Abstract
In this paper we present tableau methods for twodimensional modal logics. Although models for such logics are well known, proof systems remain rather unexplored as most of their developments have been purely axiomatic. The logics herein considered contain firstorder quantifiers with identity, and all the formulas in the language are doublyindexed in the proof systems, with the upper indices intuitively representing the actual or reference worlds, and the lower indices representing worlds of evaluation—first and second dimensions, respectively. The tableaux modulate over different notions of validity such as local, general, and diagonal, besides being general enough for several twodimensional logics proposed in the literature. We also motivate the introduction of a new operator into twodimensional languages and explore some of the philosophical questions raised by it concerning the relations there are between actuality, necessity, and the a priori, that seem to undermine traditional intuitive interpretations of twodimensional operators.
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Notes
Hazen (1978) develops a Fitchstyle natural deduction system, except that it only contains an actuality operator alongside the modal operators.
Although, as we shall see in due course, only some twodimensional operators will be said to be explicitly epistemic. In Sect. 5 we shall explore just some of those questions concerning quantification in order to set up a system intended to represent the discourse involving a priori knowledge following Chalmer’s version of twodimensional semantics. In Ray and Lampert (unpublished manuscript), we develop a Lemmonstyle natural deduction system for quantified twodimensional modal logic involving actuality and fixedly operators, although twodimensional developments of systems in the style of Fitch and Gentzen are also yet to be explored.
During the time this paper was being reviewed Gilbert (2016) was published, developing twodimensional tableaux independently from the present paper. The system considered by Gilbert is basically David and Humberstone’s propositional S5 \(\mathcal {AF}\) with the addition of a Ref operator. In addition, Gilbert presents some decidability results for different twodimensional logics, which of course will not hold for the firstorder systems herein considered. Many thanks to Shawn Standefer and an anonymous reviewer for pointing me to Gilbert’s paper.
Not everyone is happy with this. Poggiolesi and Restall (2012) accuse labelled systems of exploiting explicit semantic notions in the proof system, which is unacceptable, so they claim, on the basis that the latter should employ purely syntactic tools. We do not engage in this debate here, although a response against the charge of “semantic pollution” in labelled systems can be found in Read (2015).
We avoid using quotation marks for the sake of presentation, but usemention distinctions should be clear given the context.
Humberstone (2004, p.21).
We should point out that, originally, Crossley and Humberstone use the formula \(\mathcal {A}\varphi \supset \mathcal {FA}\varphi \), which is fine since the repeated actuality operator can be deleted in this case without loss.
Analogously, one could get necessary truths that are only a posteriori knowable, for instance, by substituting any empirical truth for \(\varphi \) in \(\mathcal {A}\varphi \supset \Box \mathcal {A}\varphi \).
The shiftably operator was previously defined in Ray and Lampert (unpublished manuscript).
Hanson (2006, p. 452), defends that, to a certain extent, deep necessity and possibility seem to capture our intuitions about necessity and possibility in general even better than superficial necessity and possibility.
In Humberstone (1982, p. 452), \(\mathcal {A}\) is called an inhibitor (in a logic without \(\mathcal {F}\)), for it “protects what is in its scope from the influence of an outlying modal operator.”
This example can be found in Sider (2010, p. 225).
Assuming, of course, a rigid actuality operator, where \(\mathcal {A}\) always takes us to the actual world. In a twodimensional logic, \(\mathcal {A}\) copies down the upper world, whatever it is, whereby it is enough for the purposes of (7) that the upper world is not w.
Sider formalizes Ref by using \(\times \). But since we use \(\times \) as a syntactic mark for when a branch of tableau closes, we choose a slightly different symbol instead.
Also, in what follows we use \(a, b, c, \ldots \) for constant symbols and \(x, y, z, \ldots \) for individual variables.
Since \(w*\in W\) is the distinguished element of W. More on this below.
Thus, by following the common practice of singling out an element of the set of worlds in the models as distinguished, we get an ordered pair, rather than the usual single distinguished world \(w*\). But this in turn might give one reason to demur: the distinguished element of the models is more than often interpreted in an intuitive way as the actual world, but certainly the pair \(\langle w*,w*\rangle \) should not be understood as the actual world! We agree. An alternative would be to define (constant domain) 2Dcentered models as \({\mathcal {M}}=\langle W,w*,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \), where \(W=Z\times Z\) and \(w*\in Z\). The reason we take the distinguished element to consist of an ordered pair is because it provides us with a more uniform account of truth in a model. Since every formula is being evaluated with respect to pairs of worlds, truth in a model must, too, be relativized to a pair. And since truth in a model is defined as truth at a distinguished point, a distinguished pair is called for. (Thus, we say that truth at \(\langle w*,w*\rangle \) can be intuitively understood as truth at the actual world with respect to itself.) If one dislikes this option we offer the alternative definition in its place. This has no direct bearing on the tableaux
In Sect. 5 we define another system in which this constraint is abandoned, and we discuss some philosophical issues related to it.
In Priest’s (2008) case, some nodes may be expressions such as ‘nrm’, where n and m are numerals such that the possible world denoted by m is raccessible from n. Fitting and Mendelsohn (1998), on the other hand, do not make use of single nodes reflecting the accessibility relations of the system in question. Rather, formulas may be prefixed by sequences of indices such as \(\sigma .n\), which means intuitively that n is a world accessible from \(\sigma \); whereby there is no need to add nodes with the single purpose of denoting an accessibility relation.
We avoid using the letter ‘o’ to denote lower indices since it may be confused with the number 0.
See, for instance, Fitting and Mendelsohn (1998, p. 49).
See, for instance, Priest (2008, p. 310).
In our case, this has to be adjusted to pairs of worlds.
This causes the rule of necessitation to fail in \(\mathbf S5 _{2D}\) as well, unless one defines general rather than local validity. In that case, \(\mathcal {A}p\supset p\) would not count as a theorem of \(\mathbf S5 _{2D}\).
This was previously shown in Ray and Lampert (unpublished manuscript) for natural deduction systems with numeric indices. Since the tableaux presented here work in a similar way, this result could be easily adapted. I am grateful to Greg Ray for helpful suggestions concerning this section.
For languages lacking a distinguishedly operator, one of the indices does not need to be different from ‘0’. Thus, general 2Dtableaux for \(\odot \)free languages might begin with the pair \(\langle 0,n\rangle \), where \(0\ne n\). However, things are trickier in languages containing \(\odot \). For example, by starting a general 2Dtableau with \(\langle 0,1\rangle \) we would be able to prove \(\Box p\supset \odot \mathcal {A}p\), which is valid when (Neutrality) is assumed, but not otherwise. Since in Sect. 5 we investigate twodimensional logics where the models are not constrained by (Neutrality), had we defined general 2Dtableaux without requiring of both indices in the tableau’s root that they must be different from ‘0’, we would have to change our definition and incorporate this restriction. Because of this, the requirement that both indices should be different from ‘0’ gives us a more general treatment to deal with general 2Dtableaux.
In “Appendix 3” we show some results concerning how \(\mathbf S5 _{2D}\) extends those systems.
See Chalmers (2004), p. 219.
These are equivalent. See Fritz (2014).
One can find the same semantics defined in Lampert (2017).
A more detailed exposition can be found in Chalmers (2004).
Although, of course, it might be dropped, generating thereby a different set of validities.
It is a simple matter to check that for any \(\mathcal {A}\)free formula \(\varphi \), we have both \(\Diamond \varphi \equiv \mathcal {C}\varphi \) and \(\Box \varphi \equiv \mathcal {D}\varphi \), the latter corresponding to axiom \(\mathcal {F}6\) in Davies and Humberstone’s S5 \(\mathcal {AF}\). See “Appendix 3”.
Chalmers thinks all superrigid expressions are semantically rigid. Although there are some wrinkles in the converse claim, if a semantically neutral expression is not equivalent to a superrigid expression, it is at least equivalent to a compound of superrigid expressions. For more details, see Chalmers (2012, p. 370).
In Chalmers’ terms, this kind of expression has a constant twodimensional intension. See Chalmers (2012, p. 370).
Williamson (2013), for example, argues that T is the correct system for a logic of knowledge.
Ultimately, these questions amount to the kind of accessibility relation one wants for a basic modal system.
This holds regardless of the chosen notion of validity—local, general, or diagonal.
In both cases, “validity” means either local, general, or threedimensional.
Tableaux for threedimensional modal logic can be easily defined by extending the twodimensional case with another numeric index. Note that we can also define a threedimensional version of \(\otimes \), whereby the third index of evaluation is copied up to the first one. If we symbolize this as \(\oplus \), its semantics can be defined as \({\mathcal {M}}\begin{array}{c} u\\ v\\ w \end{array}\vDash \oplus \varphi \) if and only if \({\mathcal {M}}\begin{array}{c} w\\ w\\ w \end{array}\vDash \varphi \). More generally, for any dimension n, we can define new actuality and Ref operators accordingly. The same is true, of course, for the other twodimensional operators mentioned here.
The answer should be negative in case a condition similar to (Neutrality) is assumed:
(3DNeutrality) If \(\varphi \) is a basic formula, then for every \(t,u,v,w,z\in Z\), \({\mathcal {M}}{\begin{array}{c} u\\ v\\ w \end{array}}\vDash \varphi \) if and only if \({\mathcal {M}}{\begin{array}{c} t\\ z\\ w \end{array}}\vDash \varphi \).
Once (3DNeutrality) is assumed, the models validate \(\varphi \equiv \mathcal {N}\varphi \) for every basic formula \(\varphi \), whence \(\mathcal {N}\) does not represent any notion of necessity in itself.
In Lampert (2017) we point out similar issues involving the apriority operator in a semantics based on epistemic twodimensional semantics, as developed by Chalmers.
We leave this case to the reader.
Smullyan (1995, pp. 58–59), presents one for firstorder logic.
Where the indices i and n may be identical to j and m, respectively.
Of course, we assume the models are constrained by (Neutrality).
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Acknowledgements
I am thankful to Shawn Standefer, Greg Restall, Lloyd Humberstone, Ted Shear, Rachel Boddy, and G. J. Mattey for comments on different versions of this paper as well as many helpful conversations. Especially, I am thankful to Greg Ray for all his help, comments, and suggestions. Different versions of this paper were presented at the Philosophy Department of the University of California, Davis; the UC Davis Logic, Language, Epistemology, and Mathematics Working Group; the annual meeting of the Association for Symbolic Logic at the University of Connecticut; NASSLLI at Rutgers University; the 5th Annual Graduate Philosophy Conference at the University of Calgary; and the annual meeting of the Canadian Society for History and Philosophy of Mathematics at the University of Calgary. Thanks are also due to two anonymous referees whose comments greatly improved the presentation of this paper. Finally, I am very grateful to Elaine Landry for support and encouragement. The present paper was supposed to be my first project under the supervision of Aldo Antonelli; I wish to dedicate it to his memory.
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Appendices
Appendix 1 Soundness and Completeness for S5\(_{2D}\)
The soundness theorem for \(\mathbf S5 _{2D}\) is an extension of corresponding proofs for modal logic presented, for instance, in Fitting and Mendelsohn (1998) and Priest (2008).
Definition 8.1
(Satisfiability) Let S be a set of doublyindexed formulas. We say S is satisfiable in \({\mathcal {M}}=\langle W ,\langle w*,w*\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \) just in case there is a function, f, assigning to each (single) index n occurring in S a possible world \(f(n)\in Z\), where \(f(0)=w*\), \(W= Z\times Z\), such that,

If \([\varphi ]^i_n\in S\), then \(\varphi \) is true at f(n) relative to f(i), i.e. \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \varphi \).

If the pairs of indices \(\langle i,n\rangle \) and \(\langle i,m\rangle \) are in S, then \(\langle f(i),f(n)\rangle {\mathcal {R}}_{\Box }\langle f(i),f(m)\rangle \).

If the pairs of indices \(\langle i,n\rangle \) and \(\langle j,n\rangle \) are in S, then \(\langle f(i),f(n)\rangle {\mathcal {R}}_{\mathcal {F}}\langle f(j),f(n)\rangle \).
Definition 8.2
A tableau branch \(\mathfrak {b}\) is satisfiable just in case the set of doublyindexed formulas on it is satisfiable in some model, and a tableau is satisfiable just in case some branch of it is satisfiable.
Lemma 8.1
A closed tableau is not satisfiable.
Proof
Suppose that \(\mathcal {T}\) is a closed tableau that is also satisfiable. By Definition 8.2, some branch \(\mathfrak {b}\) of \(\mathcal {T}\) is satisfiable, whence there is a model, \({\mathcal {M}}=\langle W ,\langle w*,w*\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \), satisfying the set S of doublyindexed formulas on \(\mathfrak {b}\) by way of the function f, by Definition 8.1. But since, by assumption, \(\mathcal {T}\) is also closed, for some formula \(\varphi \) and pair of indices \(\langle i,n\rangle \), \([\varphi ]^i_n\in S\) and \([\lnot \varphi ]^i_n\in S\). Hence, by Definition 8.1, \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \varphi \) and \({\mathcal {M}}^{f(i)}_{f(n)}\nvDash \varphi \), which is impossible. \(\square \)
Lemma 8.2
If one of the rules in \(\mathbf S5 _{2D}\) is applied to a satisfiable 2Dtableau, it results in another satisfiable 2Dtableau.
Proof
Suppose \(\mathfrak {b}\) is a branch of a satisfiable 2Dtableau \(\mathcal {T}\) and that we apply one of the rules in \(\mathbf S5 _{2D}\) to a doublyindexed formula occurring on \(\mathfrak {b}\). It is easily seen that the result is another satisfiable tableau. Since, by assumption, \(\mathcal {T}\) is satisfiable, by Definition 8.2 at least one of its branches, say, \(\mathfrak {b}*\), is satisfiable. Now, if \(\mathfrak {b}*\ne \mathfrak {b}\), then \(\mathfrak {b}*\) remains satisfiable after we apply a rule on \(\mathfrak {b}\), whence we have a satisfiable 2Dtableau by Definition 8.2. On the other hand, if \(\mathfrak {b}*=\mathfrak {b}\), we need to consider several cases. We omit the arguments for the Booleans, modal operators, quantifiers, and identity, since these are routine.^{Footnote 54} In order to carry on such arguments in the 2D framework one just needs to add a pair of indices where the first coordinate does not do any work.
Suppose \([\mathcal {A}\varphi ]^i_n\) occurs on \(\mathfrak {b}\), and we apply the rule for \(\mathcal {A}\). Since \(\mathfrak {b}\) is satisfiable, \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \mathcal {A}\varphi \), by Definition 8.1. Thus, \({\mathcal {M}}^{f(i)}_{f(i)}\vDash \varphi \), by the truth clause for \(\mathcal {A}\). Therefore, applying the rules for \(\mathcal {A}\) to a satisfiable 2Dtableau results in another satisfiable 2Dtableau. The argument for \(\lnot \mathcal {A}\) is analogous.
Suppose \([\odot \varphi ]^i_n\) occurs on \(\mathfrak {b}\), and we apply the rule for \(\odot \). Since \(\mathfrak {b}\) is satisfiable, \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \odot \varphi \), by Definition 8.1. Thus, \({\mathcal {M}}^{w*}_{f(n)}\vDash \varphi \), by the truth clause for \(\odot \), in which case \({\mathcal {M}}^{f(0)}_{f(n)}\vDash \varphi \), since ‘0’ is fixed to \(w*\). Therefore, applying the rules for \(\odot \) to a satisfiable 2Dtableau results in another satisfiable 2Dtableau. The argument for \(\lnot \odot \) is analogous.
Suppose \([\otimes \varphi ]^i_n\) occurs on \(\mathfrak {b}\), and we apply the rule for \(\otimes \). Since \(\mathfrak {b}\) is satisfiable, \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \otimes \varphi \), by Definition 8.1. Thus, \({\mathcal {M}}^{f(n)}_{f(n)}\vDash \varphi \), by the truth clause for \(\otimes \). Therefore, applying the rules for \(\otimes \) to a satisfiable 2Dtableau results in another satisfiable 2Dtableau. The argument for \(\lnot \otimes \) is analogous.
Suppose [\(\mathcal {F}\varphi \)]\(^i_n\) occurs on \(\mathfrak {b}\), and we apply the rule for \(\mathcal {F}\). Since \(\mathfrak {b}\) is satisfiable, \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \mathcal {F}\varphi \), by Definition 8.1. Also, for every pair of indices \(\langle i,n\rangle \) and \(\langle j,n\rangle \) occurring on \(\mathfrak {b}\), \(\langle f(i),f(n)\rangle {\mathcal {R}}_{\mathcal {F}}\langle f(j),f(n)\rangle \). Thus, \({\mathcal {M}}^{f(j)}_{f(n)}\vDash \varphi \), by the truth clause for \(\mathcal {F}\). On the other hand, suppose [\(\lnot \mathcal {S}\varphi \)]\(^i_n\) occurs on \(\mathfrak {b}\). Since \(\mathfrak {b}\) is satisfiable, \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \lnot \mathcal {S}\varphi \), by Definition 8.1, and so \({\mathcal {M}}^{f(i)}_{f(n)}\nvDash \mathcal {S}\varphi \). Hence, for every \(\langle f(i),f(n)\rangle {\mathcal {R}}_{\mathcal {F}}\langle f(j),f(n)\rangle \), \({\mathcal {M}}^{f(j)}_{f(n)}\nvDash \varphi \), by the truth clause for \(\mathcal {S}\), and so \({\mathcal {M}}^{f(j)}_{f(n)}\vDash \lnot \varphi \). Therefore, applying the rules for \(\mathcal {F}\) to a satisfiable 2Dtableau results in another satisfiable 2Dtableau.
Suppose [\(\mathcal {S}\varphi \)]\(^i_n\) occurs on \(\mathfrak {b}\), and we apply the rule for \(\mathcal {S}\). Since \(\mathfrak {b}\) is satisfiable, \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \mathcal {S}\varphi \), by Definition 8.1. Thus, for some \(z\in Z\), \(\langle f(i),f(n)\rangle {\mathcal {R}}_{\mathcal {F}}\langle z,f(n)\rangle \) and \({\mathcal {M}}^{z}_{f(n)}\vDash \varphi \), by the truth clause for \(\mathcal {S}\). Let g be like f except that \(g(j)=z\), where j is a new index occurring on \(\mathfrak {b}\). Since g and f agree except for j, the tableau extended by g is satisfiable. Moreover, by definition of g, \(\langle g(i),g(n)\rangle {\mathcal {R}}_{\mathcal {F}}\langle g(j),g(n)\rangle \) and \({\mathcal {M}}^{g(j)}_{g(n)}\vDash \varphi \). By choice of g, it follows that \({\mathcal {M}}^{f(j)}_{f(n)}\vDash \varphi \). The argument for \(\lnot \mathcal {F}\) is analogous. Therefore, applying the rules for \(\mathcal {S}\) to a satisfiable 2Dtableau results in another satisfiable 2Dtableau.
Finally, let \(\varphi \) be a basic formula and suppose [\(\varphi \)]\(^i_n\) occurs on \(\mathfrak {b}\), and that we apply the upper exchange rule. Since \(\mathfrak {b}\) is satisfiable, \({\mathcal {M}}^{f(i)}_{f(n)}\vDash \varphi \), by Definition 8.1. Moreover, since \(\varphi \) is a basic formula, \({\mathcal {M}}^{f(j)}_{f(n)}\vDash \varphi \), for any first coordinate f(j) of any pair of worlds, given that the models are constrained by (Neutrality). Therefore, applying the upper exchange rule to a satisfiable 2Dtableau results in another satisfiable 2Dtableau. \(\square \)
Theorem 8.1
(Soundness) If \(\varphi \) has a 2Dtableau proof, then \(\varphi \) is valid.
Proof
Suppose \(\varphi \) has a 2Dtableau proof, in which case there is a closed 2Dtableau, \(\mathcal {T}\), beginning with \([\lnot \varphi ]^0_0\). For a contradiction, assume that \(\varphi \) is not valid. Thus, there is a 2Dcentered model, \({\mathcal {M}}=\langle W ,\langle w*,w*\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \), such that \({\mathcal {M}}^{w*}_{w*}\nvDash \varphi \). Let f be a function such that \(f(0)=w*\). By Definition 8.1, \(\{[\lnot \varphi ]^0_0\}\) is satisfiable. Moreover, since its only branch is satisfiable, \(\mathcal {T}\) is also satisfiable, and so is any 2Dtableau extending it by way of the rules in \(\mathbf S5 _{2D}\), by Lemma 8.2. Therefore, \(\mathcal {T}\) is both closed and satisfiable, contradicting Lemma 8.1, whence \({\mathcal {M}}^{w*}_{w*}\vDash \varphi \). \(\square \)
Infinite Tableaux and Systematic Procedure. Some 2Dtableaux may run infinitely, and by König’s Lemma these will have at least one infinite branch. In what follows we give an example of a failed proof attempt in \(\mathbf S5 _{2D}\) generating an infinite 2Dtableau:
A 2Dmatrix for the above looks like this:
Using the same procedure as before to devise a countermodel we have the following:

Since \({\mathcal {M}}^0_0\vDash p\) and \({\mathcal {M}}^0_1\nvDash p\), it follows that \({\mathcal {M}}^0_0\nvDash \Box p\). Hence, .

Since and \({\mathcal {M}}^1_2\nvDash p\), it follows that \({\mathcal {M}}^1_1\nvDash \Box p\). Hence, .

Since \({\mathcal {M}}^2_0\vDash p\) and \({\mathcal {M}}^2_3\nvDash p\), it follows that \({\mathcal {M}}^2_2\nvDash \Box p\). Hence, \({\mathcal {M}}^2_0\nvDash p\supset \Box p\).

\(\dots \) Hence, \({\mathcal {M}}^0_0\nvDash \mathcal {S}(p\supset \Box p\)).
Even though a finite countermodel can be easily constructed to invalidate \(\mathcal {S}(p\supset \Box p\)),^{Footnote 55} we need a systematic procedure for constructing 2Dtableaux in order to prove that any tableau so constructed is such that, if it has an open branch, finite or not, there is a countermodel for it.
Now, several systematic procedures of this kind can be found in the literature, even for the modal cases. In particular, we direct the reader again to Fitting and Mendelsohn (1998, pp. 126–127,)^{Footnote 56} since our strategy will be based on it. Thus, let [\(\varphi ]^i_n\) be any doublyindexed formula of which we want to know whether or not it is satisfiable. For stage \(n=1\), introduce [\(\lnot \varphi \)]\(^i_n\) to the 2Dtableau as line 1. Next, suppose n stages have been completed. If the 2Dtableau is already closed, then we have produced a proof of [\(\varphi \)]\(^i_n\). On the other hand, if the 2Dtableau remains open, then we proceed to stage \(n+1\) in which we take an open branch \(\mathfrak {b}\) of the 2Dtableau such that \(\mathfrak {b}\) is the leftmost highest point on it, and for each doublyindexed formula \([\psi ]^j_m\) ^{Footnote 57} such that it occurs on \(\mathfrak {b}\) we extend the 2Dtableau as follows:

I
If \([\psi ]^j_m\) is a basic formula, say, \([\chi ]^k_p\), add \([\chi ]^l_p\) to the end of \(\mathfrak {b}\), where l is an upper index added by either \(\mathcal {S}\) or \(\mathcal {F}\) (if it does not already contain it).

II
If a constant, say, c, is on \(\mathfrak {b}\), add \([(c=c)]^k_p\) to the end of \(\mathfrak {b}\) for all pairs of indices \(\langle k,p\rangle \) occurring on \(\mathfrak {b}\).

III
If the pair of indices \(\langle k,p\rangle \) is on \(\mathfrak {b}\), add \([(c= c)]^k_p\) to the end of \(\mathfrak {b}\) for all constants c on \(\mathfrak {b}\).

IV
If \([\psi ]^j_m\) is \([(c= d)]^k_q\), for any constants c and d, and \([\varphi (c)]^k_p\) occurs on \(\mathfrak {b}\), add \([\varphi (d)]^k_p\) to the end of \(\mathfrak {b}\) (if it does not already contain it).

V
If \([\psi ]^j_m\) is \([\lnot \lnot \chi ]^k_p\), add \([\chi ]^k_p\) to the end of \(\mathfrak {b}\) (if it does not already contain it).

VI
If \([\psi ]^j_m\) is \([(\zeta \wedge \chi )]^k_p\), add both \([\zeta ]^k_p\) and \([\chi ]^k_p\) to the end of \(\mathfrak {b}\) (if it does not already contain them—analogously for the other conjunctive cases).

VII
If \([\psi ]^j_m\) is \([(\zeta \vee \chi )]^k_p\), split the end of \(\mathfrak {b}\) into \(\mathfrak {b}_1\) and \(\mathfrak {b}_2\), adding [\(\zeta \)]\(^k_p\) and \([\chi ]^k_p\) to the end of \(\mathfrak {b}_1\) and \(\mathfrak {b}_2\), respectively (if it does not already contain them—analogously for the other disjunctive cases).

VIII
If \([\psi ]^j_m\) is \([\Diamond \chi ]^k_p\), take the first lower numeric index, q, which is new to \(\mathfrak {b}\), and add \([\chi ]^k_q\) to the end of \(\mathfrak {b}\) (if it does not already contain it—analogously for \(\lnot \Box \)).

IX
If \([\psi ]^j_m\) is \([\Box \chi ]^k_p\), add every doublyindexed formula \([\chi ]^k_q\) to the end of \(\mathfrak {b}\) for every index q occurring on the branch (if it does not already contain it—analogously for \(\lnot \Diamond \)).

X
If \([\psi ]^j_m\) is \([\odot \chi ]^k_p\), add \([\chi ]^0_p\) to the end of \(\mathfrak {b}\) (if it does not already contain it—analogously for \(\lnot \odot \)).

XI
If \([\psi ]^j_m\) is \([\otimes \chi ]^k_p\), add \([\chi ]^p_p\) to the end of \(\mathfrak {b}\) (if it does not already contain it—analogously for \(\lnot \otimes \)).

XII
If \([\psi ]^j_m\) is \([\mathcal {A} \chi ]^k_p\), add \([\chi ]^k_k\) to the end of \(\mathfrak {b}\) (if it does not already contain it—analogously for \(\lnot \mathcal {A}\)).

XIII
If \([\psi ]^j_m\) is \([\mathcal {S} \chi ]^k_p\), take the first upper numeral index, l, which is new to \(\mathfrak {b}\), and add \([\chi ]^l_p\) to the end of \(\mathfrak {b}\) (if it does not already contain it—analogously for \(\lnot \mathcal {F}\)).

XIV
If \([\psi ]^j_m\) is \([\mathcal {F} \chi ]^k_p\), add every doublyindexed formula \([\chi ]^l_p\) to the end of \(\mathfrak {b}\) for every index l occurring on the branch (if it does not already contain it—analogously for \(\lnot \mathcal {S}\)).

XV
If \([\psi ]^j_m\) is \([(\exists x)\chi ]^k_p\), take the first constant, c, not appearing on \(\mathfrak {b}\), and add \([\chi [c/x]]^k_p\) to the end of \(\mathfrak {b}\) (if it does not already contain it—analogously for \((\lnot \forall x)\)).

XVI
If \([\psi ]^j_m\) is \([(\forall x)\chi ]^k_p\), add every doublyindexed formula \([\chi [c/x]]^k_p\) to the end of \(\mathfrak {b}\) such that c is among the first countably many individual constants available (if it does not already contain it—analogously for \((\lnot \exists x)\)).
Once this procedure is completed for the leftmost highest point on the 2Dtableau, repeat it for the next highest point closest from it until the rightmost highest point. This is where stage \(n+1\) is concluded. Now there are three possible cases to consider. Either the systematic procedure resulted in a closed 2Dtableau, thereby producing a proof of [\(\varphi \)]\(^i_n\); the procedure terminated, producing an open branch; the procedure does not terminate, producing a possibly infinite open branch.
Completeness for \(\mathbf S5 _{2D}\). We show how to construct a countermodel such that, if \(\mathfrak {b}\) is any complete open branch of a 2Dtableau, finite or not, then \(\mathfrak {b}\) is satisfiable, where \(\mathfrak {b}\) is a complete open branch of a 2Dtableau \(\mathcal {T}\) just in case every rule that can possibly be applied to it has been applied. In order to construct a countermodel we use the same procedure as in the counterexamples above.
Definition 8.3
(Equivalence class) Let \(\approx \) be an equivalence relation over the set of constants in \(\mathcal {L}_2{}_D\) such that \(c_0\approx c_1\) iff ‘[\(c_0= c_1\)]\(^i_n\)’ occurs on \(\mathfrak {b}\) for any pair of indices \(\langle i,n\rangle \). Moreover, for any c, let [c] denote the equivalence class of c relative to \(\approx \).
Definition 8.4
(Model) For the countermodel, if k is any single index occurring on \(\mathfrak {b}\), let \(Z=\{k\mid k\in \mathfrak {b}\}\), and \(W=Z\times Z\). Let \(\langle 0,0\rangle \) be a distinguished element of W. For every \(\langle i,n\rangle ,\langle i,m\rangle \in W\), set \(\langle i,n\rangle {\mathcal {R}}_{\Box }\langle i,m\rangle \), and for every \(\langle i,n\rangle ,\langle j,n\rangle \in W\), set \(\langle i,n\rangle {\mathcal {R}}_{\mathcal {F}}\langle j,n\rangle \). Next we define C as the set of all constants occurring on \(\mathfrak {b}\), and \(\mathscr {D}=\{[c]\mid c\in C\}\). Furthermore, set \(V(c,\langle i,n\rangle )=[c]\) to each \(c\in C\), and to each nplace predicate symbol, R, on \(\mathfrak {b}\), let \(V(R,\langle i,n\rangle )=\{\langle [c_0],\ldots ,[c_n]\rangle :R(c_0,\ldots ,c_n)^i_n\) occurs on \(\mathfrak {b}\)}. Given both rigidity conditions, for every constant symbol c and \(\langle i,n\rangle ,\langle j,m\rangle \in W\) we have \(V(c,\langle i,n\rangle )=V(c,\langle j,m\rangle )\). In case \(\varphi \) is atomic, if [\(\varphi \)]\(^i_n\) occurs on \(\mathfrak {b}\) then \({\mathcal {M}}^i_n\vDash \varphi \), otherwise set \({\mathcal {M}}^i_n\nvDash \varphi \).^{Footnote 58} This is enough to define a constant domain 2Dcentered model \({\mathcal {M}}=\langle W ,\langle 0,0\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \).
Lemma 8.3
(Truth lemma) Let \(\mathfrak {b}\) be a complete open branch of a 2Dtableau, and \({\mathcal {M}}\) be a constant domain 2Dcentered model, \({\mathcal {M}}=\langle W ,\langle 0,0\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \). For every doublyindexed formula, [\(\varphi \)]\(^i_n\),
Proof
By induction on \(\varphi \). We only consider the relevant twodimensional operators.
Let \(\varphi \) be \(\mathcal {A}\psi \). If [\(\mathcal {A}\psi \)]\(^i_n\) occurs on \(\mathfrak {b}\), then \([\psi ]^i_i\) is also on \(\mathfrak {b}\). By I.H., \({\mathcal {M}}^i_i\vDash \psi \), whence \({\mathcal {M}}^i_n\vDash \mathcal {A}\psi \), by the truth clause for \(\mathcal {A}\) (the argument for \(\lnot \mathcal {A}\psi \) is analogous).
Let \(\varphi \) be \(\odot \psi \). If [\(\odot \psi \)]\(^i_n\) occurs on \(\mathfrak {b}\), then \([\psi ]^0_n\) is also on \(\mathfrak {b}\). By I.H., \({\mathcal {M}}^0_n\vDash \psi \), whence \({\mathcal {M}}^i_n\vDash \odot \psi \), by the truth clause for \(\odot \) (analogously for \(\lnot \odot \psi \)).
Let \(\varphi \) be \(\otimes \psi \). If [\(\otimes \psi \)]\(^i_n\) occurs on \(\mathfrak {b}\), then \([\psi ]^n_n\) is also on \(\mathfrak {b}\). By I.H., \({\mathcal {M}}^n_n\vDash \psi \), whence \({\mathcal {M}}^i_n\vDash \otimes \psi \), by the truth clause for \(\otimes \) (analogously for \(\lnot \otimes \psi \)).
Let \(\varphi \) be \(\mathcal {F}\psi \). If [\(\mathcal {F}\psi \)]\(^i_n\) occurs on \(\mathfrak {b}\), then for every j on \(\mathfrak {b}\), \([\psi ]^j_n\) is also on \(\mathfrak {b}\). By I.H. and construction, \({\mathcal {M}}^j_n\vDash \psi \) for every \(\langle i,n\rangle {\mathcal {R}}_{\mathcal {F}}\langle j,n\rangle \), whence \({\mathcal {M}}^i_n\vDash \mathcal {F}\psi \), by the truth clause for \(\mathcal {F}\). On the other hand, let \(\varphi \) be \(\lnot \mathcal {S}\psi \). If [\(\lnot \mathcal {S}\psi \)]\(^i_n\) occurs on \(\mathfrak {b}\), then for every j on \(\mathfrak {b}\), [\(\lnot \psi \)]\(^j_n\) also occurs on \(\mathfrak {b}\). By I.H. and construction, \({\mathcal {M}}^j_n\nvDash \psi \) for every \(\langle i,n\rangle {\mathcal {R}}_{\mathcal {F}}\langle j,n\rangle \), whence \({\mathcal {M}}^i_n\nvDash \mathcal {S}\psi \), by the truth clause for \(\mathcal {S}\).
Let \(\varphi \) be \(\mathcal {S}\psi \). If [\(\mathcal {S}\psi \)]\(^i_n\) occurs on \(\mathfrak {b}\), then for some j on \(\mathfrak {b}\), \([\psi ]^j_n\) is also on \(\mathfrak {b}\). By I.H. and construction, \({\mathcal {M}}^j_n\vDash \psi \) for some \(\langle i,n\rangle {\mathcal {R}}_{\mathcal {F}}\langle j,n\rangle \), whence \({\mathcal {M}}^i_n\vDash \mathcal {S}\psi \), by the truth clause for \(\mathcal {S}\). On the other hand, let \(\varphi \) be \(\lnot \mathcal {F}\psi \). If [\(\lnot \mathcal {F}\psi \)]\(^i_n\) occurs on \(\mathfrak {b}\), then for some j on \(\mathfrak {b}\), [\(\lnot \psi \)]\(^j_n\) is also on \(\mathfrak {b}\). By I.H., \({\mathcal {M}}^j_n\nvDash \psi \) for some \(\langle i,n\rangle {\mathcal {R}}_{\mathcal {F}}\langle j,n\rangle \), whence \({\mathcal {M}}^i_n\nvDash \mathcal {F}\psi \), by the truth clause for \(\mathcal {F}\). \(\square \)
Theorem 8.2
(Completeness) If \(\varphi \) is valid, then \(\varphi \) has a 2Dtableau proof.
Proof
We prove the contrapositive. Suppose \(\varphi \) does not have a 2Dtableau proof, in which case any 2Dtableau \(\mathcal {T}\) beginning with \([\lnot \varphi ]^0_0\) remains open. Let \(\mathfrak {b}\) be a complete open branch of \(\mathcal {T}\) such that \([\lnot \varphi ]^0_0\) occurs on \(\mathfrak {b}\). By Lemma 8.3, there is a constant domain 2Dcentered model, \({\mathcal {M}}=\langle W ,\langle 0,0\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \), such that \({\mathcal {M}}^0_0\nvDash \varphi \). Consequently, \(\varphi \) is not valid. \(\square \)
Appendix 2: Soundness and Completeness Theorems for General and Diagonal Tableaux
It might be useful to indicate how the soundness and completeness theorems for local 2Dtableaux can be modified for general and diagonal 2Dtableaux. The only differences concern the main theorems for soundness and completeness, namely, Theorems 8.1 and 8.2. In what follows we show how they are adjusted for general 2Dtableaux.
Theorem 9.1
(Soundness for general 2Dtableaux) If \(\varphi \) has a general 2Dtableau proof, then \(\varphi \) is valid.
Proof
Suppose \(\varphi \) has a general 2Dtableau proof, in which case there is a closed general 2Dtableau, \(\mathcal {T}\), which begins with \([\lnot \varphi ]^n_m\), where \(n\ne m\), and both n and m are different from ‘0’. For a contradiction, assume that \(\varphi \) is not generally valid. Thus, there is a 2Dcentered model, \({\mathcal {M}}=\langle W ,\langle w*,w*\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \), and a pair \(\langle v,w\rangle \in W\), such that \({\mathcal {M}}^{v}_{w}\nvDash \varphi \). Let f be a function such that \(f(n)=v\) and \(f(m)=w\), in which case \({\mathcal {M}}^{f(n)}_{f(m)}\nvDash \varphi \). By Definition 8.1, \(\{[\lnot \varphi ]^n_m\}\) is satisfiable. Moreover, since its only branch is satisfiable, \(\mathcal {T}\) is also satisfiable, and so is any general 2Dtableau extending it by way of the \(\mathbf S5 _{2D}\) rules, by Lemma 8.2. Therefore, \(\mathcal {T}\) is both closed and satisfiable, contradicting Lemma 8.1, whence \({\mathcal {M}}^{v}_{w}\vDash \varphi \). \(\square \)
Theorem 9.2
(Completeness for general 2Dtableaux) If \(\varphi \) is valid, then \(\varphi \) has a general 2Dtableau proof.
Proof
We prove the contrapositive. Suppose \(\varphi \) does not have a general 2Dtableau proof, in which case any general 2Dtableau \(\mathcal {T}\) beginning with \([\lnot \varphi ]^n_m\), where \(n\ne m\), both n and m are different from ‘0’, and that remains open. Let \(\mathfrak {b}\) be a complete open branch of \(\mathcal {T}\) such that \([\lnot \varphi ]^n_m\) occurs on \(\mathfrak {b}\). By Lemma 8.3, there is a constant domain 2Dcentered model, \({\mathcal {M}}=\langle W ,\langle 0,0\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {F}},\mathscr {D},V\rangle \), such that \({\mathcal {M}}^n_m\nvDash \varphi \). Consequently, \(\varphi \) is not generally valid. \(\square \)
The arguments for diagonal 2Dtableaux are very similar, we just have to consider a sentence, \(\varphi \), indexed by any pair, \(\langle n,n\rangle \), where \(n\ne 0\). Besides this, the arguments are essentially the same.
Appendix 3: Different Systems
We show that sound and complete general 2Dtableau can be obtained for different systems extended by \(\mathbf S5 _{2D}\).
1.1 S5 \(\mathcal {A}\)
Crossley and Humberstone offer the following axioms for S5 \(\mathcal {A}\) with general validity, besides the usual S5 axioms:^{Footnote 59}

(A1)
\(\mathcal {A}(\varphi \supset \psi )\supset (\mathcal {A}\varphi \supset \mathcal {A}\psi )\)

(A2)
\(\mathcal {A}\equiv \lnot \mathcal {A}\lnot \varphi \)

(A3)
\(\Box \varphi \supset \mathcal {A}\varphi \)

(A4)
\(\mathcal {A}\varphi \supset \Box \mathcal {A}\varphi \).
Lemma 10.1
\(\mathbf S5 _{2D}\) with general validity extends S5 \(\mathcal {A}\) with general validity.
Proof
General 2Dtableau proof schemata can be given to prove axioms (A1)(A4). We show (A3) and (A4), and leave (A1) and (A2) for the reader. \(\square \)
Corollary 10.1
Soundness and Completeness for S5 \(\mathcal {A}\) with general validity.
Proof
By Lemma 10.1 and soundness and completeness for \(\mathbf S5 _{2D}\) general 2Dtableaux.\(\square \)
Moreover, it can be shown that \(\mathbf S5 _{2D}\) extends S5 \(\mathcal {A}\) with local validity as well by proving the axiom schema \(\mathcal {A}\varphi \supset \varphi \), which is straightforwardly done by local 2Dtableaux.
1.2 S5 \(\mathcal {AF}\)
In what follows we exhibit Davies and Humberstone’s axiomatization of S5 \(\mathcal {AF}\). Besides the basis provided by S5 \(\mathcal {A}\), we have:
 (\(\mathcal {F}\)1):

\(\mathcal {F}(\varphi \supset \psi )\supset (\mathcal {F}\varphi \supset \mathcal {F}\psi )\)
 (\(\mathcal {F}\)2):

\(\mathcal {F}\varphi \supset \varphi \)
 (\(\mathcal {F}\)3):

\(\mathcal {F}\varphi \supset \mathcal {FF}\varphi \)
 (\(\mathcal {F}\)4):

\(\varphi \supset \mathcal {F}\lnot \mathcal {F}\lnot \varphi \)
 (\(\mathcal {F}\)5):

\(\varphi \supset \mathcal {F}\varphi \), for any \(\mathcal {A}\)free formula \(\varphi \).
 (\(\mathcal {F}\)6):

\(\mathcal {FA}\varphi \equiv \Box \varphi \), for any \(\mathcal {A}\)free formula \(\varphi \).
Lemma 10.2
\(\mathbf S5 _{2D}\) with general validity extends S5 \(\mathcal {AF}\) with general validity.
Proof
Again, general 2Dtableau proof schemata can be given for axioms (\(\mathcal {F}\)1)(\(\mathcal {F}\)6).
\(\square \)
Corollary 10.2
Soundness and Completeness for S5 \(\mathcal {AF}\) with general validity.
Proof
By Lemma 10.2 and soundness and completeness for \(\mathbf S5 _{2D}\) general 2Dtableaux. \(\square \)
Theorem 10.1
Soundness for \(\mathbf{S5E }_{2D}\).
Proof
The arguments are very similar to the corresponding proofs for \(\mathbf S5 _{2D}\). We only need the following adjustments. The definition of satisfiability needs to be modified by taking into account \({\mathcal {R}}_{\mathcal {D}}\) rather than \({\mathcal {R}}_{\mathcal {F}}\), in which case for a constant domain 2Dcentered model, \({\mathcal {M}}=\langle W ,\langle w*,w*\rangle ,{\mathcal {R}}_{\Box },{\mathcal {R}}_{\mathcal {D}},\mathscr {D},V\rangle \), a set S of doublyindexed formulas, and a function f from each individual index n in S to a possible world in Z, if the pairs of indices \(\langle i,n\rangle \) and \(\langle z,z\rangle \) are in S, then \(\langle f(i),f(n)\rangle {\mathcal {R}}_{\mathcal {D}}\langle f(z),f(z)\rangle \). The rest of the definition remains the same. The soundness lemma corresponding to Lemma 8.2 varies with respect to the latter in minor details: it does not need arguments for \(\otimes \) and \(\odot \), it contains clauses for \(\mathcal {D}\) and \(\mathcal {C}\) rather than \(\mathcal {F}\) and \(\mathcal {S}\), which will be very similar, besides not containing arguments for upper exchange. We leave the details for the reader. \(\square \)
Theorem 10.2
Completeness for \(\mathbf{S5E }_{2D}\).
Proof
The arguments are, again, very similar to the ones for \(\mathbf S5 _{2D}\). However, for the countermodel corresponding to Definition 8.4, since we are not assuming rigidity with respect to the upper index, there needs to be some adjustments in our definitions. First, we need to modify Definition 8.3 of equivalent classes. Let \(\approx _{i,n}\) be an equivalence relation over the set of constants in \(\mathcal {L}_{E2D}\) such that \(c_0\approx _{i,n} c_1\) iff ‘\([c_0= c_1]^i_n\)’ occurs on \(\mathfrak {b}\) for any constants \(c_0\), \(c_1\), and pair of indices \(\langle i,n\rangle \). Moreover, for any c, let [c]\(_{i,n}\) denote the equivalence class of c relative to \(\approx _{i,n}\). Now, if k is any index occurring on \(\mathfrak {b}\), let \(Z=\{k\mid k\in \mathfrak {b}\}\), and \(W=Z\times Z\). Let \(\langle 0,0\rangle \) be a distinguished element of W. For every \(\langle i,n\rangle ,\langle i,m\rangle \in W\), set \(\langle i,n\rangle {\mathcal {R}}_{\Box }\langle i,m\rangle \), and for every \(\langle i,n\rangle ,\langle z,z\rangle \in W\), set \(\langle i,n\rangle {\mathcal {R}}_{\mathcal {D}}\langle z,z\rangle \). Let \(\mathscr {D}=\{[c]_{i,n}\mid c,\langle i, n \rangle \text { on } \mathfrak {b}\}\). Concerning rigidity, this time we only assume that for any constant symbol c and \(\langle i,n\rangle ,\langle i,m\rangle \in W\), \(V(c,\langle i,n\rangle )=V(c,\langle i,m\rangle )\), where \(V(c,\langle i,n\rangle )=[c]_{i,n}\) and to each nplace predicate symbol, R, on \(\mathfrak {b}\), let \(V(R,\langle i,n\rangle )=\{\langle [c_0]_{i,n},\ldots ,[c_n]_{i,n}\rangle :R(c_0,\ldots ,c_n)^i_n\) occurs on \(\mathfrak {b}\)}. We do not assume (Neutrality) anymore. The rest of the definition remains the same as in Definition 8.4. Furthermore, the truth lemma is basically the same except for, again, having clauses for \(\mathcal {D}\) and \(\mathcal {C}\) rather than \(\mathcal {F}\) and \(\mathcal {S}\), which are straightforward. \(\square \)
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Lampert, F. Actuality, Tableaux, and TwoDimensional Modal Logics. Erkenn 83, 403–443 (2018). https://doi.org/10.1007/s1067001798960
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DOI: https://doi.org/10.1007/s1067001798960