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A Formulae-as-Types Interpretation of Subtractive Logic

T. Crolard
2004 Journal of Logic and Computation  
We present a formulae-as-types interpretation of Subtractive Logic (i.e. bi-intuitionistic logic).  ...  Then we extend this deduction system conservatively to Subtractive Logic.  ...  Replace with: Γ, A ∨ B ∆ Γ\{H } ∆\S , (A ∨ B) → S ∨ A Formulae-as-Types Interpretation of Subtractive Logic  ... 
doi:10.1093/logcom/14.4.529 fatcat:tjyiakbzlvblxnggrksoeciszi

Subtractive logic

Tristan Crolard
2001 Theoretical Computer Science  
of intuitionistic logic with a new connector (subtraction) dual to implication.  ...  We examine the propositional calculus underlying the type system of bicartesian closed categories with coexponents and we show that this calculus corresponds to subtractive logic: a conservative extension  ...  Semantics of subtraction We are now able to extend the topological interpretation to formulas involving subtraction.  ... 
doi:10.1016/s0304-3975(99)00124-3 fatcat:kxguirzxanaatl5y2c2ojppsty

On an intuitionistic logic for pragmatics

Gianluigi Bellin, Massimiliano Carrara, Daniele Chiffi
2015 Journal of Logic and Computation  
A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.  ...  We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justifications and its relations with classical logic.  ...  The postpone term is not assigned to a formula: we may assign it to a symbol • which is not a type and thus cannot occur as a sub-formula of other formulas.  ... 
doi:10.1093/logcom/exv036 fatcat:kiqprtcp6vfrhnutwqpyadmpyy

Pragmatic and dialogic interpretations of bi-intuitionism. Part I

Gianluigi Bellin, Massimiliano Carrara, Daniele Chiffi, Alessandro Menti
2014 Logic and Logical Philosophy  
Philosophically, we extend Dalla Pozza and Garola's pragmatic interpretation of intuitionism as a logic of assertions [10] to bi-intuitionism as a logic of assertions and hypotheses.  ...  We extend the Brouwer-Heyting-Kolmogorov interpretation to bi-intuitionism by regarding co-intuitionistic formulas as types of the evidence for them: if conclusive evidence is needed to justify assertions  ...  Let us explore co-intuitionism as a logic of hypotheses and take the elementary expressions of our object language to represent types of hypotheses and the interpretation H.1 of the consequence relation  ... 
doi:10.12775/llp.2014.011 fatcat:gelonc7qlfdjvldyax3bmuaype

Programming in logic without Prolog [article]

M. H. van Emden
2014 arXiv   pre-print
To be useful in this way, logic has to provide a mechanism for the definition of new functions and new relations on the basis of those given in the interpretation of a logical theory.  ...  In this approach verification of computational results relies on a correspondence between logic interpretations and a class definition in languages like Java or C++.  ...  This research benefited from facilities provided by the University of Victoria and by the Natural Science and Engineering Research Council of Canada.  ... 
arXiv:1206.2037v5 fatcat:qup3t5j26jei5bfokqcohyn664

Assertions, Hypotheses, Conjectures, Expectations: Rough-Sets Semantics and Proof Theory [chapter]

Gianluigi Bellin
2014 Trends in Logic  
This conceptual clarification not only gives a more convincing characterization of the logic in [5] as a treatment of assertions and hypotheses, but also opens the way to distinct representations of conjectures  ...  by Dalla Pozza and Bellin [7] and others; in particular we revise the discussion of the logical properties of assertive and conjectural reasoning presented in "Towards a logic for pragmatics.  ...  Abstract forms of the continuation-passing style, e.g., as in Thielecke's work, have been typed in classical logic, suggesting an interpretation of these relations as a logical duality between classical  ... 
doi:10.1007/978-94-007-7548-0_10 fatcat:z5fsme3hlbdrtjgn6i2rcyxdam

Page 5443 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
set theory can be interpreted in type theory.  ...  The paper proceeds to develop subtractive logic, in which a syn- tactic “subtraction” operator is used to characterize disjunction.  ... 

Page 937 of Mathematical Reviews Vol. 14, Issue 10 [page]

1953 Mathematical Reviews  
The notion of interpretation is ex- tended to interpretations of a formal system $ in another system F; in such an interpretation there is associated to every formula @ of a sequence of formulas A, of  ...  of this formula to Z, as an axiom results in a system externally consistent but not omega consistent.  ... 

Page 3565 of Mathematical Reviews Vol. , Issue 92g [page]

1992 Mathematical Reviews  
We discuss a monotonic logic TML which has pred- icate formulas, temporal formulas and a special modal formula, and give a completeness theorem for it.  ...  Under the ‘subtraction’ theory of negation ‘~ p’ would be such a substitution instance, since according to the subtraction theory of negation, nothing follows from a contradiction.  ... 

Page 1291 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews  
(N-OSLO; Oslo) Interpreting higher computations as types with totality. Arch. Math. Logic 33 (1994), no. 4, 243-259.  ...  The author describes several types of models. For a model Z, L(Z) denotes the logic of the formulas that are are equivalently true in Z.  ... 

The RuleML Family of Web Rule Languages [chapter]

Harold Boley
2006 Lecture Notes in Computer Science  
The RuleML family of Web rule languages contains derivation (deduction) rule languages, which themselves have a webized Datalog language as their inner core.  ...  Combined modal logics apply special relations as operators to atoms with an uninterpreted relation, complementing the usual interpreted ones.  ...  A λ-formula quantifies variables that occur free in a functional expression much like a ∀-formula does for a relational atom.  ... 
doi:10.1007/11853107_1 fatcat:m6ywk2kaxnhtvftd2c2kxwyxui

State enumeration with abstract descriptions of state machines [chapter]

F. Corella, M. Langevin, E. Cerny, Z. Zhou, X. Song
1995 Lecture Notes in Computer Science  
We propose a theory of abstract descriptions of state machines in a many-sorted rst-order logic with abstract and concrete sorts.  ...  The theory provides a foundation for automated state enumeration methods whose complexity is independent of the width of the data path, and in particular for methods based on Multiway Decision Grahps (  ...  The denotation of a term and the truth or falsity of a formula under an interpretation and a compatible variable assignment are de ned as usual.  ... 
doi:10.1007/3-540-60385-9_9 fatcat:knpepmmuyrfftla4jijmi6d2wi

Integrating WS1S with PVS [chapter]

Sam Owre, Harald Rueß
2000 Lecture Notes in Computer Science  
This logic may not only be viewed as a highly succinct alternative to the use of regular expressions, but can also be used to encode Presburger arithmetic or quantified boolean logic.  ...  Their tool, called MONA [2], acts as a decision procedure and as a translator to finite-state automata.  ...  We would like to thank A. Møller for clarifying discussions about MONA internals, M. Sorea for comments on this paper, and S. Bensalem for providing interesting test cases.  ... 
doi:10.1007/10722167_42 fatcat:obik2mf2fvdvle5hetcievhhb4

Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

Francesco Paoli, Matthew Spinks, Robert Veroff
2008 Logica Universalis  
s logic of equilibrium [18]; iii) a new logic of "preservation of truth degrees".  ...  We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three  ...  As it is customary to do in AAL, we will not distinguish between logical languages and algebraic similarity types; so, Fm(L) will denote both the set of all formulas of L (seen as a logical language) and  ... 
doi:10.1007/s11787-008-0034-2 fatcat:ygrc4btdevd2fp6egph7z2ltoy

Equality and Monodic First-Order Temporal Logic

Anatoli Degtyarev, Michael Fisher, Alexei Lisitsa
2002 Studia Logica: An International Journal for Symbolic Logic  
It has been shown recently that monodic first-order temporal logic without functional symbols but with equality is incomplete, i.e. the set of the valid formulae of this logic is not recursively enumerable  ...  In this paper we show that an even simpler fragment consisting of monodic monadic two-variable formulae is not recursively enumerable.  ...  Intuitively, the interpretations of Ì Ä -formulae are sequences of first-order structures, or states of Å, such as Å ¼ Å ½ Å Ò An assignment in is a function from the set Ä Ú of individual variables of  ... 
doi:10.1023/a:1021352309671 dblp:journals/sLogica/DegtyarevFL02 fatcat:yvpeuqbbj5ghlhtqaak2qdpq5m
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