Abstract.  Leibniz's intuition that necessity corresponds to truth in all possible worlds enabled Kripke to define a rigorous model theory for several axiomatizations of modal logic. Unfortunately, Kripke's model structures lead to a combinatorial explosion when they are extended to all the varieties of modality and intentionality that people routinely use in ordinary language. As an alternative, any semantics based on possible worlds can be replaced by a simpler and more easily generalizable approach based on Dunn's semantics of laws and facts and a theory of contexts based on the ideas of Peirce and McCarthy. To demonstrate consistency, this paper defines a family of nested graph models, which can be specialized to a wide variety of model structures, including Kripke's models, situation semantics, temporal models, and many variations of them. An important advantage of nested graph models is the option of partitioning the reasoning tasks into separate metalevel stages, each of which can be axiomatized in classical first-order logic. At each stage, all inferences can be carried out with well-understood theorem provers for FOL or some subset of FOL. To prove that nothing more than FOL is required, Section 6 of this paper shows how any nested graph model with a finite nesting depth can be flattened to a conventional Tarski-style model. For most purposes, however, the nested models are computationally more tractable and intuitively more understandable.

This is a talk that was presented at the Φlog Conference at the University of Roskilde, Denmark, on 4 May 2002.

Semantics based on laws, facts, and contexts...

• Includes Kripke's semantics as a special case.

• Avoids a dubious ontology of possible worlds.

• Derives the accessibility relation from more basic principles instead of assuming it as a primitive.

• Puts the agent at the center of intentional modalities, such as knowing, believing, hoping, fearing, and desiring.

• Makes it possible to reason about multiple modalities and multiple interacting agents within a common framework.

• Dunn's semantics based on laws and facts instead of possible worlds.

• Contexts by Peirce and McCarthy.

• Tarski's hierarchy of metalevels.

• Nested graph models (NGMs).

• Flattening the nest.

• Classification of Modalities.

• Replacement for (or, if you like, a representation of) possible worlds.

• Each world w is replaced (or represented) by a set of facts M and a set of laws L, where L⊂M.

• p is true  iff  p∈M.

• op  iff  L |= p.

• ◊p  iff  not L |= ~p.

Define accessibility R from a world w₁ to a world w₂ to mean that the laws L₁ are a subset of the facts M₂:

R(w₁,w₂) ≡ L₁⊂M₂.

Implications:

• System S4 implies that the laws of the first world must be a subset of the laws of the second world.

• System S5 implies that all worlds accessible from a given world must have the same laws.

Result:  All modal and multimodal reasoning can be treated as metalevel reasoning about the laws and facts.

Next step is to define a theory of contexts that clearly separates the metalevel from the object level.

• Syntax.  The function of grouping, quoting, delimiting, or packaging a section of text.

• Semantics.  Describing or characterizing some real or hypothetical situation.

• Pragmatics.  The reason or purpose for quoting some section of linguistic text or characterizing some nonlinguistic situation.

Peirce used an oval to group or quote a proposition to be discussed:

Since p⊂q is equivalent to ~(p ∧ ~q), Peirce used a nest of two negations to represent implication:

For his discourse representation theory, Hans Kamp independently developed an isomorphic notation for representing context-dependent indexicals:

In 1906, Peirce introduced colors or shading to represent modal contexts:

"You can lead a horse to water, but you can't make him drink."

Corresponding conceptual graph:

"You can lead a horse to water, but you can't make him drink."

English Sentence:  "Tom believes that Mary wants to marry a sailor."

English sentence:  "There is a sailor that Tom believes Mary wants to marry."

John McCarthy introduced the predicate isTrueIn(C,p) to say that p is true in context C:

• isTrueIn(contextOf("Sherlock Holmes stories"), "Holmes is a detective").

• isTrueIn(contextOf("U.S. legal history"), "Holmes is a Supreme Court Justice").

To combine Dunn's semantics with McCarthy's contexts, introduce another predicate isLawOf:

• Facts:  M = {p | isTrueIn(C,p)}.

• Laws:  L = {p | isLawOf(C,p)}.

Truth is now a context-dependent indexical.

• The object language L₀ refers to entities in a universe of discourse D.

• The metalanguage L₁ refers to the symbols of L₀ and their relationships to D.

• The metalanguage L₁ is still first order, but its universe of discourse is enlarged from D to L₀∪D.

• The metametalanguage L₂ is also first order, but its universe of discourse is L₁∪L₀∪D.

• To avoid paradoxes, Tarski insisted that no metalanguage L_n could refer to its own symbols, but it could refer to the symbols or the domain of any language L_i where 0≤i<n.

• At every level of the Tarski hierarchy of metalanguages, the reasoning process is governed by first-order rules.

• But first-order reasoning in language L_n has the effect of higher-order or modal reasoning for every language below n.

Illustrate metalevel reasoning on a sample English sentence:

Joe said "I don't believe in astrology, but they say that it works even if you don't believe in it."

1. Mark indexicals with the # symbol, and mark nested contexts with square brackets:

   Joe said
     [#I don't believe [in astrology]
       but #they say
         [[#it works]
           even if #you don't believe [in #it]]].
   

2. Resolve indexicals:  #I = Joe; "they say" = "every person believes"; #it = astrology; #you = "every person".

   Joe said
     [Joe doesn't believe [astrology works]
       but every person x believes
         [[astrology works]
           even if x doesn't believe
             [astrology works] ]].
   

3. If Joe was sincere, he believes what he said, and a statement of the form "p even if q" implies p.

   Joe believes
     [Joe doesn't believe [astrology works]
       and every person x believes
         [astrology works] ].
   

4. Substitute "Joe" for x in "every person x":

   Joe believes
     [Joe doesn't believe [astrology works]
       and Joe believes [astrology works] ].
   

5. Substitute p for "Joe believes [astrology works]":

   Joe believes [p ∧ ~p].
   

Nested graph models can be specialized for a wide variety of purposes:

• Flat Tarski-style models.

• Kripke-style models.

• Kripke models with counterparts.

• Barcan models.

• Temporal models.

Mapping possible worlds to situations and contexts

To demonstrate that nothing more than FOL is being used, it is possible to flatten any NGM to a conventional Tarski-style model:

1. Observe that contexts are a syntactic device for partitioning the name space for individuals and relations.

2. It is possible to assign a unique name to every context.

3. Then append the name of any context x as an extra argument to every relation that occurs in x.

4. Use restricted quantifiers to limit the range of any quantifier that occurs in context x to the entities that appear in x.

5. Then erase all the context brackets.

The possibility of flattening an NGM shows that the semantics is first-order.

But the act of flattening is usually unwise:

• Readability is reduced by extra arguments on quantifiers and relations.

• Erasing context brackets mixes everything in one large pot.

• Removing separations between metalevels allows paradoxical references.

Peirce's classification of contexts

Peirce considered two ways of talking about what is actual:

• What is asserted (affirmo) is written on the front of a sheet.

• What is denied (nego) is written on the back of the sheet or inside an oval context.

But for modality, Peirce suggested a pad of paper:

• Possibility, Σ_ωp, is "inscribed" on some sheet(s) to say that it is true in some "state of affairs".

• Necessity, Π_ωp, is "inscribed" on every sheet to say that it is true in every "state of affairs".

Peirce's classification of 1906:

• Logical possibility. 
  • Σ_ωp means p is consistent or not provably false.
  • Π_ωp means p is provable.

• Subjective possibility. 
  • Σ_ωp means p is believable or not known to be false.
  • Π_ωp means p is known or not believably false.

• Objective possibility. 
  • Σ_ωp means p is physically possible.
  • Π_ωp means p is necessary according to the laws of physics.

• Interrogative mood. 
  • Σ_ωp means p is questioned.
  • Π_ωp means p is unquestionably true.

• Freedom. 
  • Σ_ωp means p is free or permissible.
  • Π_ωp means p is required or not freely false.

• Depends on a triadic relation that involves some agent.

• For an intentional context C,
        isLawOf(C,p)  ≡  (∃a:Agent)legislate(a,p,C).

• Questions to consider:

  1. How does an agent learn what laws are appropriate for some context?

  2. How does an agent learn what some other agent has legislated for some context?

  3. How do multiple agents debate, question, communicate, or negotiate what become the laws of a context?

Semantics based on laws, facts, and contexts...

• Includes Kripke's semantics as a special case.

• Avoids a dubious ontology of possible worlds.

• Derives the accessibility relation from more basic principles instead of assuming it as a primitive.

• Puts the agent at the center of intentional modalities, such as knowing, believing, hoping, fearing, and desiring.

• Makes it possible to reason about multiple modalities and multiple interacting agents within a common framework.