A Foundation for Metareasoning Part II: The Model Theory
GIOVANNI CRISCUOLO, Dipartimento di Scienze Fisiche, University of Naples, 80125 Napoli, Italy. E-mail: [email protected]
FAUSTO GIUNCHIGLIA, Dipartimento di Informatica e Studi Aziendali, University of Trento, 38100 Trento, Italy, and ITC–IRST, Centro per la Ricerca Scientifica e Tecnologica, 38050 Trento, Italy. E-mail: [email protected]
LUCIANO SERAFINI, ITC–IRST, Centro per la Ricerca Scientifica e Tecnologica, 38050 Trento, Italy. E-mail: serafi[email protected]
Abstract OM pairs are our proposed framework for the formalization of metareasoning. OM pairs allow us to generate deductively the object theory and/or the metatheory. This is done by imposing, via appropriate reflection rules, the relation we want to hold between the object theory and the metatheory. In a previous paper we have studied the proof theoretic properties of OM pairs. In this paper we study their model theoretic properties, in particular we study the relation between the models of the metatheory and the object theory; and how to use these results to refine the previous analysis.
Keywords: Metatheory, reflection, contextual reasoning.
1 Introduction The ‘Meta’ property is not a property which can be ascribed to a single theory; it is rather a relation between two theories. A theory is said to be meta of another, not necessarily distinct, theory if it is about this other theory. The latter theory is often said to be the object of the former theory. The metatheory speaks about the object theory by using a special set of predicates, called metapredicates, each of which is supposed to represent a property about the object theory. By property here we mean a set of formulas of the language of object theory. Examples of properties are: ‘being true in a model of the object theory’, ‘being a theorem in the object theory’, ‘being a theorem in the object theory extended with the set of axioms X’ etc. A reflection principle [3] expresses the correctness of the metatheory when a metapredicate is interpreted in a certain property about the object theory. A reflection principle, therefore, is a statement of the form: ‘if a metaformula, expressing a certain proposition about the object theory, is a theorem of the metatheory, then such a proposition is actually true’. The defini- tion of metatheories that satisfy a reflection principle has been one of the main research issues in the area of Formal Logic, Computer Science, and Artificial Intelligence. Such metathe-
J. Logic Computat., Vol. 12 No. 3, pp. 345Ð370 2002 c Oxford University Press 346 A Foundation for Metareasoning Part II: The Model Theory ories are usually built by adding to a theory special formulas, or special inference rules, or combinations of the two for the metapredicates. In a Part I [2] we introduced a new framework called OM pairs for a uniform specifica- tion of metatheories that satisfy different reflection principles. OM pairs allow us to specify metatheories via a set of inference rules, called reflection rules. Reflection rules are special inference rules between the object theory and the metatheory (which are supposed to be dis- tinct theories). They allow one to infer formulas in the metatheory starting from formulas in the object theory and vice versa. Once a metatheory is specified in the previous terms, it is necessary to characterize it, namely, to verify whether it satisfies a given reflection principle. This amounts to verifying, whether the metapredicate correctly represents the property expressed in the reflection prin- ciple. This characterization concerns two aspects: the set of theorems of the metatheory, and its class of models. The former problem has been studied in Part I [2]. In this paper we concentrate on the latter. We propose a model theoretic characterization of the metatheory in terms of
the class of properties about the object theory (represented as set of for- (1.1) mulas) which are the interpretations of the metapredicate in a model of the metatheory. Notice that we consider a class of properties rather than a single one. This is because, in gen- eral, there is more than one interpretation of the metapredicate that satisfies the metatheory. (1.1) can be rephrased more formally as follows:
the class of sets of formulas È in the language of the object theory, such (1.2) Ñ
that there is a model È of the metatheory that interprets the metapredi-
cate in È . The core of the paper contains a characterization of metatheories containing a single meta-
predicate , and specified via OM pairs. We show how metatheories which respect reflection principles about different properties of the object theory, such as ‘the set of formulas satis- fiable in a specific model of the object theory’, ‘the set of formulas that are provable in an extension of the object theory’ etc. can be defined via different combinations of reflection rules. The model-theoretic characterization allows us to refine the characterization of the different reflection rules given in Part I [2]. We indeed refine the partial order on the strength of reflection rules given in that paper, by proving that it is indeed a strict partial order. The consequence of this is that each combination of reflection rules represents different reflection principles. The paper is structured as follows: in Section 2 we recall the basic definition of OM pair given in Part I [2]. In Section 3 we characterize the models of the metatheory. The main idea is to define them as sets of object level formulas. This allows us to study the relations between the models of the metatheory and the object theory and, therefore, to make precise statements like ‘the metatheory speaks about the object theory’ and ‘the object theory is the model of the metatheory’. Section 4 refines the analysis described in Section 3 of [2] and gives a strict ordering characterization of the reflection principles formalizables by OM pairs, based on the strength of the derivability relations and the sets of theorems generated. In Section 5, we generalize the notion of duality, given in [2], to models, and we show that it preserves satisfiability. A Foundation for Metareasoning Part II: The Model Theory 347 2OMpairs
In this section we briefly recall the main definition of OM pairs given in Part I. Å
We start from two arbitrary theories Ç and , presented as axiomatic formal systems, i.e.
Ç Ä ª ¡ Å Ä ª ¡ Ç Å
Ç Ç Å Å Å
Ç and . is called the object theory, is called
Ä Ä Ç Å ª ª Ç Å
Å Ç Å
the metatheory. Ç ( )isthelanguage of ( ), ( )istheset of axioms of ( ),
¡ ¡ Ç Å Å
and Ç ( )isthedeductive machinery (set of inference rules) of ( ). We call Meta-
Ä ¡ ¡
Ç Å formulas, the formulas of Å . We suppose that and are set of natural deduction
style inference rules, as defined in [6].
Ç Å ÌÀ ´Ç µ
O and M, are the derivability relations defined by and , respectively; and
Å µ Ç Å ÌÀ´ are the set of theorems of and respectively. We then consider new kinds of
inter-theory inference rules, called bridge rules [4], whose premisses and conclusions belong Ä
to different languages. Thus we may have a bridge rule with premisses in Ç and conclusion
Ä Ä Ä
Å Ç
in Å ; and, vice versa, a bridge rule with premisses in and conclusion in . Bridge
Ç µ ÌÀ ´Å µ rules extend deductively ÌÀ´ and , that is, new object and metatheorems may be proved by applying bridge rules. The set of theorems of the metatheory may be extended whenever the set of theorems of the object theory is extended, and vice versa. We focus on
the following bridge rules, and we call them reflection rules.
´ µ ´ µ
Ò
Ò
ÊÙÔ ÊÙÔ
Ê Û Ê Û
´ µ ´ µ
For each of the reflection rule above, we consider two versions, the unrestricted version, which can be applied with no restriction and the restricted version (denoted by adding the index), which can be applied under the following condition:
RESTRICTIONS: Rules labelled with index Ö are applicable if the premiss does not depend
on any assumptions in the same theory.
Ä Ä Å For the rest of the paper we make the following simplifying hypotheses: Ç and are
1
Ä Ä Ç propositional and Å is the propositional metalanguage of . DEFINITION 2.1 (Propositional metalanguage)
Given a logical language Ä, its propositional metalanguage is the propositional language
´Ä µ ´ µ Ä , whose set of atomic wffs is the set .
DEFINITION 2.2 (OM pair)
Ç Å ´Êʵ Ç Ä ª ¡ Å
Ç Ç
An Object-Meta Pair (OM pair) is a triple where Ç ,
Ä ª ¡ Ä Ä ´Êʵ
Å Å Å Ç Å , contains the propositional metalanguage of ,and is a set of
reflection rules.
Å ´Êʵ
Given an object theory Ç , a metatheory , and a set of reflection rules , we say that
Ç Å ´Êʵ Ç Å ´Êʵ OM is the OM pair composed of and connected by . Notation-
ally, when it contains more than one bridge rule we represent ´Êʵ by listing its elements
Ò
´Êʵ ÊÙÔ · Ê Û
separated by a ·. One example of is .
Ä Ä Ä Ä
Å Ç Å
Awff Ç is derivable from a set of assumptions ,inOM,in
¡ ¡ Å
symbols OM , if there is a deduction in OM of from that applies Ç , ,theaxioms
Å
of Ç and , and the reflection rules of OM. is provable in (a theorem of ) OM, abbreviated
as OM , if it is derivable from the empty set.
ÌÀ ´Ç µ Ä Ä ÌÀ ´Å µ Ä
Ç Ç ÇÅ Å
ÇÅ is the set of formulas of provable in OM. is
Ä ÌÀ´Ç µ ÌÀ ´Ç µ ÌÀ´Å µ ÇÅ the set of formulas of Å provable in OM. Trivially and 1In Part I [2] we have shown how the results in the propositional case can be generalized to the first-order case.
348 A Foundation for Metareasoning Part II: The Model Theory
ÌÀ ´Å µ Ç Å ´Êʵ
ÇÅ . Given an OM pair OM we use the term object theory of
¼ ¼
´Êʵ Ç Ä ª ¡ Ç
OM (generated by ) to denote any theory Ç whose set of theorems
Ç
ÌÀ ´Ç µ ´Êʵ
is ÇÅ and the term metatheory of OM (generated by ) to denote any theory
¼ ¼
Å Ä ª ¡ ÌÀ ´Å µ Ç ÌÀ ´Ç µ
Å ÇÅ ÇÅ
Å whose set of theorems is . Trivially TH( ’)
Å
Å ÌÀ ´Å µ and TH( ’)= ÇÅ .
We consider the OM pairs with the following combinations of reflection rules:
Ò Ò
½¾ Ê Û · ÊÙÔ ·Ê Û ½
Ö Ö
½¿ ÊÙÔ · Ê Û ¾ Ê Û
Ö Ö
Ò
½ ÊÙÔ · Ê Û · ÊÙÔ ¿ ÊÙÔ
Ö Ö Ö
Ò
½ ÊÙÔ · Ê Û · Ê Û Ê Û
Ö Ö
Ò
½ ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ
Ö
Ò
½ ÊÙÔ · Ê Û · Ê Û ÊÙÔ · Ê Û
Ö Ö Ö
Ò Ò Ò
½ ÊÙÔ · Ê Û · ÊÙÔ ·Ê Û ÊÙÔ · Ê Û
Ö Ö Ö Ö Ö
Ò Ò Ò
½ ÊÙÔ · Ê Û · ÊÙÔ ·Ê Û ÊÙÔ · Ê Û ·Ê Û
Ö Ö Ö Ö Ö
Ò
¾¼ ÊÙÔ · Ê Û ÊÙÔ · Ê Û ·ÊÙÔ
Ö Ö Ö
Ò Ò Ò
¾½ ÊÙÔ · Ê Û · ÊÙÔ ½¼ ÊÙÔ · Ê Û ·ÊÙÔ ·Ê Û
Ö Ö Ö Ö Ö
Ò Ò
¾¾ ÊÙÔ · Ê Û ½½ Ê Û · ÊÙÔ Ö
3 A model-theoretic characterization
The goal of this section is to understand the relation between the models of the metatheory Ä
and the object theory. For our purpose it is enough to consider the case in which Å is the
Ä ´ µ
propositional metalanguage of Ç , i.e. all its atomic wffs are of the form for some
Ä Å
wff of Ç . Being a propositional theory, the models of are truth assignments for the
Ä Ñ Ä Ë Ä
Å Ñ Ç atomic wffs of Å . Any interpretation of identifies a subset according to
the following definition:
Ë Ä Ñ´ ´ µµ Ç
Ñ True Ä
We can therefore consider the set of interpretations of Å to be the powerset of the wffs in Ä
Ç .
For any set ´Êʵ of reflection rules, we characterize:
´Êʵ Ç
the models of the metatheory generated by in terms of ;
´Êʵ Å the set of theorems of the object theory generated by in terms of the models of . The first result allows us to make precise in which sense the metatheory speaks about the object theory. The second result allows us to make precise the sense in which the object
theory is a model of the metatheory.
Ñ Ä Ñ
We use the following notation. Let be an interpretation of Å , we say that is a model
Ñ Ñ of , in symbols , to mean that satisfies all wffs in (according to the classical
definition of satisfiability).
Ä
Furthermore we say that a set of wffs is a classical propositional theory if it contains
all the classical tautologies and it is closed under modus ponens. is consistent if .
Ä
is maximal if for any wff , or . We also use the following notation. For
Ä ª ¡ Ä Ì · Ä ª ¡
any theory Ì and any , denotes the extended theory .
Ä ÊÙÔ ´ µ ´ µ Ä Å
Foranysetofwffs Ç , denotes the set of wffs .
´ µ ÊÙÔ ´ µ Ä
We sometimes use instead of . Analogously for any set of wffs Å ,
Ê Û ´ µ ´ µ Ä
denotes the set of wffs Ç . A Foundation for Metareasoning Part II: The Model Theory 349 3.1 The models of the metatheory
Let us start by considering the metatheories generated by combinations of reflection rules
ÌÀ ´Å µ
which are characterizable axiomatically, i.e. the metatheories for which ÇÅ
Å ·¨µ ¨ ÌÀ´ , with being some schematic metaformula. ([2, Section 4] gives an in depth
analysis of the axiomatic characterizability of metatheories.) In this case the (obvious) idea
Ä ª ¨ Å is to look for the subclass of interpretations of Å which satisfies and .
THEOREM 3.1
Å ´Êʵ
Let OM be an OM pair composed of Ç and connected by the set of reflection rules ;
´Êʵ Ê Û Ñ ÌÀ ´Å µ ª
ÇÅ Å
1. if Å ,then is a model of , if and only if it is a model of ;
Ö
´Êʵ ÊÙÔ Ñ ÌÀ ´Å µ ª
ÇÅ Å
2. if Å ,then is a model of , if and only if it is a model of ,
Ö
Ç µ Ñ and ÌÀ´ .
Item (i) of Theorem 3.1 states that any model of Å is a model of the metatheory of OM
and vice versa. Ñ is a model of the metatheory of OM independently of the object theory. ª
Item (ii) states that the models of the metatheory of OM are all the models of Å which contain the theorems of the object theory.
PROOF. [Theorem 3.1] These results are a direct consequence of [2, Theorem 4.1]. This
theorem states the (obvious) result that reflection down leaves the metatheory unchanged, and
ÌÀ ´Å µ ÌÀ ´Å µ
we have ÇÅ , while reflection up adds to the metatheory all the theorems
´ µ ÌÀ´Ç µÌÀ ´Ç µ of the form , with ÇÅ .
THEOREM 3.2
Å ´Êʵ
Let OM be an OM pair composed of Ç and connected by the set of reflection rules .
Ñ Ä ÌÀ ´Å µ Ñ ª
ÇÅ Å
An interpretation of Å is a model of if and only if, and:
ÊÙÔ · Ê Û ÌÀ ´Ç µ Ñ Ñ
1. If ´Êʵ is then, and is a classical propositional theory.
Ö
Ò
ÊÙÔ ·Ê Û·ÊÙÔ ÌÀ´Ç µ Ñ Ñ
2. If ´Êʵ is ,then and is a consistent classical
Ö Ö
propositional theory.
ÊÙÔ · Ê Û ÌÀ´Ç µ Ñ Ñ 3. If ´Êʵ is ,then and is a maximal classical propositional
theory.
Ò
ÊÙÔ · Ê Û · ÊÙÔ ÌÀ´Ç µ Ñ Ñ
4. If ´Êʵ is ,then and is a consistent and maximal Ö
classical propositional theory.
Ò Ò
·Ê Û ÊÙÔ ÊÙÔ · Ê Û Ç · Ñ
5. If ´Êʵ is either or , then either is consistent,or
Ö
Ñ .
ª
To understand Theorem 3.2, let us consider the case with Å . In this situation the
ÌÀ ´Å µ Ç ´Êʵ
models of ÇÅ are completely determined by and . (These observations can
ª ª Å
be easily generalized. In fact, from Theorem 3.2, Å restricts us to the models of .)
ÌÀ ´Å µ
Consider item (1). The models of ÇÅ are the sets of theorems of the extensions of
the object theory, i.e. all the theories which can be obtained by adding a set of axioms to
´ µ
Ç . Therefore, if is a logical consequence of a set of formulas ,then is also a
´ µ ÌÀ ´Å µ logical consequence of . Consider item (2). The models of ÇÅ are the sets of
theorems of the consistent extensions of the object theory. Notice that, as there are no consis-
Ç ÌÀ ´Å µ
tent extensions of an inconsistent theory, if is inconsistent then ÇÅ is inconsistent
ÌÀ ´Å µ
as well. Consider item (3). The models of ÇÅ are the maximal (possibly inconsis-
ÌÀ ´Å µ tent) extensions of the object theory. Consider item (4). The models of ÇÅ are the 350 A Foundation for Metareasoning Part II: The Model Theory
maximal consistent extensions of the object theory. Since the set of maximal consistent sets
´ µ
containing the object theory is isomorphic to the set of models of the object theory,
Ñ ÌÀ ´Å µ can be interpreted as ‘ is true in ’. Finally, consider item (5). The models of ÇÅ
are the sets of formulas which can be added to the object theory without making it inconsis-
Ç
tent. If, for instance, is derivable from in , the object theory extended by and is
Ñ ÌÀ ´Å µ
inconsistent. This corresponds to the fact that in any model of ÇÅ it is never the
Ñ case that and both belong to . A further example: if the object theory is inconsistent (i.e. all formulas are provable) then it cannot be consistently extended. In this case the only model of the metatheory of OM is the empty set.
PROOF. [Theorem 3.2] This proof is based on the result of Theorems 4.2 and 4.3 in [2]. In
these theorems the metatheory of OM is characterized in terms of the following formulas:
´ µ ´ ´ µ ´ µµ
(K)
´ µ ´ µ
(Comp)
´ µ
(nTbot)
´ ÓÒ×µ ´ µ ´ µ ´ µ
½ Ç ·
½ Ò
Ä
where are understood as schematic variables (parameters) ranging over Ç .
ÌÀ ´Å µ ÌÀ´Å · ÊÙÔ ´ÌÀ´Ç µµ · ´Ã µµ
Item (1). In this case ÇÅ , as proved by [2,
Ñ ÌÀ ´Å µ Ñ ª Ñ ÊÙÔ ´ÌÀ ´Ç µµ Å
Theorem 4.2]. Then, ÇÅ if and only if , and
Ñ ´Ãµ Ñ ÌÀ ´Å µ Ñ ª ÌÀ´Ç µ Ñ Ñ Å . Suppose that is a model of ÇÅ ,then and .
is a classical propositional theory as it contains all the propositional tautologies (contained
Ç µ Ñ ´Ãµ
in ÌÀ´ ), and it is closed under modus ponens since . Vice versa, suppose that
Ñ ª Ñ ÌÀ´Ç µ Ñ ª Å
Å and is a classical propositional theory which contains ,then ,
ÊÙÔ ´ÌÀ´Ç µµ ÌÀ ´Ç µ Ñ Ñ ´Ãµ Ñ by hypothesis, Ñ as ,and as is closed under
logical consequence.
ÌÀ ´Å µÌÀ´Å · ÊÙÔ ´ÌÀ ´Ç µµ · ´Ã µ·´ µµ
Item (2). In this case ÇÅ nTbot , as proved
Ñ ÌÀ ´Å µ Ñ ª Ñ ÊÙÔ ´ÌÀ´Ç µµ Å
by [2, Theorem 4.2]. Then, ÇÅ if and only if, , ,
Ñ ´Ãµ Ñ ´ µ Ñ ÌÀ ´Å µ Ñ ª Ñ Å
and nTbot . From item (1),if ÇÅ ,then ,and
Ç µ Ñ ´ µ
is a classical propositional theory that contains ÌÀ´ .Furthermore nTbot implies
Ñ Ñ Ñ ª Ñ
that and therefore that is consistent. Vice versa, if Å and is a consistent
Ç µ Ñ ÊÙÔ ´ÌÀ ´Ç µµ
classical propositional theory which contains ÌÀ´ then, by item (1),
´Ãµ Ñ Ñ Ñ ´ µ
and Ñ . The consistency of implies that and therefore that nTbot .
ÌÀ ´Å µÌÀ´Å · ÊÙÔ ´ÌÀ ´Ç µµ · ´Ã µ · ´ ÓÑÔµµ
Item (3). In this case ÇÅ , as proved
Ñ ÌÀ ´Å µ Ñ ª Ñ ÊÙÔ ´ÌÀ´Ç µµ Å
by [2, Theorem 4.2]. Then, ÇÅ if and only if , ,
Ñ ´Ãµ Ñ ´ ÓÑÔµ Ñ ÌÀ ´Å µ
and . From item (1) of this theorem, if ÇÅ ,then
Ñ ª Ñ ÌÀ´Ç µ
Å ,and is a classical propositional theory that contains .Furthermore
´ ÓÑÔµ Ñ Ñ Ñ
Ñ implies that, for any object wff , or ,i.e. is maximal.
Ñ ª Ñ
Vice versa, if Å and is a maximal classical propositional theory which contains
Ç µ Ñ ÊÙÔ ´ÌÀ´Ç µµ Ñ ´Ãµ
ÌÀ´ , then, by item (1) of this theorem, and .Furthermore
Ñ Ñ Ñ ´ ÓÑÔµ
Ñ being maximal implies that or and therefore that .
ÌÀ ´Å µ ÌÀ´Å ·ÊÙÔ´ÌÀ´Ç µµ · ´Ã µ · ´ ÓÑÔµ · ´ µµ
Item (4). In this case ÇÅ nTbot , as proved by [2, Theorem 4.2]. Item (4) is therefore a consequence of items (2) and (3) of
this theorem.
ÌÀ ´Å µÌÀ´Å · ´ ÓÒ×µµ
Item (5). In this case ÇÅ , as proved by [2, Theorem 4.3].
Ñ ÌÀ ´Å µ Ñ
Let us start with the only if direction. Let be a model of ÇÅ . Then, is a model
A Foundation for Metareasoning Part II: The Model Theory 351
ª
½ Ò · of Å , and by [2, Theorem 4.3], for any such that
O Ò
½
Ñ Ñ Ò
if ½ ,then
Ç · Ñ
· Ñ
Suppose that is inconsistent. Then, for any formula , Ç , this implies that
Ñ
· Ò there is a finite number of wffs ½ such that . This implies
O Ò
½
Ñ Ñ
that .Asthisistrueforanyformula , must be the empty set.
Ñ ÌÀ ´Å µ
As far as the only if direction, to prove that ÇÅ we exploit again [2, Theo-
rem 4.3] and prove that
Ñ Å · ´ µ ´ µ ´ µ
Ò Ç ·
½ (3.1)
Ò
½
Ñ Ñ ´ µ Ñ ª
If then for any formula , and since satisfies Å , then (3.1) is true.
Ñ Ñ ª Ñ ´ µ ´ µ
½ Ò
Suppose instead and Å . Then, suppose that .
Ç · Ñ Ç · Ñ Ò
If is derivable in ½ ,then cannot be in ,otherwise would be
´ µ Ñ ´ µ inconsistent. This implies Ñ ,thatis , which implies (3.1).
Let us now consider the combinations of reflection rules which are not characterizable
ÊÙÔ ·Ê Û Ü Ä
axiomatically. We start with . For any set of interpretations of Å we define:
Ö Ö
Ì
Ñ Ü Ñ Ü Ä
Ç for all
Ë
Ü Ä Ñ Ü Ñ
Ç there exists an
THEOREM 3.3
Å ÊÙÔ ·Ê Û
Let OM be an OM pair composed of Ç and connected by . The set of models
Ö Ö
ÌÀ ´Å µ Ä Ü Å
of ÇÅ is the largest subset of interpretations of that satisfies (3.2) in :
Ì
Ü Ñ ª ÌÀ´Ç · Ü µ Ñ
Å (3.2)
Ñ ÌÀ ´Å µ ª Å Theorem 3.3 states that is a model of ÇÅ iff it satisfies and contains
the set of formulas which are logical consequences of the intersection of all the models of
ÌÀ ´Å µ
ÇÅ . This theorem does not provide an explicit definition of the set of models of
2
ÌÀ ´Å µ Ñ ÌÀ ´Å µ ÇÅ ÇÅ . Indeed, to prove that a certain interpretation is a model of by
exploiting Theorem 3.3, one needs, first to prove that a certain set Å of interpretations is
ÌÀ ´Å µ Å
contained in the set of models of ÇÅ (by showing that is a solution of (3.2)), and
Å then to prove that Ñ . Let us see some examples.
EXAMPLE 3.4
ª Å Ñ Ä Ç
If Å is the empty set, then a set which satisfies (3.2), is the set of which contains
Ì
ÌÀ´Ç µ Ñ Å ª ÌÀ´Ç · ŵ
. Indeed any satisfies an empty set of axioms Å ,and
Ì
Ç µ Ñ Å ÌÀ´Ç µ ÌÀ´ , because .
EXAMPLE 3.5
ª ´Ô µ Å Ñ Ä Ç
If Å contains the single axiom ,thenaset that satisfies (3.2) is the set of
Ç · Ôµ which contains ÌÀ´ .
EXAMPLE 3.6
ª ´Ô µ ´Õ µ Å
If Å contains the single axiom ,thenaset that satisfies (3.2) is the set
Ñ Ä ÌÀ´Ç µ Ô Ñ ÌÀ´Ç µ Õ Ñ of Ç such that or .
2 ·
This is the model theoretic counterpart of the fixpoint characterization of the metatheory generated by ÊÙÔ Ö
ÊÛ . Ö 352 A Foundation for Metareasoning Part II: The Model Theory
EXAMPLE 3.7
ª ´Ô µ Å Ä Ç
If Å is the axiom ,thenaset that satisfies (3.2) is the set of subsets of
Ç µ Ô
which contain ÌÀ ´ and does not contain .
ÌÀ ´Å µ
In the previous examples, notice that the models of ÇÅ might not be closed under logical consequence. To force this closure we have to relax the restriction on the application of reflection up.
PROOF. [Theorem 3.3] The proof is in two steps. In the first step we prove that the set Å of · Ê Û
models of the metatheory generated by ÊÙÔ satisfies (3.2). In the second step we
Ö Ö
prove that any other solution of (3.2) is contained in Å.
First step
ÌÀ ´Ç µ ÌÀ ´Å µ ÇÅ From [2, Theorem 4.4] we have that ÇÅ and are the smallest sets of wffs
which satisfy the following equations:
Ü ÌÀ ´Ç ·Ê Û´Ü µµ Å
Ç (3.3)
Ü ÌÀ ´Å ·ÊÙÔ´Ü µµ Ç Å (3.4)
We have therefore that
ÌÀ ´Å µÌÀ´Å ·ÊÙÔ´ÌÀ´Ç · Ê Û ´ÌÀ ´Å µµµµµ
ÇÅ ÇÅ
Å ÌÀ ´Å µ Ñ Å ª Å
Let be the set of models of ÇÅ . if and only if it satisfies and satisfies
ÊÙÔ ´ÌÀ´Ç · Ê Û ´ÌÀ ´Å µµµµ ÌÀ´Ç ·
all the formulas in ÇÅ . We have therefore that
Ê Û ´ÌÀ ´Å µµ Ñ
ÇÅ ,andthat
Å Ñ ª ÌÀ´Ç · Ê Û ´ÌÀ ´Å µµµ Ñ
Å ÇÅ
´ µ ÌÀ ´Å µ Ñ Å Ñ
Since belongs to ÇÅ if and only if for all , , we can conclude
Ì
Ê Û ´ÌÀ ´Å µµ Å
that ÇÅ . We have therefore that:
Å Ñ ª ÌÀ´Ç · ŵ Ñ Å
Second step
Å Ä
Let us prove that the is the largest set of interpretations of Å which satisfies (3.2),
Æ Å
namely, that if Æ is a solution of (3.2), then . To do this let us consider the following
Å Å ½
sequence of theories ¼ (originally defined in the proof of [2, Theorem 4.4]):
Å Å
¼ (3.5)
Ç Ç
¼ (3.6)
Å Å · ÊÙÔ ´ÌÀ´Ç µµ
·½
(3.7)
Ç Ç · Ê Û ´ÌÀ´Å µµ
·½
(3.8)
¼ Å Å
For any ,let be the set of models of . We prove by induction on that for all
¼ Æ Å
, .
Base case
Æ Ñ Ñ ª Å ¼ Å . A Foundation for Metareasoning Part II: The Model Theory 353
Step case
Since Æ satisfies (3.2),
Æ Ñ ª ÌÀ´Ç · Ƶ Ñ
Å
Ì Ì
Æ Å Å Æ
The induction hypothesis implies that , and therefore that
Æ Ñ ª ÌÀ´Ç · Å µ Ñ
Å
Ì
Å Å Å Ê Û ´ÌÀ´Å µµ
Since is the set of models of ,wehavethat . If we substitute
Ì
Ê Û ´ÌÀ ´Å µµ Å
with in the previous equation we obtain:
Æ Ñ ª ÌÀ´Ç · Ê Û ´ÌÀ´Å µµµ Ñ
Å
Ñ ª ÊÙÔ ´Ç · Ê Û ´ÌÀ ´Å µµµ
Notice that any interpretation that satisfies Å and is a model
Å ·½
of , and therefore, we have that
Æ Å
·½
¼ Æ Å
Thus we have shown that for any , . This implies that:
Æ Å
¼
Ë
ÌÀ ´Å µ Å Å
Since ÇÅ , the set of models of is the intersection of the models of
¼
Ì
Å Å Å Æ Å
each ,i.e. . Thus we conclude that .
¼
Ò Ò
·Ê Û ÊÙÔ ·Ê Û ·Ê Û
The set of models of the metatheories generated by ÊÙÔ and
Ö Ö Ö Ö
·Ê Û
can be characterized similarly to ÊÙÔ . For a set of wff ,let be defined as
Ö Ö
.
THEOREM 3.8
Ò
Å ÊÙÔ ·Ê Û
Let OM be an OM pair composed of Ç , connected by . The set of models
Ö Ö
ÌÀ ´Å µ Ä Ü Å
of ÇÅ is the largest subset of interpretations of which satisfies (3.9) in :
Ì
Ü Ñ ª ÌÀ´Ç · ܵ Ñ
Å (3.9)
THEOREM 3.9
Ò
Å ÊÙÔ · Ê Û · Ê Û
Let OM be an OM pair composed of Ç , connected by .Thesetof
Ö Ö
ÌÀ ´Å µ Ä Å models of ÇÅ is the largest subset of interpretations of which satisfies (3.10) in
Ü:
Ñ ª Ñ Ç
Å is the set of theorems of an extension of
Ë
Ü (3.10)
Ü Ñ If ,then
The proofs of these two theorems are analogous to that of Theorem 3.3. Ò
Ê Û implements a weak form of negation as failure. The models of the metatheory
Ö
Ò
ÊÙÔ · Ê Û · Ê Û Ç ª
generated by indeed are the extensions of , which satisfy Å and
Ö Ö
Ç contain whenever is not provable in any extension of . 354 A Foundation for Metareasoning Part II: The Model Theory
3.2 The object theory in terms of the models of Å
Our goal here is to understand the meaning of in terms of properties of the object theory, i.e.
which is the property of the object theory represented by . This allows us to make precise the sense in which the object theory can be considered a model of the metatheory. We start from the weakest combinations of reflection rules and then move to those of increasing strength.
THEOREM 3.10
Å ´Êʵ
Let OM be an OM pair composed of Ç and connected by the set of reflection rules .
Å ÌÀ ´Å µ
Let be the set of models of ÇÅ . Then:
Ì
´Êʵ Ê Û ÌÀ ´Ç µÌÀ´Ç · ŵ ÇÅ
1. If Ç ,then .
Ö
´Êʵ ÊÙÔ ÌÀ ´Ç µ ÌÀ´Ç µ ÇÅ
2. If Ç ,then . Ö The underlying intuitions are the same as those described above for Theorem 3.1.
THEOREM 3.11
Å ´Êʵ
Let OM be an OM pair composed of Ç and connected by a set of reflection rules .
Å ÌÀ ´Å µ
Let be the set of models of ÇÅ . Then:
Ì
Å ÌÀ ´Ç µ Ê Û ´Êʵ ÇÅ
1. If Ç ,then .
Ö
Ì
ÊÙÔ ´Êʵ ÌÀ ´Ç µ Å ÇÅ
2. If Å ,then .
Ö
Ì
Ê Û ´Êʵ ÊÙÔ ´Êʵ ÌÀ ´Ç µ Å
Å ÇÅ
3. if Ç and ,then .
Ö Ö
PROOF. [Theorem 3.11] Item (1) is a straightforward consequence of item (1) of Theorem
ÌÀ ´Ç µ ÊÙÔ ´ µ
3.10. Item (2) is provable as follows. Suppose that ÇÅ , then, by ,
Ö
Ì
ÌÀ ´Å µ Å
ÇÅ and therefore, . Item (3) is the conjunction of (1) and (2). Ê Û
If the object theory is generated by a set of reflection rules weaker than ÊÙÔ or
Ö Ö (in the sense of Theorem 3.11), then the set of object theorems is not completely determined by the metatheory. Vice versa, from Theorem 3.11 we can conclude that the object theory
(more precisely, its set of theorems) generated by a set of reflection rules which are at least as Ê Û
strong as ÊÙÔ and , is completely determined by the set of models (theorems) of the
Ö Ö
metatheory. Finally notice that in general, according to Theorem 3.11, an interpretation Ñ of
Ä Ñ ÌÀ ´Ç µ ÌÀ ´Å µ
ÇÅ ÇÅ Å with is not necessarily a model of . Consider the following example.
EXAMPLE 3.12
Let OM be composed of the empty object theory and the metatheory with a set of axioms
ª ´Ô µ ´Õ µ ÊÙÔ ·Ê Û
Å , connected by . A model of the metatheory of OM
Ö
Ô Õ Ñ ÌÀ´Ç · Ôµ Ñ ÌÀ ´Ç · Õ µ ¾ contains either or . In particular ½ and are both
models of the metatheory of OM. On the other hand the intersection of these models (which
ÌÀ ´Ç µ Ô Õ ÌÀ ´Ç µ ÌÀ ´Å µ
ÇÅ ÇÅ is ÇÅ ) does not contain , nor . Therefore, is not a model of .
Theorem 3.11 provides the characterization of the object theory for all the combinations of
Ò Ò
· Ê Û ÊÙÔ · Ê Û
reflection rules with the exception of ÊÙÔ and . For these two special Ö cases we have the following result.
THEOREM 3.13
Å ´Êʵ
Let OM be an OM pair composed of Ç and connected by the set of reflection rules .
Å ÌÀ ´Å µ
If is the set of models of ÇÅ , then:
A Foundation for Metareasoning Part II: The Model Theory 355
Ì
Ò
ÊÙÔ ·Ê Û ÌÀ ´Ç µÌÀ´Ç · ŵ
1. If (RR) is ,then ÇÅ ;
Ö
Ì
Ò
ÌÀ´Ç · ѵ ÊÙÔ ·Ê Û ÌÀ ´Ç µ
2. If (RR) is ,then ÇÅ .
ѾÅ
PROOF. [Theorem 3.13] Item (1). From [2, Theorem 4.3] we have
ÌÀ ´Ç µ ÌÀ ´Ç · Ê Û ´ÌÀ ´Å µµµ ÇÅ
ÇÅ (3.11)
Ì Ì
Ê Û ´ÌÀ ´Å µµ Å ÌÀ ´Ç µ ÌÀ ´Ç · ŵ ÇÅ Since ÇÅ , (3.11) implies that .To
prove item (2), by [2, Theorem 4.3], we have
Ï Ï
Ò Ò
ÌÀ ´Ç µÌÀ´Ç · ´ µ ÌÀ ´Å µµ
ÇÅ
ÇÅ (3.12)
½ ½
Ì
ÌÀ ´Ç µ ÌÀ´Ç · ѵ ½
To prove that ÇÅ , we show that for any extra axiom
ѾÅ
Ì
Ç ÌÀ ´Ç µ ÌÀ´Ç · ѵ
ÇÅ ½ Ò
Ò added to to obtain ,wehavethat .For
ѾÅ
Ñ Å ´ µ ´ µ ÌÀ ´Å µ Ñ ´ µ ´ µ
Ò ÇÅ ½ Ò
each ,if ½ then ,
½ Ò Ñ ÌÀ´Ç · ѵ
½ Ò
which implies that there is an such that ,and .
Ñ Å ÌÀ ´Ç µ ÌÀ´Ç · ѵ ÌÀ ´Ç µ ÇÅ
Therefore, for each , ÇÅ ; and therefore,
Ì
Ç · ѵ
ÌÀ´ .
ѾÅ
Ì Ì
ÌÀ´Ç · ѵ ÌÀ ´Ç µ ÌÀ´Ç · ѵ
Vice versa, let us prove ÇÅ .Let ,
ѾŠѾÅ
¼
Ñ Å ¥ Ç · Ñ Å
and for each let Ñ be a proof of in .Let be the following theory:
´ µ
Ñ Ñ Å
,and Ñ is the finite (possibly
´ ´ µµ
Ñ
Å ·ÌÀ ´Å µ·
ÇÅ empty) set of formulas occurring as axioms in
¥
Ñ
¼
´ µ ´ µ Å Ñ
where Ñ is the conjunction of the formulas in . Let us prove that is
¼
Ñ Å Ñ ÌÀ ´Å µ
unsatisfiable. By contradiction let be a model of . is a model of ÇÅ and
¼
Ñ Ñ ´ ´ µµ Ñ Å Ñ
therefore, Ñ . This implies that and therefore that .
¼ ¼ Å
Å unsatisfiable implies that is finitely unsatisfiable, i.e. there is a finite sequence of sets
ÌÀ ´Å µ· ´ ´ µµ ´ ´ µµ
Ñ ÇÅ Ñ Ñ
Ñ such is unsatisfiable.
½ Ò ½ Ò
This implies that
´ ´ µµ ´ ´ µµ ÌÀ ´Å µ
Ñ Ñ ÇÅ
½ Ò
Ò Ñ
and therefore that for each ½ , with ,
´ µ ´ µ ÌÀ ´Å µ
½ Ò ÇÅ
We have therefore that
ÌÀ ´Ç µ
½ Ò ÇÅ
ÌÀ ´Ç µ
A possible proof of in ÇÅ is therefore the following: .
.
Î .
$ % $ %
Ò ½
´ µ
½ Ò
¾
¥ ¥
Ò ½
´ µ ´ µ
½ Ò
E
ÌÀ ´Ç µ
We can therefore conclude that ÇÅ . 356 A Foundation for Metareasoning Part II: The Model Theory
TABLE 1. Metamodels and object theorems generated by a set of reflection rules
´Êʵ ÌÀ ´Å µ ÌÀ ´Ç µ ÇÅ
Models of ÇÅ
Ä ÌÀ ´Ç µ
Any set of Ç -wffs
Ì
Ê Û Ä ÌÀ ´Ç · ŵ
Any set of Ç -wffs
Ö
ÊÙÔ ÌÀ´Ç µ Ñ ÌÀ ´Ç µ
Ö
Ì
Ê Û Ä ÌÀ ´Ç · ŵ
Any set of Ç -wffs
ÊÙÔ ÌÀ´Ç µ Ñ ÌÀ ´Ç µ
Ì
·Ê Û Å
ÊÙÔ Fixpoint equation (3.2)
Ö Ö
Ì
Ò
·Ê Û Ñ ÌÀ´Ç µ ÌÀ ´Ç · ŵ
ÊÙÔ is consistent with
Ö
Ì
·Ê Û Ç Å
ÊÙÔ Any extension of
Ö
Ì
Ò
·Ê Û·ÊÙÔ Ç Å
ÊÙÔ Any consistent extension of
Ö Ö
Ì
Ò
Å ·Ê Û·Ê Û
ÊÙÔ fixpoint equation (3.10)
Ö Ö
Ì
Ò
·Ê Û·ÊÙÔ Ç Å
ÊÙÔ Any consistent extension of
Ö
Ì
Ò
·Ê Û·Ê Û Ç Å
ÊÙÔ Any maximal extension of
Ö
Ì
Ò
·Ê Û Ñ ÌÀ´Ç µ ÌÀ´Ç · ѵ
ÊÙÔ is consistent with
ѾÅ
Ì
Ç Å
ÊÙÔ · Ê Û Any maximal extension of
Ì
Ò Å
ÊÙÔ · Ê Û · ÊÙÔ Any maximal consistent Ö
extension of Ç
Ò Ò
·Ê Û ÊÙÔ · Ê Û
Notice that, ÊÙÔ and generate the same metatheory but they do not Ö generate the same object theory. Consider the following example.
EXAMPLE 3.14 Å
Let OM be the OM pair composed of the empty object theory Ç and the metatheory with
Ò
´Ô µ ´Õ µ ÊÙÔ ·Ê Û
axiom , connected by the reflection rules .LetOMÖ be the OM Ò
pair obtained from OM by adding the restriction to ÊÙÔ . Since they have the same set of Å
metatheorems, OM Ö and OM have the same set of models defined as follows:
Ñ Ñ Ô Ñ Õ Ñ Å is consistent and or (3.13)
The set of object theorems of OMÖ differs from that of OM. Indeed, according to Theorem
3.13, we have the following. The set of object level theorems of OMÖ is equal to the theorems Å
provable from Ç extended with the intersection of all the elements of , which is the empty
ÌÀ ´Ç µ ÌÀ´Ç µ
set. Therefore, ÇÅ is the set of the propositional tautologies. On the Ö
other hand the set of object level theorems of OM is equal to the intersection of the theorems
Ñ Ñ Å
provable from Ç extended with for some . This means, for instance, that, since
Õ Ñ Ô Õ Ç · Ñ Ñ
either Ô or belongs to any , is provable from for all . Thisimpliesthat
Ô Õ ÌÀ ´Ç µ
ÇÅ . Table 1 summarizes the results proved in this section.
4 A strict ordering characterization
In Part I [2] reflection rules have been partially ordered; however, we still do not have explicit
´Êʵ ´Êʵ ´Êʵ ´Êʵ
results stating that if , it is not the case that .Inthis
¾ ¾ section we show that all the combinations of reflection rules described in this paper generate A Foundation for Metareasoning Part II: The Model Theory 357 different object-meta relations, and therefore the partial order described in the previous paper can be transformed in a strict partial order. Let us recall the definition of partial orders between combinations of reflection rules given in the previous paper.
DEFINITION 4.1 ¾
For each two OM pairs OM½ and OM ,
´Ç µ ÌÀ ´Ç µ ÌÀ
Ç ¾
1. OM½ OM if ; ¾
OM½ OM
´Å µ ÌÀ ´Å µ ÌÀ
Å ¾
2. OM½ OM if ; ¾
OM ½ OM
¾
3. OM½ OM if OM OM ;
½ ¾
Ç Å
¾ ½ ¾ ¾ ½
4. OM½ OM if OM OM and OM OM ,where ;
Ç Å
¾ ½ ¾ ¾ ½ 5. OM½ OM if OM OM and not OM OM ,where .
Orders on sets of reflection rules can be defined on the basis on the orders of OM pairs.
DEFINITION 4.2
Let OM´Êʵ be an OM pair composed of an object theory and a metatheory connected by
´Êʵ ´Êʵ
the set of reflection rules ´Êʵ . For any combination of reflection rules , :
½ ¾
´Êʵ ´Êʵ ´Êʵ ´Êʵ
1. Ç if, for any OM and OM , with the same object theory and
½ ¾ ½ ¾
´Êʵ ´Êʵ
metatheory, OM Ç OM ;
½ ¾
´Êʵ ´Êʵ ´Êʵ ´Êʵ
2. Å if, for any OM and OM with the same object theory and
½ ¾ ½ ¾
´Êʵ ´Êʵ
metatheory, OM Å OM ;
½ ¾
´Êʵ ´Êʵ ´Êʵ ´Êʵ
3. if, for any OM and OM with the same object theory and
½ ¾ ½ ¾
´Êʵ ´Êʵ
metatheory, OM OM ;
½ ¾
Ç Å ´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ
4. For any , , if both and
½ ¾ ½ ¾ ¾
´Êʵ .
½
Ç Å ´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ
5. For any , ,if and not
½ ¾ ½ ¾ ¾
´Êʵ . ½
The orders defined above are not completely independent of one another. For instance, if
´Êʵ ´Êʵ
two combinations of reflection rules are equivalent (i.e. ) then they gener-
½ ¾
´Êʵ ´Êʵ ´Êʵ ´Êʵ Å
ate the same object theory and metatheory (i.e. Ç and ).
½ ¾ ½ ¾
The relations among the three orders are summarized in Figure 1. In this figure, an arrow
´Êʵ ´Êʵ ´Êʵ ´Êʵ
connects to , if and only if implies . The correctness of
½ ¾ ½ ¾ this latter graph is straightforward.
THEOREM 4.3 (Correctness of the graph in Figure 2)
´Êʵ ´Êʵ
If the graph of Figure 2 contains an arc labelled with from to then
½ ¾
´Êʵ
´Êʵ .
½ ¾ To prove the previous theorem we need the following lemma:
LEMMA 4.4 (Effective assumptions)
´Êʵ
Ç Å Å
Let OM be an OM pair with the set of reflection rules. Let Ç and be a
Ä Ä Å
set of wffs of Ç and respectively. Then:
´Êʵ
Å Å
1. if the reflection up rules in are restricted, then Ç OM if and only if
Å Å OM ;
358 A Foundation for Metareasoning Part II: The Model Theory
ª Â
ª Â
ª Â
ª Â ©
Ç Å
Ç Å
È
È
È
ÊÈÕ © Ê © µ
Ç Å
FIGURE 1. Relations among orders
´Êʵ
Å Ç
2. if the reflection down rules in are restricted, then Ç OM if and only if
Ç Ç OM . To prove (1), it is enough to observe that only way to ‘switch’ from object to meta in order to infer a metaformula from object assumptions, is by applying restricted reflection rules. However, these rules are applicable only if all the assumptions in the object theory are discharged. A similar argument can be used to prove (2).
PROOF. [Theorem 4.3] The graph in Figure 2 is obtained from the analogous graph in [2] by
Å
refining the relations between reflection rules. Since Ç (resp. and ) are defined as
Ç Å
the conjunction of and (resp. the conjunction of and and the conjunction of
and ), the additional information encoded in the graph in Figure 2, with respect to the previous version, consists of statements asserting that two combinations of reflection rules
differ either in the object and/or metatheory they generate, or in their derivability relation.
´Êʵ ´Êʵ ´Êʵ ´Êʵ Å
Following Definition 4.2, we prove Ç (resp. ), by
½ ¾ ½ ¾ ¾
showing that there are two OM pairs OM½ and OM composed of the same object and ´Êʵ
metatheories and connected by ´Êʵ and respectively, which generate a different
½ ¾
object (resp. meta) theory. In practice, we look for an object (resp. meta) wff not provable
´Êʵ ´Êʵ
¾
in OM ½ andprovableinOM . is automatically proved as a consequence of
½ ¾
´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ
Å Ç
Ç or . We prove without proving
½ ¾ ½ ¾ ½ ¾
´Êʵ ´Êʵ
Å ¾
or by providing two OM pairs OM½ and OM and a set of object and meta
½ ¾
¾ wffs such that is not derivable from in OM ½ ,and is derivable from in OM . We proceed by considering the arcs of the graph in Figure 2 from the bottom up. We say that a (meta or object) theory is an empty (meta or object) theory meaning that it has no
axioms.
ÊÙÔ ÊÙÔ
Å ½ ¾
1. Å . (We prove .) Consider the OM pairs OM and OM composed
Ö Ö
of the empty object theory and the empty metatheory connected by no reflection rules and
ÊÙÔ ´ µ
by respectively. No wff of the form is a theorem of the metatheory of OM½ ,
Ö
´ µ
while, for instance, is a metatheorem of OM¾ .
Ê Û Ê Û
Ç ½ ¾
2. Ç . (We prove .) Consider the OM pairs OM and OM composed
Ö Ö
´ µ
of the empty object theory and the metatheory with the only axiom connected by Ê Û
no reflection rules and by respectively. Since the object theory of OM½ is equal to
Ö
Ç Ê Û
, is not provable, Instead is provable (by ) in the object theory of OM¾ .
Ö
ÊÙÔ ÊÙÔ ÊÙÔ ÊÙÔ
½ ¾
3. . (We prove .) Consider two OM pairs OM and OM
Ö Ö
Å ÊÙÔ
composed of the empty object theory Ç and the empty metatheory , connected by
Ö
ÊÙÔ Å ´ µ
and respectively. The metatheory of OM½ is equal to and therefore is not
A Foundation for Metareasoning Part II: The Model Theory 359
ÊÙÔ
ÊÛ
Ò
ÊÙÔ
Ö
Ç
Å
Ç
ÊÙÔ
Å
ÊÛ
Ç
Å
¡
¡
¡
ÊÙÔ ÊÙÔ
Ö Ö
¡
ÊÛ ÊÛ
Ò Ò
¡
ÊÙÔ ÊÛ
¡
Ç
Ç Ç
Å
¡ Á
Å Å
¡
¡
ÊÙÔ ÊÙÔ
¡
Ò
Ö Ö
ÊÙÔ
ÊÛ ÊÛ
¡
ÊÛ
Ò Ò
ÊÛ ÊÙÔ
Ö Ö
¡
Ç
¡
Á Á
Ç Å
¡
Ç Ç
Å
Å Å
¡
¡
Ò
ÊÙÔ ÊÙÔ
Ö Ö
¡
ÊÛ ÊÛ
¡
Ý
Ç
¡
Å
Á Á
¡
Ç
Å
¡
¡
ÊÙÔ
Ö
ÊÙÔ ÊÛ
ÊÛ
Ö
Ç
Á Á
Å
Ç Ç Ç
Å Å Å
ÊÙÔ ÊÛ
Ö Ö
Á
Ç Ç
Å Å
FIGURE 2. Strict ordering of (RR)s
360 A Foundation for Metareasoning Part II: The Model Theory
´ µ
provable in it. Furthermore, by Lemma 4.4, is not derivable in the metatheory
´ µ
from the assumption of the object theory. In OM¾ instead, is derivable (by
ÊÙÔ ) from the assumption in the object theory.
Ê Û Ê Û
4. . Analogous to the previous case.
Ö
ÊÙÔ ÊÙÔ · Ê Û ÊÙÔ ÊÙÔ ·Ê Û Å
5. Å . (We prove .) Consider two OM
Ö Ö Ö Ö Ö Ö
Ç Å ¾
pairs OM½ and OM composed of the empty object theory and the metatheory with
´ µ ÊÙÔ ÊÙÔ · Ê Û
the axiom connected by and respectively. By Theorem 3.1,
Ö Ö Ö
Ñ ÌÀ ´Ç µ Ç
the interpretation is a model of the metatheory of OM½ .As is
Ñ ´ µ
consistent, there is an object wff distinct from such that and therefore
´ µ ¾
is not a theorem of the metatheory of OM½ .InOM, instead, each formula
´ µ
is provable as follows:
´ µ
Ê Û
Ö
ÊÙÔ
Ö
´ µ
ÊÙÔ ÊÙÔ ·Ê Û ÊÙÔ ÊÙÔ ·Ê Û Ç
6. Ç .(Weprove .) In the above proof we can
Ö Ö Ö Ö Ö Ö
see that is provable in the object theory of OM ¾ while it is not provable in the object
theory of OM½ , because there is no rule going from the metatheory to the object theory.
Ê Û ÊÙÔ · Ê Û Ê Û ÊÙÔ · Ê Û Ç
7. Ç . (We prove .) Consider two OM pairs
Ö Ö Ö Ö Ö Ö
Ç Å ¾
OM½ and OM composed of the empty object theory and the metatheory with the
´ µ Ê Û ÊÙÔ ·Ê Û
axiom connected by and respectively. From [2, Theorem
Ö Ö Ö
ÌÀ´Ç · Ê Û ´ÌÀ ´Å µµµ
4.1] we have that the set of object theorems of OM½ is .Since
Å ´ µ
does not prove any theorem of the form , the set of object theorems of OM½ is
ÌÀ´Ç µ Ç
. In particular is not provable in and therefore it is not provable in OM ½ .In
OM¾ instead is a theorem of the object theory. Indeed:
I
ÊÙÔ
Ö
´ µ ´ µ
E
´ µ
Ê Û
Ö
Ê Û ÊÙÔ · Ê Û Ê Û ÊÙÔ ·Ê Û Å
8. Å . (We prove .) Consider the OM
Ö Ö Ö Ö Ö Ö ¾
pairs OM½ ,andOM composed by the empty object theory and metatheory connected by
ÊÙÔ ÊÙÔ · Ê Û ´ µ
and respectively. is not provable in OM½ as the empty set is
Ö Ö Ö
´ µ ¾
a model of the metatheory of OM½ , while is provable in OM .
ÊÙÔ ÊÙÔ · Ê Û ÊÙÔ ÊÙÔ · Ê Û
9. . (We prove .) Consider two OM pairs
Ç Å ¾
OM ½ and OM composed of the empty object theory and the empty metatheory , ÊÙÔ · Ê Û
connected by ÊÙÔ and respectively. By [2, Theorem 4.1] we have that the
ÌÀ´Ç µ
set of theorems of the object theory of OM½ is and therefore is not provable in
½
the object theory of OM½ . Therefore, by Lemma 4.4, in OM is not derivable in the
´ µ
object theory from .InOM¾ , instead, can be derived in the object theory from
´ µ Ê Û
in the metatheory (via a single application of ).
ÊÙÔ ·Ê Û ÊÙÔ · Ê Û ÊÙÔ ·Ê Û ÊÙÔ · Ê Û Å
10. Å .(Weprove .) Con-
Ö Ö Ö Ö Ö Ö
Ç ¾ sider two OM pairs OM½ and OM composed of the empty object theory and the empty
A Foundation for Metareasoning Part II: The Model Theory 361
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û
metatheory Å , connected by and respectively. By Theo-
Ö Ö Ö
Ñ ÌÀ´Ç µ Ñ
rem 3.3 the set of models of the metatheory of OM½ is . Consider the
Ñ ÌÀ´Ç µ Ñ ´ µ ´Ô µ
model of the metatheory of OM½ . Clearly ,
´ µ ´Ô µ ´ µ ´Ô µ
and therefore is not a theorem of OM½ .However, is
a theorem of OM¾ .
ÊÙÔ · Ê Û ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û Ç
11. Ç .(Weprove .) Consider
Ö Ö Ö Ö Ö Ö
Ç ¾
two OM pairs OM ½ and OM composed of the empty object theory and the meta-
´Ô µ ´Õ µ ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û
theory Å with axiom , connected by and ,
Ö Ö Ö
respectively. By Theorem 3.3 the set of models of the metatheory of OM½ is:
Ñ ÌÀ´Ç µ Ñ Ô Ñ Õ Ñ
and or (4.1) Ñ
By Theorem 3.11 the set of theorems of the object theory of OM½ is the intersection of
ÌÀ´Ç µ Ô Õ ÌÀ´Ç µ Ç
for each Ñ in (4.1), which is . The formula is not provable in as ,
Õ
being empty, proves only classical tautologies. This implies that Ô is not a theorem of
Ô Õ ¾
the object theory of OM½ .InOM , instead, is provable as follows:
´Ô µ ´Õ µ
Ê Û Ê Û
Ô Õ
I I
Ô Õ Ô Õ
ÊÙÔ ÊÙÔ
Ö Ö
´Ô µ ´Õ µ ´Ô Õ µ ´Ô Õ µ
E
´Ô Õ µ
Ê Û
Ô Õ
Ò Ò
Ê Û ÊÙÔ · Ê Û Ê Û ÊÙÔ ·Ê Û Å
12. Å . (We prove .) Consider two OM
Ö Ö
Ç ¾
pairs OM½ and OM composed of the empty object theory and the empty metatheory
Å ÌÀ´Å µ
. By [2, Theorem 4.1], the set of theorems of the metatheory of OM½ is Ä
(i.e. the set of tautologies of Å ), and by [2, Theorem 4.1], the set of metatheorems of
ÌÀ´Å · ´ ÓÒ×µµ Ä Å
OM¾ is which contains wffs which are not tautologies of (e.g.
´Ô µ ´Õ µ ´ Ô Õ µ
).
Ò Ò
Ê Û ÊÙÔ · Ê Û Ê Û ÊÙÔ ·Ê Û Ç
13. Ç . (We prove .) Consider two OM pairs
Ö Ö
Ç Å ¾
OM½ and OM composed of the empty object theory and the metatheory with the
´ µ ´Ô µ
axiom . By Theorem 3.1 the set of models of the metatheory of OM½
Ñ Ñ Ô Ñ
is the set of models of Å , i.e. the set of models such that or .By Ç
Theorem 3.11 the set of object theorems of OM½ is the set of theorems of extended
with the intersection of the models of Å . Since this intersection is empty, the object
Ç Ô
theory of OM ½ is equal to and, therefore, it does not prove . In the object theory of Ô
OM¾ , instead, is provable as follows:
I
Ò
ÊÙÔ
Ö
´ µ ´ µ
E
´ µ ´Ô µ ´Ô µ ´Ô µ
E
´Ô µ
Ê Û
Ô
Ò
ÊÙÔ ·Ê Û ÊÙÔ · Ê Û · Ê Û ÊÙÔ ·Ê Û ÊÙÔ · Å
14. Å . (We prove
Ö Ö Ö Ö Ö
Ò
Ê Û · Ê Û ¾
.) Consider two OM pairs OM½ and OM composed of the empty object Ö
362 A Foundation for Metareasoning Part II: The Model Theory
Å ´Ô µ Ô
theory Ç and the metatheory , with the axiom .Since is not provable form
ÌÀ´Ç µ
the empty object theory, the interpretation Ñ is a model of the metatheory Ô
(according to Theorem 3.2). Since also is not provable in the empty object theory (i.e.
Ô ÌÀ´Ç µ Ñ ´ Ô µ ´ Ô µ
), and therefore is not provable in the metatheory
´ Ô µ ´ Ô µ ¾
of OM½ .InOM instead is provable. A proof of is the following:
´Ô µ
Ò
Ê Û
Ö
Ô
ÊÙÔ
Ö
´ Ô µ
Ò
ÊÙÔ ·Ê Û ÊÙÔ · Ê Û · Ê Û ÊÙÔ ·Ê Û ÊÙÔ · Ê Û · Ç
15. Ç . (We prove
Ö Ö Ö Ö Ö
Ò Ê Û
). Consider the two OM pairs of the previous case. The object theory of OM½ does
Ö
Ô
not prove , Indeed, by Theorem 3.1, the object theory of OM½ is the intersection of the Ñ
models of the metatheory of OM½ . The model considered in the previous case is such
Ô Ñ Ô ¾
that . Therefore is not provable in the object theory of OM ½ .InOM instead
Ò
Ô Ê Û ´Ô µ
is provable (by applying to ).
Ö
Ò
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û ·ÊÙÔ ÊÙÔ ·Ê Û ÊÙÔ · Ê Û · Å
16. Å . (We prove that
Ö Ö Ö Ö Ö
Ò
ÊÙÔ Ç ¾
.) Consider two OM pairs OM½ and OM composed of the empty object theory
Ö
Å ´ µ
and the empty metatheory . The metatheory of OM½ does not prove . Indeed
Ñ Ä ÌÀ ´Å µ
the model which interprets in the set of wffs Ç , is a model of .The
OM ½
´ µ
metatheory of OM ¾ instead proves . A proof is the following.
I
Ò (4.2)
ÊÙÔ
Ö
´ µ
Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ · Ê Û
17. Ç . See Example 3.14.
Ö
Ò Ò Ò
ÊÙÔ · Ê Û · Ê Û ÊÙÔ ·Ê Û·Ê Û ÊÙÔ · Ê Û · Ê Û Å
18. Å .(Weprove
Ö Ö Ö Ö Ö
Ò
ÊÙÔ ·Ê Û·Ê Û ¾
.) Consider two OM pairs OM½ and OM composed of the empty
Ö
Å ÊÙÔ ·
object theory Ç and the empty metatheory , connected by the reflection rules
Ö
Ò Ò
· Ê Û ÊÙÔ ·Ê Û ·Ê Û
Ê Û and respectively. By Theorem 3.9, the set of
Ö Ö
models of the metatheory of OM½ is equal to the set of interpretations which are the
Ñ ÌÀ´Ç µ
sets of theorems of an extension of Ç . The interpretation is, therefore, a
Ô Ô Ç
model of the metatheory of OM½ . Since neither nor is provable in , then neither
´Ô µ Ñ ´ Ô µ Ñ ´Ô µ ´ Ô µ
Ñ nor and therefore . This implies that
´Ô µ ´ Ô µ
is not valid, and therefore not provable, in the metatheory of OM½ .In
´Ô µ ´ Ô µ
the metatheory of OM¾ , instead, is provable as follows:
´Ô µ
Ò
Ê Û
Ô
ÊÙÔ
Ö
´Ô µ ´ Ô µ
I I
´Ô µ ´Ô µ ´Ô µ ´ Ô µ ´Ô µ ´ Ô µ
E
´Ô µ ´ Ô µ
Ò Ò Ò
ÊÙÔ · Ê Û · Ê Û ÊÙÔ · Ê Û · Ê Û ÊÙÔ · Ê Û · Ê Û Ç
19. Ç . (We prove
Ö Ö Ö Ö Ö
Ò
ÊÙÔ ·Ê Û·Ê Û ¾
.) Consider two OM pairs OM½ and OM composed of the empty
Ö
Å ´Ô µ ´ Ô µ ´Õ µ
object theory Ç , and the metatheory with the axiom ,
Ò Ò
·Ê Û ·Ê Û ÊÙÔ · Ê Û ·Ê Û
connected by the reflection rules ÊÙÔ and
Ö Ö Ö A Foundation for Metareasoning Part II: The Model Theory 363
respectively. By Theorem 3.9 it can be proved that the set of models of the metatheory of Ñ
OM½ is equal to the set of interpretations which are the set of theorems of an extension
Õ Ñ Ô Ñ Ô Ñ Ô Ô Ç
of Ç ,and if or . Since neither nor is provable in , then the
Ñ ÌÀ´Ç µ
interpretation is a model of the metatheory of OM½ . Theorem 3.11 implies
that the set of theorems of the object theory of OM½ is equal to the intersection of each
Ñ Ô
model of the metatheory. Since Õ does not belong to a model ,then is not provable
Õ ´Ô µ ´ Ô µ ¾
in the object theory of OM ½ .InOM, instead, is provable as is a Õ
theorem of the metatheory of OM ¾ (see the previous example) and can be derived by Ê Û
Eand .
Ò Ò Ò
ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ ·Ê Û·ÊÙÔ
20. . (We prove
Ö Ö Ö Ö Ö
Ò
ÊÙÔ · Ê Û · ÊÙÔ ¾
.) Consider two OM pairs OM½ and OM composed of the empty
Ö
Å ÊÙÔ ·
object theory Ç , and the empty metatheory , connected by the reflection rules
Ö
Ò Ò
ÊÙÔ ·Ê Û·ÊÙÔ
Ê Û · ÊÙÔ and respectively. By Theorem 3.2 the metatheory of
Ö Ö
Ñ ÌÀ´Ç µ
OM½ has at least a model (e.g. ), and therefore it is consistent. By Lemma 4.4, the non-derivability of in the metatheory implies the non-derivability of in the
metatheory starting from in the object theory. This derivation is instead possible in
OM ¾ :
I
Ò
ÊÙÔ ÊÙÔ
Ö
´ µ ´ µ
E
Ò Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û ·ÊÙÔ ÊÙÔ ·Ê Û ÊÙÔ · Å
21. Å . (We prove
Ö Ö
Ò
Ê Û · ÊÙÔ ¾
.) Consider two OM pairs OM½ and OM composed of the empty object
Ò
Å ÊÙÔ ·Ê Û
theory Ç , and the empty metatheory , connected by the reflection rules
Ò
· Ê Û · ÊÙÔ Ñ
and ÊÙÔ respectively. By Theorem 3.2, the interpretation is a
Ö
´ µ
model of the metatheory of OM½ . This model does not satisfy . This implies
´ µ ´ µ ¾ that is not a theorem of the metatheory of OM½ .InOM instead, is
provable (by ÊÙÔ ).
Ö
Ò Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û· ÊÙÔ ÊÙÔ ·Ê Û ÊÙÔ · Ê Û · Ç
22. Ç .(Weprove
Ö Ö
Ò
ÊÙÔ ¾
.) Consider two OM pairs OM½ and OM composed of the empty object theory
Å ´ µ ´ µ
Ç , and the metatheory with axiom , connected by the reflection
Ò Ò
·Ê Û ÊÙÔ ·Ê Û·ÊÙÔ
rules ÊÙÔ and respectively. By Theorem 3.13 we have Ö
that the set of theorems of the object theory of OM½ is equal to the intersection of the
Ç · ѵ Ñ Å
sets ÌÀ´ for each model of . According to Theorem 3.2 the intrepretation
Ñ
, is a model of the metatheory of OM½ . This implies that the set of theorems of
ÌÀ´Ç µ Ç
the object theory of OM½ is . Note that is empty and therefore consistent. The
object theory of OM¾ , instead, is inconsistent, as is provable in the object theory of
OM ¾ .
Ò Ò
ÊÙÔ · Ê Û · Ê Û ÊÙÔ · Ê Û ÊÙÔ ·Ê Û·Ê Û ÊÙÔ ·
23. .(Weprove
Ö Ö
Ê Û ÊÙÔ ÊÙÔ
.) The reasoning is similar to that done for the case .
Ö
Ò Ò
ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ · Ê Û ·
24. Å . (We prove that
Ö Ö Ö
Ò Ò
ÊÙÔ ÊÙÔ · Ê Û · ÊÙÔ
½ ¾
Å .) Consider two OM pairs OM and OM composed of
Ö Å
the empty object theory Ç , and the empty metatheory , connected by the reflection rules
Ò Ò
· Ê Û · ÊÙÔ ÊÙÔ · Ê Û · ÊÙÔ
ÊÙÔ and respectively. According to Theorem 3.2
Ö Ö
Ñ ÌÀ´Ç µ
the interpretation is a model of the metatheory of OM½ . Note that, for any
Ô Ô Ç Ñ ´Ô µ proposition Ô, neither nor is provable in . From this we can infer that
364 A Foundation for Metareasoning Part II: The Model Theory
´ Ô µ Ñ ´Ô µ ´ Ô µ ´Ô µ ´ Ô µ and Ñ ,i.e. . This implies that is
not provable in the metatheory of OM½ . The same formula, however, is provable in the
metatheory of OM ¾ (see proof above).
Ò
ÊÙÔ · Ê Û ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ · Ê Û ÊÙÔ · Å
25. Å . (We prove that
Ö
Ò
Ê Û · ÊÙÔ ¾
.) Consider two OM pairs OM½ and OM composed of the empty object
Ö
Å ÊÙÔ · Ê Û
theory Ç , and the empty metatheory , connected by the reflection rules Ò
and ÊÙÔ · Ê Û · ÊÙÔ , respectively. According to Theorem 3.2 the interpretation
Ö
ÑÑ Ä ½
Ç is a model of the metatheory of OM . Since this model does not satisfy
´ µ ´ µ
, is not provable in the metatheory of OM½ . As showed before
´ µ
is a theorem of the metatheory of OM¾ .
Ê Û ÊÙÔ ·Ê Û Ê Û Ê Û Ê Û ÊÙÔ ·Ê Û
Ç Ç
26. . Note that and , implies
Ö Ö Ö Ö Ö
Ê Û ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û Ç
that Ç . From the fact that ,wehave
Ö Ö Ö Ö Ö
ÊÙÔ ÊÙÔ ·Ê Û ´Êʵ ´Êʵ ´Êʵ ´Êʵ
Ç
that Ç .Since and implies
Ö
½ ¾ ½ ¾
´Êʵ ´Êʵ ÊÙÔ ÊÙÔ ·Ê Û
, we conclude that .
Ö
½ ¾
Ò Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û ·ÊÙÔ ÊÙÔ ·Ê Û Ç
27. Ç . Suppose, by contradiction, that
Ö Ö Ö Ö
Ò Ò Ò
ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ ·Ê Û·ÊÙÔ ÊÙÔ · Ê Û · ÊÙÔ
. From the fact that Ç ,
Ö Ö Ö Ö Ö
Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û·ÊÙÔ
we have that Ç . However, from the fact that
Ö Ö
Ò Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ · Ê Û ÊÙÔ ·Ê Û ÊÙÔ · Ç
Ç , and from the fact that
Ö Ö
Ò Ò Ò
Ê Û · ÊÙÔ ÊÙÔ ·Ê Û ÊÙÔ · Ê Û · ÊÙÔ
,wehavethat Ç and, therefore, that
Ö Ö
Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û·ÊÙÔ
Ç . This contradicts what we have proved before.
Ö Ö
Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ · Ê Û · ÊÙÔ
We conclude therefore that Ç .
Ö Ö Ö
Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û·ÊÙÔ
28. Å . We reason by contradiction as before. Suppose
Ö Ö Ö
Ò Ò
ÊÙÔ · Ê Û ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ ·Ê Û·
that Å . Then from the fact that
Ö Ö Ö Ö
Ò Ò Ò Ò
ÊÙÔ ÊÙÔ · Ê Û · ÊÙÔ ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û·ÊÙÔ Å
Å ,wehavethat .
Ö Ö Ö Ö
Ò Ò Ò
ÊÙÔ ÊÙÔ · ·Ê Û ÊÙÔ · Ê Û
On the other hand from the fact that Å and that
Ö
Ò Ò Ò
Ê Û ÊÙÔ ·Ê Û·ÊÙÔ ÊÙÔ ·Ê Û ÊÙÔ · Ê Û · ÊÙÔ Å
,wehavethat Å
Ö Ö Ö
Ò Ò
ÊÙÔ ·Ê Û ÊÙÔ ·Ê Û·ÊÙÔ
and therefore that Å . This contradicts what we
Ö Ö
Ò Ò
ÊÙÔ · Ê Û ÊÙÔ · Ê Û · ÊÙÔ
have proved before. We conclude therefore that Å .
Ö Ö Ö
Further relations between sets of reflection rules can be inferred from the structure of the graph in Figure 2. Indeed we can exploit the following properties.
PROPOSITION 4.5
Let and be two relational symbols describing a relation among two sets of reflection
Å Ç ´Êʵ ´Êʵ ´Êʵ
rules. Let Ü stand for , ,or .Let , and be sets of reflection rules.
½ ¾ ¿
Then the following properties hold:
´Êʵ
IMPLICATION If ´Êʵ ,and is reachable from in graph of Figure 1, then
½ ¾
´Êʵ
´Êʵ .
½ ¾
´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ
Ü Ü
TRANSITIVITY-1 If Ü and ,then .
½ ¾ ¾ ¿ ½ ¿
´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ
Ü Ü
TRANSITIVITY-2 If Ü and ,then .
½ ¾ ¾ ¿ ½ ¿
´Êʵ ´Êʵ
IDENTITY Ü .
½
´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ ´Êʵ
Ü
SUBSTITUTIVITY If Ü and ,then .
½ ¾ ¾ ¿ ½ ¿
Ü
Ü Ü
These properties follow immediately from the definitions of Ü and , with
Ç Å
Ç Å Ç Å . Table 2 links the holding of of any two sets of A Foundation for Metareasoning Part II: The Model Theory 365
TABLE 2. Strict ordering among (RR)s
½ ¾ ¿ ½¼ ½½ ½¾ ½¿ ½ ½
ÊÙÔ ÊÙÔ ÊÙÔ ÊÙÔ ÊÙÔ
Ò Ò
Ö Ö Ö Ö
ÊÙÔ ÊÙÔ ÊÙÔ ÊÙÔ ÊÙÔ
Ö Ö Ö
ÊÛ ÊÛ ÊÛ ÊÛ ÊÛ
ÊÛ ÊÙÔ ÊÛ ÊÙÔ
Ö Ö
ÊÛ ÊÛ ÊÛ ÊÛ ÊÛ
Ò Ò Ò Ò Ò
Ö
ÊÙÔ ÊÛ ÊÙÔ ÊÛ ÊÙÔ
Ö Ö Ö
Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç
½
Å Å Å Å Å Å Å Å Å Å Å Å Å Å
Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç
¾ ÊÛ
Å Å Å Å Å Å Å Å Å Å Å
Å Å Å
Ö
Ç Ç Ç Ç Ç Ç Ç Ç Ç
¿ ÊÙÔ
Ç Ç Ç Å Å Ç Å Å Å Å Å Ç Å Å
Ö
Ç Ç Ç Ç Ç Ç Ç Ç Ç
Ç Ç
ÊÛ
Å Å Å Å Å Å Å Å Å Å Å Å
Å Å
Ç Ç
Ç Ç Ç Ç Ç Ç Ç
ÊÙÔ
Å Å
Ç Ç Ç Ç Ç
Å Å Å Å Å Å Å
Ç Ç Ç Ç Ç Ç Ç
ÊÙÔ
Ö
Å Å Å Å Å Å Å
ÊÛ
Ö
Ò
Ç Ç Ç Ç
ÊÙÔ
Ö
Ç Å Ç Å Ç Å Ç Å
ÊÛ
Ç Ç Ç Ç Ç Ç
ÊÙÔ
Ö
Å Å Å Å Å Å
ÊÛ
ÊÙÔ
Ö Ç Ç
ÊÛ
Ç Ç Å Ç Ç Å
Ò
ÊÙÔ
Ö
ÊÙÔ
Ö Ç Ç Ç
ÊÛ
½¼
Å Å Å
Ò
ÊÛ
Ö
ÊÙÔ
Ö Ç
Ç
ÊÛ
½½
Å
Ç Ç Ç
Ò
Å
ÊÙÔ
ÊÙÔ
Ö Ç Ç
ÊÛ
½¾
Å Å
Ò
ÊÛ
Ò
Ç Ç
ÊÙÔ
Ç
½¿
Å Å
Å Ç Ç Ç Ç
ÊÛ
Å
Ç
ÊÙÔ
Ç
½
Å
ÊÛ
Å
ÊÙÔ
ÊÛ
½
Ç Ç
Ò
ÊÙÔ
Ö
´Êʵ ´Êʵ
reflection rules. A relational symbol occurs in row column if and only
½ ¾
´Êʵ
if ´Êʵ . The results in Table 2 can be derived from the facts explicitly stated in
½ ¾ the graph in Figure 2 plus the properties IMPLICATION, TRANSITIVITY-1, TRANSITIVITY-2, IDENTITY and SUBSTITUTIVITY.
EXAMPLE 4.6
Ê Û ÊÙÔ ·Ê Û
In order to show that Å , which is not implicitly stated in the graph of
Ö Ö
Figure 2 by a labelled arc, we reason as follows:
½ Ê Û ÊÙÔ · Ê Û
Å is stated in the graph of Figure 2;
Ö Ö Ö
¾ Ê Û Ê Û
Å is stated in the graph of Figure 2;
Ö
¿ Ê Û ÊÙÔ ·Ê Û
Å is derivable from 1. and 2. by SUBSTITUTIVITY.
Ö Ö
THEOREM 4.7 (Completeness of the graph in Figure 2)
The graph in Figure 2 is complete. That is, for each two sets of reflection rules ´Êʵ and
½
´Êʵ ´Êʵ ´Êʵ
Ç Å Ç Å
and for each relational symbol ,
¾ ½ ¾
´Êʵ
iff ´Êʵ is derivable from the facts explicitly stated in the graph in Figure 2, and
½ ¾
from IMPLICATION, TRANSITIVITY-1, TRANSITIVITY-2, IDENTITY and SUBSTITUTIVITY.
The proof of Theorem 4.7 can be done by showing that, if [ ] does not occur in the 366 A Foundation for Metareasoning Part II: The Model Theory
dual ´)
¹ µ
OM dual ´OM
Ê Ê
dual ´)
¹
Ñ
ѵ dual ´
FIGURE 3. A duality graph for satisfiability
´Êʵ ´Êʵ ¾
slot at row column of Table 2, then there are two OM pairs OM½ and OM ,
½ ¾
¾ ½ ¾ such that OM ½ OM [OM OM ]. The proof, not reported because it is routine and tedious, exploits the same techniques exploited in the proof of Theorem 4.3.
Let us discuss, top to bottom, left to right, some of the results in Table 2. The results in
row 1 (´Êʵ ) are obvious. It is sufficient to notice that a single reflection up rule does ½ not modify the object theory. Trivially, the dual result holds in the case of reflection down only. The results in rows 2 and 3 have similar explanations. The relations generated by a reflection up rule cannot be compared with those generated by a reflection down rule. A restricted rule generates the same meta and object theories as its corresponding unrestricted rule. It only allows more derivations, all those which (start in one theory and finish in another and) contain at least an application of a reflection rule in presence of open assumptions. The effects of dropping the restriction appear when we have at least one reflection up rule and
one reflection down rule. In row 4 the only non-trivial result is in column 6. As from row
Ê Û ÊÙÔ
3 column 4, Ê Û and generate the same meta and object theories. Adding to
Ö Ö
Ê Û Ê Û ÊÙÔ ·Ê Û
increases the theorems of the metatheory, this proves Å .To
Ö Ö Ö
Ê Û ÊÙÔ ·Ê Û ª Ç
prove that Ç it is sufficient to consider the case where and
Ö Ö
ª ´ µ ´ µ Ê Û ÊÙÔ ·Ê Û
Å . Finally, to prove it is sufficient to
Ö Ö
further consider the fact that any derivation which contains an application of reflection down · Ê Û
with open assumptions is not a derivation of the OM pair with ÊÙÔ . Inrow5the
Ö Ö result in column 6 can be proved with arguments similar to those for row 4, column 6.
5 A duality property for satisfiability As already stated in Part I [2], duality principles usually state the preservation of certain log- ical properties, e.g. provability or satisfiability, under appropriate syntactic transformations on formulas. Examples of duality principles for propositional logic and modal logic can be found in [5] and [1] respectively. In this section we consider satisfiability, in particular
we prove (under certain general conditions) the correctness of the graph in Figure 3, where
µ dual´ is a transformation (defined below) on formulas/ models and OM pairs. We recall from Part I [2] the notions of dual reflection rule, dual formula, and dual OM pair. See Part I [2] for an explanation of the underlying intuitions.
DEFINITION 5.1 (Dual reflection rule)
´µ For any positive [negative] reflection rule , dual is the corresponding negative [positive] A Foundation for Metareasoning Part II: The Model Theory 367 reflection rule.
DEFINITION 5.2 (Dual formula)
Ä ´µ
For any formula Å , dual is defined accordingly to 1Ð4:
µ
1. dual´ , does not contain any occurrence of ;
´ µµ ´ µ
2. dual´ ;
Æ µ ´µ Æ ´ µ Æ
3. dual´ dual dual ,where is either or or ;
µ ´µ
4. dual´ dual .
µ Notice that, dual´ only modifies formulas containing predicate, and preserves the struc-
ture of the connectives. This is not a surprise, as we are interested in a duality principle
´ µµ ´ µ
for metaproperties. To define dual´ we reason as follows: since is deriv-
Ò
´ ´ µµ ´ÊÙÔ µ ÊÙÔ
able by ÊÙÔ from , dual must be derivable by dual (i.e. ) from .
´ µµ
This implies that dual´ must be an element of the set of formulas derivable from
Ò
ÊÙÔ ´ ´ µµ ´ µ
by . In symbols dual O . We have chosen
´ µµ ´ µ dual´ . An important property of dualization is that the dual of the dual of an entity (formula, OM
pair) is the entity itself (or equivalent, giving some appropriate notion of equivalence). To
´ µ ´ µ
satisfy this requirement, if is of the form , must be equivalent to
µ ´ ´µµ ´ ,whichisdual dual . This is not true in all OM pairs. We therefore restrict
ourselves only to those OM pairs which satisfy the following property:
´ µ ´ µ OM (5.1) We call the OM pairs which satisfy Condition (5.1), classical.
DEFINITION 5.3 (Dual OM pair)
Ç Å ´Êʵ
If OM is classical OM pair, then the dual of OM, is the OM pair
µ Ç ´Å µ ´´Êʵ µ ´Å µ
dual´OM dual dual ,where,dual , is the theory obtained by sub-
ª Å ´ª µ ´ µ ´ µ Å stituting the axioms Å of with dual and by adding .
Let us define the dualization for the interpretations of the language of the metatheory. We
´ µ Ñ
require that an interpretation Ñ satisfies if and only if the dual interpretation of
´ µ
satisfies . This implies that:
Ñ ´Ñµ
dual (5.2)
´Ñµ
By replacing Ñ with dual and with in (5.2) we obtain:
´Ñµ ´ ´Ñµµ dual dual dual (5.3)
Combining (5.2) and (5.3) we have that
Ñ ´ ´Ñµµ
dual dual (5.4)
´Ñµµ Ñ Finally we require that dual´dual satisfies the same formulas as (this is in order to satisfy the principle that the dual of the dual of an entity is the entity itself). Hence from (5.4)
we can conclude:
Ñ Ñ (5.5)
Notice that restriction (5.5) is the semantical counterpart of restriction (5.1) for classical
µ OM pairs. In the following, dual´ is defined only for those interpretations which satisfy condition (5.5). We are now ready to give the notion of dual interpretation. 368 A Foundation for Metareasoning Part II: The Model Theory
DEFINITION 5.4 (Dual interpretation)
For each interpretation Ñ of the metalanguage that satisfies condition (5.5), the interpretation
ѵ Ñ dual´ .
A dual interpretation preserves satisfiability.
THEOREM 5.5 (Duality preserves satisfiability)
Ñ
For each interpretation Ñ of the metalanguage that satisfies condition (5.5), ,ifand
ѵ ´µ only if dual´ dual .
PROOF. We proceed by induction on the complexity of the meta wffs. Let us start by proving
´ µ ´Ñµ ´ µ Ñ ´ µ Ñ
that Ñ , if and only if dual .If ,then .By
Ñ ´Ñµ
condition (5.5) . By definition of dual interpretation dual and therefore
ѵ ´ µ ´Ñµ ´ µ ´Ñµ
dual´ .Viceversa,ifdual ,then dual , then,
Ñ Ñ
by definition of dual interpretation, . Condition (5.5) implies that and
´ µ ´ µ therefore Ñ . The step cases are routine as dual distributes over the connectives.
Part I [2] shows that, for classical OM pairs, duality also preserves derivability, that is that:
´ µ
Å Ç Ç Å ´ µ Ç
Ç OM dual dual OM (5.6)
´ µ ´ µ
Å Å Ç Å ´ µ Å
Ç OM dual dual OM dual (5.7)
Ç Å Å where Ç and are a set of object wffs and a set of meta wffs respectively. Com-
bining this latter result with Theorem 5.5 we obtain that the class of models of the metatheory µ generated by the OM pair dual´OM is the dual of the class of models of OM. More formally:
THEOREM 5.6
For any classical OM pair OM, an interpretation Ñ is a model of the metatheory of OM if
ѵ ´ µ
and only if dual´ is a model of the metatheory of dual OM .
´ µ
PROOF.LetÑ be a model of OM and a theorem of the metatheory of dual OM .By[2,
µ Ñ ´µ ´Ñµ
Theorem 5.5], dual´ is a theorem of OM and therefore dual . By (5.7), dual
´µµ ´ ´µµ ´Ñµ dual´dual . The fact that dual dual is equivalent to ensures that dual .
Theorem 5.6 allows us to extend the results about the models of the metatheory proved in the previous section. The generalization can be done for the set of reflection rules which are the dual of the combinations that generate classical OM pairs. We summarize these results in
the following table, constructed from Table 1.
´Êʵ ÌÀ ´Å µ
Models of ÇÅ
Ò Ò
ÊÙÔ Ç
· ÊÛ The set of formulas consistent with an extension of
Ö
Ò Ò
ÊÙÔ Ç ·ÊÙÔ
· ÊÛ The set of formulas consistent with a consistent extension of
Ö Ö
Ò Ò
ÊÙÔ ·ÊÛ
· ÊÛ Fixpoint equation dual of (3.10)
Ö Ö
Ò Ò
ÊÙÔ Ç ·ÊÙÔ
· ÊÛ The set of formulas consistent with a consistent extension of
Ö
Ò Ò
ÊÙÔ Ç ·ÊÛ
· ÊÛ Any consistent maximal extension of or the empty set
Ö
Ò Ò
· ÊÛ Ç
ÊÙÔ Any consistent maximal extension of or the empty set
Ò Ò
· ÊÛ ·ÊÙÔ Ç
ÊÙÔ Any consistent maximal extension of Ö A Foundation for Metareasoning Part II: The Model Theory 369 6Conclusion This paper follows from Part I [2], which investigates the proof theory of OM pairs. The core result of this paper is the definition of a model theory where the models of the metatheory, defined as sets of formulas, are put in correspondence with the object theory, and vice versa. This paper extends the results presented in Part I [2]. In particular, the partial ordering of OM pairs defined in Part I [2] is strengthened to a strict ordering, and the principle of duality for derivability introduced in Part I [2] is extended to satisfiability. OM pairs are our proposed formalism for studying the foundations of metareasoning. OM
pairs separately represent the three main factors of metalevel reasoning (i.e. the object theory, Å the metatheory, and the reflection principles) using three distinct components, namely, Ç ,
and ´Êʵ . In many approaches a fourth factor is considered, namely, the amalgamation of the object and metatheory. The object theory and the metatheory can be considered as a single theory, or the metatheory can be embedded in the object theory, or the metatheory can contain
the object theory. We have shown in Part I [2] that OM pairs can also formalize this aspect
Å Ç
Ç Ç Å
by using bridge rules of the form Å ,and .
Å The first advantage of using OMÇ pairs is that it is possible to reconstruct and classify many different and heterogeneous forms of metareasoning in terms of these four parameters. Examples on how different approaches can be reconstructed and compared using OM pairs, are given in the related work section of [2]. To support this comparison, we have studied the proof and model theoretic properties of many possible parameter configurations. For most of the combinations of reflection rules and kinds of object and metatheories (empty, horn, etc.), we have provided an axiomatic characterization, a semantic characterization and a Natural Deduction calculus. A second advantage is that the explicit representation of these four components allows us to study the effects of each single component on the overall system, for instance, on the theorems or the models of the object theories and metatheories. To this extent, as an example, we have studied the effects of embedding the metatheory into the object theory, when they are connected by different reflection principles. We have proved that amalgamating them does not lead to inconsistent theories, ... and so on. In order to complete our study of OM pairs, there is at least a further topic that must be accomplished. In these two papers, indeed, we limit ourself by considering reflection rules on closed formulas. There are, however, interesting reflection principles stating the correctness of the metatheory with respect to properties of the object theory on open formulas or terms. Examples of such properties are: ‘being an open formula that represents a certain subset of the domain’, ‘being a term of the object theory that denotes a certain individual’, .... Reflection principles on these properties have been studied in Formal Logics as non-local reflection principles, in Meta Logic Programming as non-ground representations, while in Knowledge Representation they are often tightly connected to the problem of quantifying in. To formalize these more general reflection principles, we have to consider reflection rules on open formulas. OM pairs with reflection rules defined on non-closed formulas will be the subject of further studies. However, it is our opinion that many of the results described in this paper will scale to these new kinds of reflection rules. 370 A Foundation for Metareasoning Part II: The Model Theory Acknowledgements The research described in this paper owes a lot to the openness and sharing of ideas which exists in the Automated Reasoning Area at ITC-IRST and in the Mechanized Reasoning Group of the University of Trento and Genoa. Discussions with Massimo Benerecetti, Paolo Bouquet, Alessandro Cimatti, Chiara Ghidini, Enrico Giunchiglia, Silvio Imb«o and Paolo Traverso, have provided useful insights. We also thank the anonymous reviewer for very useful feedback.
References [1] B. F. Chellas. Modal Logic – an Introduction. Cambridge University Press, 1980. [2] G. Criscuolo, F. Giunchiglia, and L. Serafini. A foundation for metareasoning, Part I: The proof theory. Journal of Logic and Computation, 12, 167Ð208, 2002. [3] S. Feferman. Transfinite recursive progressions of axiomatic theories. Journal of Symbolic Logic, 27, 259Ð316, 1962. [4] F. Giunchiglia. Contextual reasoning. Epistemologia, special issue on I Linguaggi e le Macchine, XVI:345Ð 364, 1993. Short version in Proceedings IJCAI’93 Workshop on Using Knowledge in its Context, Chambery, France, pp. 39Ð49. Also IRST-Technical Report 9211-20, IRST, Trento, Italy, 1993. [5] S.C. Kleene. Introduction to Metamathematics. North Holland, 1952. [6] D. Prawitz. Natural Deduction - A proof theoretical study. Almquist and Wiksell, Stockholm, 1965. Received 30 October 1997