## The Proof-Theory and Semantics of Intuitionistic Modal Logic (1994)

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@TECHREPORT{Simpson94theproof-theory,

author = {Alex K. Simpson},

title = {The Proof-Theory and Semantics of Intuitionistic Modal Logic},

institution = {},

year = {1994}

}

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### Abstract

Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary meta-theory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are self-justifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic first-order logic. It is also established that, in many cases, the natural deduction systems induce well-known intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their

### Citations

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Communication and Concurrency
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- 1989
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- 1996
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- 1985
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274 |
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Citation Context ...2.1 Natural deduction for intuitionistic logic 2.1.1 The natural deduction system The natural deduction system for intuitionistic first-order logic was introduced by Gentzen in his classic 1935 paper =-=[36]-=-. This system provides a very attractive formalization of intuitionistic logic for three intimately related reasons. First, it formalizes ordinary intuitive (intuitionistic) reasoning very closely. Se... |

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Citation Context ...c came in the late 1950s and early 1960s with the development of an intuitive semantics based on ‘possible worlds’ by Kripke (after whom the semantics is often named), Kanger and Hintikka (see, e.g., =-=[50,47,44]-=-). In philosophy, possible world semantics has been used in support of elaborate metaphysical arguments (see, e.g., Kripke [52]); however, the philosophical controversy over modal logic is far from se... |

201 |
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- 1992
(Show Context)
Citation Context ...ties of processes arising from process calculi such as Milner’s CCS [58]. Processes are modelled by labelled transition systems, which are, in turn, models of (multi-)modal logic (see, e.g., Stirling =-=[75]-=-). Thus one naturally ascribes modalChapter 9. Conclusions and further work 180 properties to processes, and one may speak of a process satisfying a formula of a modal program logic such as Hennessy-... |

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Citation Context ...eems to have been independently rediscovered by Gonzalez [39] (1985) and Tapscott [77] (1987). Similar ideas are used in a non-modal context in the ‘multilanguage systems’ of Giunchiglia and Serafini =-=[38]-=- (1994). The application of the approach to intuitionistic modal logics has been a recent occurrence. Indeed, we developed our systems N (T ) without knowledge of any other work in this direction. How... |

179 |
Labelled Deductive Systems
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- 1996
(Show Context)
Citation Context ... for first-order logic with relativized quantifiers [35]. The use of prefixed formulae and relational assumptions means that our systems N (T ) are ‘labelled deductive systems’ in the sense of Gabbay =-=[34]-=-. Labelled deductive systems are proposed by Gabbay as providing a unifying framework forChapter 4. Natural deduction for intuitionistic modal logics 83 giving proof systems to many different logics.... |

155 |
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Citation Context ...ions of modal model, satisfaction and validity in the standard informal meta-theory of ordinary classical mathematics. (Actually, only the so-called normal modal logics arise in this way, see Chellas =-=[13]-=-. However, we take possible world semantics as funda-Chapter 3. Intuitionistic modal logic 34 mental. Therefore we shall only be interested in normal modal logics.) Modal models determine a natural ‘... |

144 |
Elements of intuitionism
- Dummett
- 2000
(Show Context)
Citation Context ...hich applies to non-mathematical reasoning as well as to mathematical reasoning. For a general introduction to both the philosophy and mathematics of intuitionism and intuitionistic logic see Dummett =-=[16]-=-. In this thesis we study various intuitionistic modal logics obtained by extendingChapter 1. Introduction 3 intuitionistic logic with intensional operators. We give three reasons for considering suc... |

141 |
Contraction-free sequent calculi for intuitionistic logic
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- 1992
(Show Context)
Citation Context ...itionistic propositional connectives are not complete without contraction. However, there are ways of modifying the implication rules to overcome this problem (see, e.g., the recent paper of Dyckhoff =-=[18]-=-). We do not know if such techniques can be extended to the modal sequent calculi we consider. It is rather unfortunate that our decidability proof is restricted to such a limited class of intuitionis... |

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- 1990
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Citation Context ...ls are closely related to the transition systems of computer science. This connection has led to many applications of 1Chapter 1. Introduction 2 modal logic in computer science such as dynamic logic =-=[49]-=- and Hennessy-Milner logic [42]. For a general introduction to modal logic see Hughes and Cresswell [46]. Intuitionism arose as a school of mathematics founded by the Dutch mathematician L. E. J. Brou... |

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Ideas and results in proof theory
- Prawitz
- 1971
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Citation Context ...tuitionistic first-order logic without equality. Henceforth, we write IL for intuitionistic first-order logic, and NIL for its natural deduction system. The reader is referred to the works of Prawitz =-=[65,66]-=- for a more thorough treatment of natural deduction for IL. We assume given some first-order signature. We use: x,y...to range over variables; t,...to range over terms; a,b,...to range over atomic for... |

132 | Metamathematical Investigation of Intuitionistic Arithmetic and Analysis - Troelstra - 1973 |

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Semantical Analysis of Intuitionistic Logic I
- Kripke
- 1965
(Show Context)
Citation Context ...PL is the decidability of theoremhood (cf. Dummett [16, p. 146]). 2.2 The semantics of intuitionistic logic The semantics we shall use for intuitionistic first-order logic was introduced by Kripke in =-=[51]-=-, and was inspired by his earlier semantics for modal logic (which we shall discuss in Section 3.1). Kripke motivated the semantics as giving an intuitive account of the intuitionistic connectives. Ho... |

109 |
The logical basis of metaphysics
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- 1991
(Show Context)
Citation Context ... of intuitionistic logic has also found recent philosophical application. Dummett has argued that the proof theory justifies intuitionistic logic as the underlying logic of an anti-realist philosophy =-=[17]-=-. His argument gives an account of intuitionism which is substantially different from Brouwer’s and which applies to non-mathematical reasoning as well as to mathematical reasoning. For a general intr... |

104 | Simple Consequence relations
- Avron
- 1991
(Show Context)
Citation Context ...ical Framework (LF) [41]. The various sideconditions on the modal rules (and on (Rχ)) are all handled easily by the binding mechanisms of LF. Thus our inference rules are ‘pure’ in the sense of Avron =-=[2]-=-. For the difficulties that arise in representing other forms of proof system for modal logic in LF see Avron et al [3].Chapter 5 Meta-logical completeness In this short chapter we prove the meta-the... |

93 | Categorical logic
- Pitts
- 2000
(Show Context)
Citation Context ...on top of transition systems between partial states [60] (1990). Pitts has proposed using constructive logic with certain ‘evaluation’ modalities to reason about functional programs with side-effects =-=[63]-=-Chapter 3. Intuitionistic modal logic 45 (1991). Lastly, in hardware design, an application of an intuitionistic ♦ modality to abstract from constraints has been suggested by Mendler [56] (1991). In ... |

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- Honsell, Pollack
- 1992
(Show Context)
Citation Context ...ding mechanisms of LF. Thus our inference rules are ‘pure’ in the sense of Avron [2]. For the difficulties that arise in representing other forms of proof system for modal logic in LF see Avron et al =-=[3]-=-.Chapter 5 Meta-logical completeness In this short chapter we prove the meta-theoretical soundness and completeness (see Section 3.4) of the natural deduction system N (T ). Recall the translation, (... |

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63 |
Semantical Analysis of Modal Logic I: Normal Propositional Calculi. In: Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9:67{96
- Kripke
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Citation Context ...c came in the late 1950s and early 1960s with the development of an intuitive semantics based on ‘possible worlds’ by Kripke (after whom the semantics is often named), Kanger and Hintikka (see, e.g., =-=[50,47,44]-=-). In philosophy, possible world semantics has been used in support of elaborate metaphysical arguments (see, e.g., Kripke [52]); however, the philosophical controversy over modal logic is far from se... |

59 |
Provability interpretations of modal logics
- Solovay
- 1976
(Show Context)
Citation Context ...f mathematical logic in its own right (see, e.g., van Bentham [6]). Second, the semantics enabled modal logic to be applied to interesting mathematical problems such as Solovay’s Completeness Theorem =-=[72]-=-. Lastly, possible world models are closely related to the transition systems of computer science. This connection has led to many applications of 1Chapter 1. Introduction 2 modal logic in computer s... |

56 |
Modal Logics for Communicating Systems
- Stirling
- 1987
(Show Context)
Citation Context ...r Servi’s models of intuitionistic S5. In computer science, various applications of intuitionistic modal logics have been described. In Stirling’s 1987 paper on modal logics for communicating systems =-=[74]-=-, an intuitionistic modal logic was used to capture a bisimulation preorder on diverging processes (an idea attributed to Gordon Plotkin). A similar application appears also in Hennessy and Plotkin [4... |

38 |
Symbolic Logic, An Introduction
- Fitch
- 1952
(Show Context)
Citation Context ...l logic (both classical and intuitionistic). The work most directly related to ours is that on natural deduction systems for modal logic using relative truth. The approach was initiated by Fitch, see =-=[26]-=- (1952) and [27] (1966), who gave proof systems for several classical modal logics. His systems were generalized by Siemens to cover a still wider range of classical modal logics [71] (1977). A discus... |

37 |
Proof Theory and Meaning
- Sundholm
- 1986
(Show Context)
Citation Context ...l deduction both justifies the philosophy that the meaning of a logical constant is given by its inference rules and is a prerequisite for the philosophy to apply [17]. See Sundholm’s survey article, =-=[76]-=-, for a general discussion of this argument and for further references. When we come to modal logic, we shall be interested in propositional modal logic. Accordingly, we briefly review intuitionistic ... |

31 |
A framework for intuitionistic modal logics
- Plotkin, Stirling
- 1986
(Show Context)
Citation Context ...nistic modal logic as the natural deduction system. It turns out that many of the intuitionistic modal logics we induce occur already in work of Ewald [20], Fisher Servi [24] and Plotkin and Stirling =-=[64]-=-. However, their original definitions were semantic and not intuitionistically motivated. Further, despite being well known, it was an open question whether the logics were decidable. (A flawed proof ... |

23 | Intuitionistic necessity revisited
- Bierman, Paiva
- 1996
(Show Context)
Citation Context ...tting the false rule altogether.) Prawitz’ main result was a normalization theorem for the ♦- free fragment. He gave no further analysis of the induced intuitionistic modal logics. In a recent paper, =-=[7]-=- (1993), Bierman and de Paiva reformulated Prawitz’ system for S4 (without ♦) to give a simpler account of normalization. (They also gave a categorical semantics to the proof theory.) A different appr... |

23 | Axiomatizations for some Intuitionistic Modal Logics - Servi - 1984 |

20 |
Intuitionistic tense and modal logic
- Ewald
- 1986
(Show Context)
Citation Context ...introduced in Section 3.3. (We shall do this in Chapter 8. It is not completely trivial.) The desired implication would then follow from the known completeness of IK with respect to birelation models =-=[20,24,64]-=-. Alternatively, as mentioned on page 60, Stirling has given an unpublished semantic proof of the meta-logical completeness of IK (also relying on the completeness of IK relative to birelation models)... |

20 |
Tableau methods of proof for modal logics
- Fitting
- 1972
(Show Context)
Citation Context ...truth was Kanger’s system for S5 using ‘spotted’ formulae [47] (1957). The approach has been generalized by FittingChapter 7. Normalization and its consequences 145 with his prefixed tableau systems =-=[28]-=- (1972) and [29, Ch. 8] (1983) (which are closely related to cut-free sequent calculi). The sequent calculi presented in Section 7.2 differ in two ways from those in the aforementioned work. First, th... |

20 | Cut-free Sequent and Tableau Systems for Propositional Normal Modal Logics
- Goré
- 1992
(Show Context)
Citation Context ...al sequents of formulae, rather than prefixed formulae. With such systems, cut-elimination and the subformula property yield very easy proofs of decidability. For a survey of such techniques see Goré =-=[40]-=- (1992). However, as in the case of natural deduction, we do not know how to apply such techniques to any of the intuitionistic modal logics we are interested in. (Again, it is the ♦ modality that cau... |

20 |
Constructive modal logics I
- Wijesekera
- 1990
(Show Context)
Citation Context ...at that has been considered. The notions of model proposed by Gabbay [33, §7] (1975), Ono [61] (1977), Sotirov [73] (1984), Boˇzić andDoˇsen [8,15] (1984–5), Yokota [84,85,86] (1985–6) and Wijesekera =-=[82]-=- (1990) all differ from each other as well as from those of Ewald et al, although there are also many similarities. We shall discuss some of the differences below. (Sotirov [73] also contains a useful... |

19 |
Relating theories of the λ-calculus
- Scott
(Show Context)
Citation Context ... or not one accepts its underlying philosophical tenets. Third, there are deep connections with computation theory that have recently been exploited in computer science (see Martin-Löf [54] and Scott =-=[69]-=- for two very different applications). The proof theory of intuitionistic logic has also found recent philosophical application. Dummett has argued that the proof theory justifies intuitionistic logic... |

18 |
Automated Proof Search in Non-Classical Logics
- WALLEN
- 1990
(Show Context)
Citation Context ...L. Although we have not analysed the problem of proof search, proof systems based on relative truth have been important in the theorem proving literature for classical modal logics (see, e.g., Wallen =-=[81]-=-). But the most striking use of our specialized modal systems will be given by our later proof that, for certain T ,the consequence relation of N (T ) is decidable (Section 7.3). There are other forms... |

17 |
On Modal Logics with an Intuitionistic Base, Studia Logica 36
- Servi
- 1977
(Show Context)
Citation Context ...va [57] (1980)). In a later paper, Font and Verdú considered the abstract algebraic properties of (again ♦-free) intuitionistic analogues of S4 and S5 [31] (1989). In the first of a series of papers, =-=[21]-=- (1977), Fischer Servi proposed a way of determining the correct intuitionistic analogue of a classical modal logic. Motivated by Gödel’s translation of intuitionistic propositional logic into classic... |

16 |
Semantics for a class of Intuitionistic Modal Calculi
- Servi
- 1980
(Show Context)
Citation Context ...induced cut-free sequent calculi. The sequent calculi are then used to prove the decidability of a number of the logics. In Chapter 8 we consider a semantic framework due to Ewald [20], Fischer Servi =-=[23]-=- and Plotkin and Stirling [64]. We prove the finite model property for those logics proved decidable in the Chapter 7. (This fact gives an alternative, but more complicated, proof of decidability.) In... |

16 |
Y.: Proofs and types. Cambridge tracts in theoretical computer science
- Girard, Taylor, et al.
(Show Context)
Citation Context ...les. This time the rewrite rules in question are the so-called permutative reductions (we use the terminology of Prawitz [66], Girard writesChapter 2. Intuitionistic logic 17 ‘commuting conversions’ =-=[37]-=-). As there are 7 elimination rules and 3 indirect rules, there are 21 cases to be eliminated by permutative reductions (although the symmetry between (∧E1) and (∧E2) means that there are only 18 esse... |

14 |
On some Intuitionistic modal logics
- Ono
- 1977
(Show Context)
Citation Context ...f this approach to the ♦ operator is considered in the recent paper of Masini [55](1993). We shall discuss these approaches in more detail in Section 4.6.Chapter 3. Intuitionistic modal logic 43 Ono =-=[61]-=- (1977) continued the tradition of considering intuitionistic analogues of S4 and S5. He analysed several inequivalent variants of the two logics, especially the ♦-free fragments. He gave these both a... |

12 |
Correspondence theory
- Bentham
- 1984
(Show Context)
Citation Context ...to other fields. First, the semantics is mathematically natural. Thus the model theory of modal logic has become an interesting subfield of mathematical logic in its own right (see, e.g., van Bentham =-=[6]-=-). Second, the semantics enabled modal logic to be applied to interesting mathematical problems such as Solovay’s Completeness Theorem [72]. Lastly, possible world models are closely related to the tr... |

12 | Geometric logic in computer science
- Vickers
- 1993
(Show Context)
Citation Context ...intuitionistic logic In this section we consider some properties of so-called geometric theories in intuitionistic logic. Geometric theories have played an important part in topos theory (see Vickers =-=[80]-=- for an introduction and references). However, we shall be interested in them for proof-theoretical reasons. It turns out that geometric theories are exactly the theories expressible by natural deduct... |