Abstract
We recall a largely forgotten intellectual project: that of providing a formal theory of situations that does justice to informal ideas about situations and information flow with the ‘situation theory’ community of the late 1980s and early 1990s. Instead of defending specific desiderata, and in the spirit of Barwise’s ‘Branch Points’, we record some difficulties that defined the project by posing a series of twelve questions. Drawing on the theory of channels and information flow (Barwise and Seligman, late 1990s), with some modifications and extensions, we provide a version of situation theory that answers some of these questions. One of the main extensions is to allow probabilistic constraints. We also consider a more recent proposal by van Benthem to capture many of situation theory’s insights using a modal logic closely related to dependency logic and use this as an alternative but comparable way of answering our questions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The translation from this third century B.C. Chinese text is abridged with modifications from Thesaurus Linguae Sericae http://tls.uni-hd.de/home_en.lasso
- 2.
John Barwise later came to associate this essential aboutness of descriptive language with the views of the English philosopher J. Austin, expressed mostly clearly in [3] and called the resulting notion of proposition ‘austinian’. This was an essential ingredient of Barwise and Etchemendy’s account on the Liar Paradox in [8].
- 3.
Considered by Barwise in [6] as Choice 6.
- 4.
Persistence is studied in classical model theory as the question of which formulas are satisfied in a model whenever they are satisfied in a submodel.
- 5.
Alternatively, we could include a ‘parameter’ \(\mathsf {dom}\) for the domain of quantification to give a parametric infon \(\langle \!\langle \mathrm {everyone_\mathsf {dom}\ is\ dancing}\rangle \!\rangle \), allowing the value of the parameter to be ‘anchored’ in the context of a particular situation. Then \(s\) supports this infon iff it supports that everyone in \(\mathsf {dom}[s]\) is dancing, where \(\mathsf {dom}[s]\) is a set of individuals anchored to \(\mathsf {dom}\) by \(s\). This would allow persistence to hold for non-parametric infons but fail for parametric ones. Of course, one could also make similar a distinction between logically simple infons and complex ones, involving quantification, with varying persistence properties. These and other ideas were explored at the time.
- 6.
The move to non-well-founded set theory is motivated by such considerations. If a situation \(s\) is modelled as the set of infons it supports and if those infons are modelled also as sets in such a way that the things they are about occur as hereditary members, related to the infon by the transitive closure \(\in ^*\) of the membership relation, then we have that \(s\in ^*\langle \!\langle \mathrm {everyone_s\ \text{ is } \text{ dancing }}\rangle \!\rangle \in s\) and so a counterexample to the well-foundedness of \(\in \). This can easily be avoided, as pointed out by Paul King [20] and others, by choosing to model the component structure of infons in a way that does not require them to be hereditary members, but such a sidestep doesn’t really help. The more important issue concerning the identity of situations, mentioned above, is independent of the way in which they are modelled. Nonetheless, the study of non-well-foundedness reveals various solutions in the form of strengthened principles of individuation, such as Peter Aczel’s Anti-foundation Axiom in [1], which was subsequently used to obtain the SOST models of situation theory mentioned in the introduction.
- 7.
Barwise gives the example of the Cantorian conception of set. A more quotidian example is our system of directions: up, down, left, right. When extended around the surface of the earth, incoherence is less than a hemisphere away. See Gupta [18]. Schemes will be discussed more in Sect. 35.1.3, below.
- 8.
A line of reasoning that purports to undermine this attempt at conceptual relativism goes as follows. In making the objection to identifying \(\sqsubseteq \) and \(\unlhd \) explicit we reveal its rather awkward presupposition: that situations are individuated and even ordered in a way that is independent of any scheme of individuation. This smacks of blind faith or incoherence. Yet this attack can be undermined by considering that the way of individuating situations and ordering them into parts need only be different from the way presupposed by \(\models \) and \(\sqsubseteq \). When considering multiple schemes of individuation one would have to distinguish between multiple \(\sqsubseteq \) relations. Then the distinction between \(\unlhd \) and \(\sqsubseteq \) is just that of the ordering given by two schemes: an internal one and an external one.
- 9.
This sort of observation led to some parallels between research on situation theory and relevant logic, summarisd in [22].
- 10.
van Benthem’s account, to be discussed in Sect. 35.3.3 has this flavour.
- 11.
Such talk of information access and processing was commonplace in discussions of situation theory at the time but never made formally precise. A good discussion is by Israel and Perry in [19].
- 12.
By ‘outright’ contradiction, I mean a contradiction in the metalanguage, in which we are explaining what ‘support’ means. However attractive impossible situations might be to someone of paraconsistent leanings, he or she must therefore come up with a different explanation for ‘supports’, at least when talking to a classical logician.
- 13.
Objects of interest to us, such as trees, cites and other people typically have rather unclear boundaries and their properties are vague.
- 14.
The logical and topological interpretation may coincide when, for example, the space is the Stone space of the Boolean algebra of logically equivalent formulas. This was used in [26] to use classifications in the analysis of the logic of diagrammatic reasoning.
- 15.
We are restricted here to a classical notion of state; quantum processes are quite different.
- 16.
In [12] ‘local logic’ refers only to what we are calling sound local logics; there a logic \(L\) is ‘sound’ iff \(N_L=\fancyscript{A}{^{\scriptscriptstyle {\vee }}}\), so that a ‘sound and complete’ logic is just a the natural logic of a classification. Present purposes dictate a slightly more general approach, but many of the results of [12] carry through.
- 17.
Concretely, \(\fancyscript{A}+\fancyscript{B}\) is the classification of pairs of tokens \(\langle a,b \rangle \) with \(a\in \fancyscript{A}{^{\scriptscriptstyle {\vee }}}\) and \(b\in \fancyscript{B}{^{\scriptscriptstyle {\vee }}}\) of types from the disjoint union \(\fancyscript{A}{^{\scriptscriptstyle {\wedge }}}+\fancyscript{B}{^{\scriptscriptstyle {\wedge }}}\) such that \(\langle a,b \rangle \models _{\fancyscript{A}+\fancyscript{B}}\alpha \) iff \(a\models _{\fancyscript{A}}\alpha \), and \(\langle a,b \rangle \models _{\fancyscript{A}+\fancyscript{B}}\beta \) iff \(a\models _{\fancyscript{A}}\beta \). More abstractly, it is the sum in the category of classifications and infomorphisms, to be introduced in Sect. 35.2.2.
- 18.
Not every infomorphism is an interpretation, in the usual sense, since there is no requirement of compositionally.
- 19.
A number of papers by Vaughan Pratt, such as [24], explore the state-event interpretation of Chu spaces.
- 20.
\(\overleftarrow{f}\) and \(\overrightarrow{f}\) may not preserve lattice joins and meets.
- 21.
Specifically, \((cd)_{\fancyscript{A}}=p_{[c]}c_{\fancyscript{A}}\) and \((cd)_{\fancyscript{B}}=p_{[d]}c_{\fancyscript{B}}\).
- 22.
\(I+J\) is the disjoint union of \(I\) and \(J\). The construction of \(cd\) is a generalisation of the binary case. Just take the co-limit of
- 23.
Indeed, any core logic \(L\in \mathsf {Log}([c])\), such as that is given by the construction of a channel from a chain, can be factored into the pushing process in a similar way.
- 24.
\(\sigma _{\fancyscript{A}}{^{\scriptscriptstyle {\wedge }}}\) and \(\sigma _{\fancyscript{B}}{^{\scriptscriptstyle {\wedge }}}\) are the inclusions of \(\fancyscript{A}{^{\scriptscriptstyle {\wedge }}}\) and \(\fancyscript{B}{^{\scriptscriptstyle {\wedge }}}\) in the disjoint union \(\fancyscript{A}{^{\scriptscriptstyle {\wedge }}}+\fancyscript{B}{^{\scriptscriptstyle {\wedge }}}\). And \(\sigma _{\fancyscript{A}}{^{\scriptscriptstyle {\vee }}}\) and \(\sigma _{\fancyscript{B}}{^{\scriptscriptstyle {\vee }}}\) are the projections of \(\fancyscript{A}{^{\scriptscriptstyle {\vee }}}\times \fancyscript{B}{^{\scriptscriptstyle {\vee }}}\) to \(\fancyscript{A}{^{\scriptscriptstyle {\vee }}}\) and \(\fancyscript{B}{^{\scriptscriptstyle {\vee }}}\), respectively.
- 25.
There is a lot to investigate about extension logics, such as the relationship to structural properties. Here I conjecture that a logic satisfies \(\mathsf {WIC}\) iff it is the image of a sound and complete extension logic perhaps with some additional properties.
- 26.
Null in the underlying measure, which we can assume to be, e.g. Borel measure, so that the singletons and countable unions of them are all null. Typical probability measures, such as the uniform distribution, or any normal distribution are all of this kind.
- 27.
A potential for confusion here is to mistake the \(\models \) of situation theory with the relation \(\models _{[s]}\) of the classification \([s]\) but we trust that any difficulties can be resolved in context. Of course, the classification of situations by the infons they support is another classification, of which the situation theoretic \(\models \) is the classification relation.
- 28.
as every reader of Douglas Adam’s Hitchhiker’s Guide knows.
- 29.
The sum \([s]+[t]\) is defined as a limit by infomorphisms
and
and these are used to find the ‘correlates’, e.g. \(\langle \!\langle \alpha ;+\rangle \!\rangle \) corresponds to \(\langle \!\langle \iota _s{^{\scriptscriptstyle {\wedge }}}(\alpha );+\rangle \!\rangle \).
- 30.
It was the realisation that the theory of [12] was unable to account for probabilistic structure that led to the present variant.
- 31.
Notice that in \(s\models (t\models \sigma )\), the infon \(\sigma \) is an infon of the classification \([t]\) not of \([s]\). In fact, the subexpression \((t\models \sigma )\) does not denote an infon at all.
- 32.
Compare Rosenschien’s notion of ‘physical information’ in [25], which was designed for the purpose of comparing local and global description of information content when designing robots.
- 33.
More precisely, \(V(s_1,\ldots ,s_n)= \langle V(s_1),\ldots , V(s_n) \rangle \) and \(g(x_1\ldots x_n)=\langle g(x_1),\ldots ,g(x_n) \rangle \).
- 34.
Here ‘\(\mathbf t\subseteq \mathbf s\)’ is just the syntactic requirement that names in the string \(\mathbf t\) also occur in the string \(\mathbf s\).
- 35.
There are two relevant differences between analytic and general constraints. One is that analytic constraints carry information about the situation itself, as captured by this equivalence. But this other is that there is a different source for their modal force of analytic constraints. A fuller treatment of them should replace \(U\) with another modal operator with a wider range of gobal state functions.
- 36.
van Benthem notes (p. 6) that this is related to Beth’s theorem in first-order logic that any implicit definition, e.g., of the local state of \(s\) in terms of the local state of \(u\), relative to a theory can be given an explicit definition.
- 37.
It is not yet clear to me whether or not the translation into the guarded fragment of first-order predicate logic, given by van Benthem, can be extended to the language with \(\unlhd \).
- 38.
One could add the relation \(\sqsubseteq \) to the language in a fairly straightforward way, but it clearly has a second-order interpretation and I suspect that this would add greatly to the complexity of the logic.
- 39.
In fact there are two kinds of equivalence between distinct situations that may be considered in these models. The first is necessary local-state equivalence, whereby \(g(s)=g(t)\) for all \(g\in G\). Such situations have perfectly synchronised identical local states and so support the same infons. The second concerns the predicates used to classify the local states. Even if \(s\) and \(t\) have quite different local states, they may still satisfy the same predicates and so support the same infons.
- 40.
For example, if for each situation \(s\) there is a function \(P_s\) on the set of predicates and the valuation \(V\) is restricted so that for \(x\subseteq y\), \(y\in V(P_s\alpha )\) iff \(x\in V(\alpha )\) then we recover Persistence in the form: \(s\unlhd t\ \wedge \ s\models \sigma \ \rightarrow \ t\models P_s\sigma \). And then we can represented the part-whole relationship by the informorphism
defined by \(\iota _{s,t}{^{\scriptscriptstyle {\wedge }}}(\alpha )=P_s\alpha \) and \(\iota _{s,t}{^{\scriptscriptstyle {\vee }}}(g)=g(V(s))\).
References
Aczel P (1988) Non-well-founded sets. CSLI Publications, Stanford, CSLI lecture notes
Aczel P (1990) Replacement systems and the aziomatisation of situation theory. In: Cooper R, Mukai K, Perry J (eds) Situation theory and its aplications, vol I. CSLI Publications, Stanford
Austin J (1950) Truth. Proc Aristotelian Soc Suppl 24:111–128
Bar-Hillel Y, Carnap R (1952) An outline of a theory of semantic information. Technical report 247, Research Laboratory for Electronics, MIT.
Barwise J (1981) Scenes and other situations. J Philos 78(7):369–397
Barwise J (1989) Notes on branch points in situation theory. In: The situation in logic, CSLI Lecture Notes, Chap 11, p 255277. CSLI Press, Stanford, CA.
Barwise J (1989) Situations, facts, and true propositions. In: The situation in logic, CSLI Lecture Notes, Chap 10, p 221254. CSLI Press, Stanford, CA.
Barwise J, Etchemendy J (1987) The liar: an essay in truth and circularity. Oxford University Press Inc, New York, NY
Barwise J, Perry J (1983) Situations and attitudes. MIT Press, Cambridge, MA
Barwise J, Seligman J (1993) Imperfect information flow. In: Eighth annual IEEE symposium on logic in computer, Science, pp 252–260.
Barwise J, Seligman J (1994) The rights and wrongs of natural regularity. Philos Perspect 8:331
Barwise J, Seligman J (1997) Information flow: the logic of distributed systems. Cambridge University Press, Cambridge
van Benthem J (2006) Information as correlation versus information as range. Technical report PP-2006-07, ILLC, University of Amsterdam
van Benthem J (2010) Information transfer across Chu spaces. Logic J IGPL 8(6):719–731
van Benthem J, Israel D (1999) Review of Information flow: the logic of distributed systems, Jon Barwise and Jerry Seligman. J Logic Lang Inf 8(3):390–397
Dretske F (1981) Knowledge and the flow of information. Basil Blackwell, Oxford
Dretske F (1981) The pragmatic dimension of knowledge. Philos Stud 40:363–378
Gupta A (1999) Meaning and misconceptions. In: Jackendoff R, Bloom P, Wynn K (eds) Language, logic, and concepts: essays in memory of John Macnamara. The MIT Press, Cambridge, pp 15–41
Israel D, Perry J (1991) Information and architecture. In: Barwise J, Gawron JM, Plotkin G, Tutiya S (eds) Situation theory and its applications, vol 2. CSLI Press, Stanford, CA
King PJ (1994) Reconciling Austinian and Russellian accounts of the Liar paradox. J Philos Logic 23:451–494
Lunnon R (1991) Many sorted universes, SRD’s, and injective sums. In: Barwise J, Gawron P, Plotkin G, Tutiya S (eds) Situation theory and its applications, vol II. CSLI Publications, Stanford, pp 51–79
Mares E, Seligman J, Restall G (2011) Situations, constraints and channels. In: van Benthem J, ter Meulen A (eds) Handbook of logic and language. Eslevier, Burlington, pp 329–344
Plotkin G (1990) An illative theory of relations. In: Cooper R, Mukai K, Perry J (eds) Situation theory and its aplications, vol I. CSLI Publications, Stanford
Pratt V (2002) Event-state duality: the enriched case. In: Proceedings of the CONCUR’02, Brno.
Rosenschein SJ (1989) Synthesizing information-tracking automata from environment descriptions. In: The principles of knowledge representation and reasoning KR’89.
Seligman J (1993) An algebraic appreciation of diagrams. In: The 9th Amsterdam colloquium, pp 607–625.
Seligman J (2009) Channels: from logic to probability. In: Sommaruga G (ed) Formal theories of information: from Shannon to semantic information theory. Springer, Berlin, pp 193–233
Seligman J, Moss LS (2011) Situation theory. In: van Benthem J, ter Meulen A (eds) Handbook of logic and language. Elsevier, North Holland, pp 253–328
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Seligman, J. (2014). Situation Theory Reconsidered. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_35
Download citation
DOI: https://doi.org/10.1007/978-3-319-06025-5_35
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06024-8
Online ISBN: 978-3-319-06025-5
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)