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A Formulae-as-Types Interpretation of Subtractive Logic

2004
*
Journal of Logic and Computation
*

We present

doi:10.1093/logcom/14.4.529
fatcat:tjyiakbzlvblxnggrksoeciszi
*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*...##
###
Subtractive logic

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*. ...

##
###
On an intuitionistic logic for pragmatics

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*. ...

##
###
Pragmatic and dialogic interpretations of bi-intuitionism. Part I

2014
*
Logic and Logical Philosophy
*

Philosophically, we extend Dalla Pozza and Garola's pragmatic

doi:10.12775/llp.2014.011
fatcat:gelonc7qlfdjvldyax3bmuaype
*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 ...##
###
Programming in logic without Prolog
[article]

2014
*
arXiv
*
pre-print

To be useful in this way,

arXiv:1206.2037v5
fatcat:qup3t5j26jei5bfokqcohyn664
*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. ...##
###
Assertions, Hypotheses, Conjectures, Expectations: Rough-Sets Semantics and Proof Theory
[chapter]

2014
*
Trends in Logic
*

This conceptual clarification not only gives

doi:10.1007/978-94-007-7548-0_10
fatcat:z5fsme3hlbdrtjgn6i2rcyxdam
*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 ...##
###
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]

2006
*
Lecture Notes in Computer Science
*

The RuleML family

doi:10.1007/11853107_1
fatcat:m6ywk2kaxnhtvftd2c2kxwyxui
*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. ...##
###
State enumeration with abstract descriptions of state machines
[chapter]

1995
*
Lecture Notes in Computer Science
*

We propose

doi:10.1007/3-540-60385-9_9
fatcat:knpepmmuyrfftla4jijmi6d2wi
*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. ...##
###
Integrating WS1S with PVS
[chapter]

2000
*
Lecture Notes in Computer Science
*

This

doi:10.1007/10722167_42
fatcat:obik2mf2fvdvle5hetcievhhb4
*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. ...##
###
Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

2008
*
Logica Universalis
*

s

doi:10.1007/s11787-008-0034-2
fatcat:ygrc4btdevd2fp6egph7z2ltoy
*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 ...##
###
Equality and Monodic First-Order Temporal Logic

2002
*
Studia Logica: An International Journal for Symbolic Logic
*

It has been shown recently that monodic first-order temporal

doi:10.1023/a:1021352309671
dblp:journals/sLogica/DegtyarevFL02
fatcat:yvpeuqbbj5ghlhtqaak2qdpq5m
*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*...
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