory can be derined in terms of the two primitives equality and set membership. However, since set theory permits very general constructions, such provers find it hard to decide, even in apparentlytrivialcases, how existentially quantified set variables should be instantiated. Extending results of Behmann (1922) we give algorithms for instantiating set- and map-valued variables in simple set-theoretic formulae, and also for calculating the truth-values of these formulae.

### Review of: R. L. Epstein and W. A. Carnielli. Computability. Computable Functions, Logic, and the Foundations of Mathematics, 2nd Edition

"... rom some important papers on the subject. This includes, aside from `On the infinite', sections from Turing's `On computable numbers' (1936), Post's `Finite combinatory processes' (1936) in the part on computable functions, from G odel's `On formally undecidable proposi ..."

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rom some important papers on the subject. This includes, aside from `On the infinite', sections from Turing's `On computable numbers' (1936), Post's `Finite combinatory processes' (1936) in the part on computable functions, from G odel's `On formally undecidable propositions' (in a section discussing the relevance of the incompleteness theorems to Hilbert's programme in Chapter 24), and selections from Church, Turing, G odel, and Kalm ar in Chapter 25 (`Church's Thesis'). The final chapter (Chapter 26), `Constructivist Views of Mathematics' gives the book a nice ending by tying the discussion of computability and undecidability to the general philosophical issues of the nature of mathematics. Selections from Brouwer's `Intuitionism and formalism' (1913), Bishop's Foundations of Constructive Analysis' (1967), and a section on strict finitism (with readings from van Dantzig and David Isles) provide a basis for an informed discussion of these issues. The technical material is well-organi

### JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION

"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."

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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.