## Abstract

The starting point of this chapter is the remarkable fact that proof procedures for wide classes of hybrid logics can be given in a uniform way, and moreover, this encompasses proof procedures like natural deduction and tableau systems which are suitable for actual reasoning. A focus of the chapter is such proof procedures. Axiom systems, which are not meant for actual reasoning, are only mentioned in passing. We present a relatively small selection of procedures rather than trying to be encyclopedic. This allows us to give a reasonably detailed treatment of the selected procedures. Another focus of the chapter is the origin of hybrid logic in Arthur Prior’s philosophical work.

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## Notes

- 1.
The chapter is composed of material adapted from the author’s book (Braüner 2011). The author wishes to acknowledge the financial support received from The Danish Natural Science Research Council as funding for the projects HyLoMOL (2004–2008) and HYLOCORE (2009–2013).

- 2.
The word “actual” has here a broad meaning, not restricted to actual human reasoning. The logic does not care whether it is a human that carries out the reasoning, or the reasoning takes place in a computer, or in some other medium.

- 3.
This should not be confused with the term “hybrid systems” which in the computer science community is used for systems that combine discrete and continuous features.

- 4.
Considering a nominal as a symbol that refers to something is not the only way to view nominals. Two different views on nominals can be identified in the works of Arthur Prior, as is clear from the quotation below where Prior discusses the addition of nominals to a temporal version of modal logic called tense logic.

We might …equate the instant

*a*with a conjunction of all those propositions which would ordinarily be said to be true at that instant, or we might equate it with some proposition which would ordinarily be said to be true at that instant only, and so could serve as an index of it (Hasle et al. 2003, p. 124).In the second half of the sentence, the nominal

*a*is viewed as a proposition that can serve as an index of an instant, which is clearly in line with considering a nominal as a symbol that refers to an instant. On the other hand, in the first half of the sentence, the nominal*a*is viewed as a description of the content of an instant. The alternative view on nominals expressed in the first half of the sentence quoted above can also be found in a number of other places in Prior’s works, for example the following.The essential trick is to treat the instant variables as a special sort of

*propositional*variables, by identifying an ‘instant’ with the totality of what would ordinarily be said to be true*at*that instant, …(Hasle et al. 2003, p. 141).See the discussion of Prior’s work in Sect. 1.2 of the present handbook chapter, in particular Footnote 8 of that section. Moreover, see the discussion in Patrick Blackburn’s paper (2006), the last paragraph of page 353, including Footnote 7, and the first complete paragraph of page 362, in particular Footnote 11. Incidentally, note that the description of the content of an instant as the conjunction of all propositions true at that instant is similar to a maximal consistent set of formulas.

- 5.
In fact, the paper (Blackburn and Seligman 1995) gives a result (Proposition 4.5 on p. 264) indicating that the \(\forall \) binder has a surprisingly local character when it is not accompanied by satisfaction operators or some similar machinery. Informally, this result says that the \(\forall \) binder is then insensitive to the information at points outside the submodel generated by the point of evaluation, that is, it cannot detect the truth-values of formulas at such points.

- 6.
Further discussion of this point can be found in a number of places, notably the paper Blackburn (2006). This paper also discusses hybrid-logical versions of

*bisimulations*, which give a mathematical way to illustrate the local character of the Kripke semantics. See also the paper Simons (2006) which discusses a number of logics of location involving what we here call satisfaction operators. - 7.
All results in the present handbook chapter can be generalized to cover an arbitrary, finite number of modal operators, but in the interest of simplicity, we shall stick to one modal operator unless otherwise is specified.

- 8.
It appears that this criticism presupposes a view on nominals according to which a nominal is a symbol that refers to something, like a first-order variable does. As remarked in Footnote 4 in Sect. 1.1.1, there is an alternative view on nominals according to which a nominal is viewed as a description of the content of an instant. It is not clear whether the criticism applies if this alternative view on nominals is adopted.

- 9.
Sylvan actually argues that it is not necessary to reduce first-order earlier-later logic (the B-series conception of time) to tense logic (the A-series conception), or vice versa. Sylvan points out that Prior regarded tense-logical postulates as being capable of giving the meaning of statements like ‘time is continuous’ and ‘time is infinite both ways’ (cf. Prior 1967, p. 74). To this Sylvan responds as follows.

Time is an item, a theoretical object, which bears both the tensed and the temporally ordered properties which the item in question genuinely has. …

Part of the elegance of such a simple characterization of Time is that it neatly decouples the stable sense of ‘time’ …from various vexed issues as to exactly which properties the item genuinely has (and so from what Time is ‘really’ like). Whichever it should have, under evolving or under alternative theories, the item can remain abstractly one and the same. Naturally, tight coupling remains between the item and its properties; but it is not a meaning connection, it is a theory-dependent linkage (Sylvan 1996, p. 114).

Sylvan sees Prior’s goal to reduce the B-series talk to A-series talk as part of a more general, and in Sylvan’s view overdeveloped, reductionist inclination of analytic philosophers, which also encompasses philosophers having the converse reduction as a goal, that is, having the goal of reducing A-series to B-series (cf. Sylvan 1996, p. 112).

- 10.
A variation of the \(\downarrow \) binder (called the “freeze” quantifier) was actually introduced already in 1989 in connection with real-time logics, see the paper by Alur and Henzinger (1989). See also the survey Alur and Henzinger (1992). The \(\downarrow \) binder and the freeze quantifier were discovered independently of each other.

- 11.
To prove these results, the paper Blackburn and Seligman (1995) introduces a proof technique called the spy-point technique, which later has been used in many other connections.

- 12.
The interpolation theorem for propositional logic says that for any valid formula

*ϕ*→*ψ*there exists a formula*θ*containing only the common propositional symbols of*ϕ*and*ψ*such that the formulas*ϕ*→*θ*and*θ*→*ψ*are valid. Interpolation theorems for other logics are formulated in an analogous fashion. - 13.
An unexplored line of work is to find out whether interpolation can be proved in other ways, for example using proof systems like the linear reasoning systems which in Fitting (1984) are used to prove interpolation for some particular propositional and first-order modal logics.

- 14.
See ten Cate (2004) for semantic characterizations of frame classes definable by pure hybrid-logical formulas.

- 15.
Like frame classes definable by pure formulas, frame classes definable by geometric theories can be given a semantic characterization, see the remark at the end of Sect. 1.4.3.

- 16.
This was pointed out to the author by Balder ten Cate (personal communication).

- 17.
According to Prawitz’ terminology (Prawitz 1978), a natural deduction introduction rule for a connective is

*explicit*if the connective in question is exhibited exactly once, namely in the conclusion of the rule. Thus, according to this terminology, the hybrid-logical introduction rule for the modal operator is not explicit. - 18.
This is actually a generally occurring problem (with a generally applicable solution) since the same problem crops up (and is solved in the same way) in connection with normalisation for intuitionistic hybrid logic, cf. Braüner and de Paiva (2006), and normalisation for first-order hybrid logic, cf. Braüner (2005). See also Braüner (2011). In the first case the reduction rule for the modal operator \(\diamond \), which in intuitionistic hybrid logic is primitive, not defined, might generate new maximum formulas on the form \(@_{a}\diamond e\), and in the second case, the reduction rule for the quantifier \(\forall \) might generate new maximum formulas on the form @

_{ a }*E*(*t*) where*E*(*t*), called the existence predicate, is an abbreviation for ∃*y*(*y*=*t*) which in turn is an abbreviation for \(\neg \forall y\neg (y = t)\). - 19.
Labelled systems have the labelling machinery at the metalevel, whereas hybrid-logical systems have machinery with similar effect at the object level. A third option is chosen in Fitting’s paper (1972b) where a curious modal-logical axiom system is given in which labelling machinery is incorporated directly into the object language itself. In that system sequences of formulas of ordinary modal logic, delimited by a distinguished symbol ∗ , are used as names for possible worlds. To be more specific, a sequence \({\ast}\diamond \phi _{1},\ldots,\diamond \phi _{n},\diamond \phi _{n+1}{\ast}\) is used as the name of a world accessible from the world named by \({\ast}\diamond \phi _{1},\ldots,\diamond \phi _{n}{\ast}\) and in which the formula

*ϕ*_{ n + 1}is true, if there is one. It is allowed to form object language formulas by prefixing ordinary modal-logical formulas with such sequences. Intuitively, a prefixed formula \({\ast}\diamond \phi _{1},\ldots,\diamond \phi _{n}{\ast}\psi\) says that the formula*ψ*is true at the world named by the prefix. Prefixed formulas can be combined using the usual connectives of classical logic. - 20.
This terminology is used in a somewhat different sense than is common: Our destructive rules preserve information in the sense that if a conclusion of a destructive rule has a model, then this model is a model for the premise of the rule as well, that is, no models are included (note that this is opposite of soundness which says that no models are excluded). In the usual sense destructive rules are rules that do not preserve information (see Fitting 1972a).

- 21.
An occurrence of a satisfaction statement @

_{ a }*ϕ*or the negation of a satisfaction statement \(\neg @_{a}\phi\) in a tableau can be seen as a formula*ϕ*together with a pair consisting of the representation of a possible world (the nominal*a*) and the representation of a truth-value (depending on whether the satisfaction statement is negated or not). Note in this connection that in the possible worlds semantics, the semantic value assigned to a formula is a function from possible worlds to truth-values, and set-theoretically, such a function is a set of pairs of possible worlds and truth-values (called the graph of the function). Hence, the pairs of nominals and representations of truth-values associated with formulas in the tableau system can be considered representations of elements of functions constituting semantic values. Thus, the tableau rules step by step build up semantic values of the formulas involved, similar to the way in which the accessibility relation step by step is built up (there is a difference however; the accessibility relation can be*any*relation, but the semantic value of a formula has to be a*function*, that is, a relation where no element of the domain is related to more than one element of the codomain, and this is exactly what is required of an open branch in a tableau, namely that no satisfaction statement is related to more than one truth-value). - 22.
This was pointed out to the author by Jens Ulrik Hansen.

- 23.
This was pointed out to the author by Thomas Bolander.

- 24.
Also other criteria could be considered, one important example being interpolation, that is, the criterion that a proof system should lend itself to the calculation of interpolants, perhaps after being enhanced with further machinery, like the tableau system for first-order hybrid logic which in the paper Blackburn and Marx (2003) is used as the basis of an algorithm that calculates interpolants. See the remarks on interpolation in Sect. 1.3. Note that there are two steps: The first step is the requirement of a logic (which here is a formal language together with a semantics) that it satisfies interpolation. This might be proved semantically, independent of any proof systems. If the logic does satisfy interpolation, then the second step is the requirement of a proof system for the logic that the proof system in question can be used as the basis for calculating interpolants.

- 25.
It should be mentioned that there are a number of natural deduction and Gentzen style formulations for modal logic that do not fit this categorisation well. Notable here are formulations in terms of Nuel Belnap’s display logic and Kosta Dŏsen’s higher-level sequents. However, these formulations differ considerably from Gentzen’s original natural deduction and sequent systems and they are more complicated from a technical point of view. (Although it has to be acknowledged that display sequents as well as higher-level sequents were introduced as natural generalisations of Gentzen’s notion of a sequent, intended to allow a uniform sequent-style formulation of many different logics.) An overview can be found in Wansing (1994). Also notable are modal hypersequent systems, see Avron (1996) as well as the handbook chapter Fitting (2007).

- 26.
In fact, Prawitz’ systems for S4 and S5 deviate from most standard systems since his introduction rules for □ make use of “non-local” side-conditions, that is, side-conditions that do not just refer to the premises of the rules and to undischarged assumptions, but to the whole derivations of the premises.

- 27.
- 28.
This can actually be generalized: These two features also enable the formulation of natural deduction systems for intuitionistic hybrid logics satisfying the criterias, cf. Braüner and de Paiva (2006) and Braüner (2011), but in that case the features are interpreted intuitionistically, that is, they are interpreted as statements in intuitionistic first-order logic and intuitionistic hybrid logic. See also Braüner (2006). In the case of first-order hybrid logic, cf. Braüner (2005) and Braüner (2011), we can furthermore express that an individual

*t*exists at a world*a*, that is, the formula @_{ a }*E*(*t*) is true, which enables the formulation of natural deduction systems for first-order hybrid logics satisfying the criteria (here*E*(*t*) is the existence predicate which is defined as an abbreviation for ∃*y*(*y*=*t*)). - 29.
In the paper (Brünnler 2006) Kai Brünnler compares labelled and unlabelled Gentzen systems for modal logic. A system of the latter kind is a system that does not use labels, which he makes more precise by calling a Gentzen system

*pure*if each sequent has an equivalent object language formula. Clearly, what we here call standard proof systems for modal logic are pure: In natural deduction terminology, a derivation of a modal-logical formula*ϕ*from a set of modal-logical formulas \(\Gamma \) is equivalent to the modal-logical formula \(\bigwedge \Gamma \rightarrow \phi\). On the other hand, labelled natural deduction systems for modal logic are clearly not pure, but it is remarkable that hybrid-logical natural deduction systems actually are pure in Brünnler’s sense. - 30.
Belnap’s paper (1962) is a response to Prior’s paper (1960) in which Prior raises doubt as to whether the meaning of logical connectives can be explained in terms of derivation rules along the lines of natural deduction introduction and elimination rules. In his paper, Prior introduces a logical connective

*tonk*with introduction rules similar to the standard natural deduction introduction rules for disjunction and elimination rules similar to the standard natural deduction elimination rules for conjunction. An effect of extending a formal system with*tonk*together with the mentioned rules is that any formula becomes derivable, which obviously is absurd. In his response to Prior’s paper, Belnap suggests imposing certain restrictions on derivation rules, thereby excluding Prior’s rules for*tonk*from the permissible rules for a connective. According to Belnap, the crucial restriction is*conservativity*: When a formal system is extended with a new logical connective together with a set of derivation rules, then for any formula built using only the original connectives, it is required that if the formula is derivable in the extended system, then it is also derivable in the original system, that is, it is derivable without using the derivation rules for the new connective. - 31.
A frequently discussed issue in proof-theoretic semantics is which restrictions to impose on the derivation rules for a connective, that is, which sets of derivation rules to take as permissible. This discussion can be traced back to Prior (1960) and Belnap’s (1962) papers, cf. the previous footnote. A number of restrictions on derivation rules have been proposed, one proposal being conservativity, cf. the previous footnote, another proposal being the inversion principle (cf. the paper Prawitz (1971)).

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Braüner, T. (2014). Hybrid Logic. In: Gabbay, D., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 17. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6600-6_1

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