Proceedings of the Second International Conference. San Mateo, CA: Morgan Kaufmann, 1991. Reflections about Reflection. Giuseppe Attardi. Dipartimento di ...
Appears in: Allen, J. A., Fikes, R., and Sandewall, E. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of the Second International Conference. San Mateo, CA: Morgan Kaufmann, 1991.
Reflections about Reflection
Giuseppe Attardi Dipartimento di Informatica Università di Pisa Corso Italia 40, I–56125 Pisa, Italy
Abstract In the line of a syntactic treatment of modalities, many proposals have been presented consisting in an extension of a logic calculus with a meta-language for expressing facts about terms and sentences, and with an axiomatization of provability. These proposals differ in the use of the inference rules used to link the object level and meta-level theories; the formulation of these reflection rules is crucial since it affects the consistency of the extended calculus. We argue that reflection rules resulting in a conservative extension are too weak. On the other hand, well known results show that non conservative extensions often run into paradoxes. We propose a non conservative extension where useful theorems can be proved while consistency is retained.
1 INTRODUCTION The advantages of a syntactic approach to the representation of truth, knowledge and belief have been largely discussed in the literature [(McCarthy 1979), (Moore, 1977), (Konolige, 1982), (Perlis, 1985), (Perlis, 1988), (Davies, 1990)]. Along this line of research several proposals are based on extending a logic calculus with a meta-language, for expressing facts about terms and sentences, and with an axiomatization of provability. Provability in an agent’s theory is seen as one of the primitive relations for the formalization of belief and knowledge. These proposals differ in the degree of connection between object-theory and meta-theory, ranging from a semantic connection of completely separate theories as in (Konolige, 1982), to the reflection principles of FOL
Maria Simi Dipartimento di Matematica e Informatica Università di Udine Via Zanon 6, I–33100 Udine, Italy
(Weyhrauch, 1980), which provide a bridge between the two, still distinct, theories, to the amalgamated solutions, where there is just one theory encompassing object and meta-level. In the “separated” approach, in order to express nested belief, a hierarchy of languages has to be constructed, where each language is a meta-language for the theory below in the hierarchy. Self reference is not allowed and the construction of paradoxical sentences is blocked. As a consequence also non paradoxical self referential or mutually referential sentences cannot be represented. This is seen as the major drawback of this approach (see for example (Perlis, 1985), (Perlis, 1988), (Davies, 1990)): self referential statements about truth, belief or knowledge are in fact naturally found in common sense reasoning. Among the amalgamated solutions we can further distinguish among conservative extensions (like the one proposed by Bowen and Kowalski (Bowen and Kowalski, 1982) and non conservative ones such as the Omega extension in (Attardi and Simi, 1984). In the case of amalgamated solutions, a proper formulation of the reflection rules is crucial since it affects the consistency of the extended calculus. Reflection rules resulting in a conservative extension are very weak; in principle they do not add anything to the reasoning that can be performed in the object level or meta-level theories. On the other hand, well known negative results (Montague, 1963) show that non conservative extensions often run into paradoxes. We will propose a formulation of the inference rules which lays in the middle ground between the BowenKowalski proposal and the Omega proposal, resulting in a non conservative but consistent extension.
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In section 2 we set the ground for the following discussion making precise our assumptions and defining the conventions to be used in the rest of the paper. In sections 3 and 4 we present an analysis of different formulations of the reflection rules, especially in the context of the amalgamated approach, and discuss the implications on the resulting theories. In section 5, 6 and 7 we propose a new version of the reflection rules, demonstrate with an example its expressiveness, and prove the consistency of the resulting logic. Finally, in section 8 and 9, we discuss related work and comment on the results.
2 META-APPARATUS AND BASIC ASSUMPTIONS In the following we will assume that the meta-language is a first-order language whose domain of discourse are the syntactic expressions of the object language. We rely on the existence of a naming device for terms and sentences of the object language. Any of the mechanisms proposed in the literature will do, such as having terms of the meta-language for terms and sentences of the object language. We will use the convention of quoting expressions when their names should be used. Moreover, the meta-theory shall include an axiomatization of provability as in (Bowen and Kowalski, 1982). This can be done, for example, by considering a set of inference rules in the style of natural deduction and writing a meta axiom for each of the inference rules. For instance, for the rule of implication introduction, formulated as follows: Γ ∪ A_ B Γ _ (A ⇒ B) we will have a corresponding meta axiom: PR(‘Γ ∪ A’, ‘B’) ⇒ PR(‘Γ’, ‘A ⇒ B’) Here and in the rest of the paper we use the conventions that Γ, Γ1, Γ2 are metavariables for set of statements and A, B are metavariables for statements. In correspondence, ‘Γ’, ‘Γ1’, ‘Γ2’ are sets of quoted statements and ‘A’, ‘B’ are quoted statements. Finally, T means derivable according to the axioms and rules of the calculus T. Given the different meanings in which the term “conservative extension” has been used in the literature, we feel the need to make precise its use in the context of this paper. Let us call OT the object level theory and MT the corresponding meta-level theory including an axiomatization of provability (the language of the two theories may or may not be the same); with RR we will
denote the reflection rules used to connect the two theories above. We will speak about “conservative” or “non conservative” with respect to the extension of the theory OT+MT as a result of adding the reflection rules, i.e. OT+MT+RR is a conservative extension of OT+MT when any theorem derivable from OT+MT+RR is also derivable from OT+MT, or in other words RR are derived inference rules and as such do not add new theorems. This use complies with the classical definition given for example in (Shoenfield, 1967) according to which an extension is conservative if any formula of the original theory which is a theorem of the extension is also a theorem of the original theory. In our case the language of the original theory and of the extension are the same. Different seems to be the use in (Bowen and Kowalski, 1982) who talk about the amalgamation being a conservative extension in the sense that “no new theorems are provable in the amalgamation that were not already provable in either L or M”. We believe that this definition is too strong. In fact their own amalgamation is not conservative according to this definition: given a formula A provable in L but not in M and a formula B provable in M but not in L, the formula A ∧ B is provable in the amalgamation but neither in L nor in M. The Bowen and Kowalski amalgamation is however a conservative extension according to our definition. Still another perspective is taken, in the discussion about reflection presented in (Giunchiglia and Smaill, 1989). While we talk about extensions of the theory OT+MT through the addition of reflection rules, they talk about extensions of OT with meta axioms or inference rules.
3 SEPARATION OF THEORIES In Konolige’s so called “syntactic approach” (Konolige, 1982), propositional attitudes are expressed in a metalanguage (ML) as relations between agents and sentences of the object language (OL). The meta theory includes an axiomatization of truth and of the provability relation in the object level theory. In order to deal with nested beliefs, a hierarchy of such languages and theories has to be constructed and a complex technical machinery is required for linking the levels appropriately. Each theory in this hierarchy includes the language of the theory below, the language for talking about it, and an axiomatization of truth and provability. The reasoning of an agent corresponds to the reasoning in one of these meta theories, at the appropriate level according to the degree of nesting of his beliefs; for example three levels are needed for the
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reasoning of the third wise man in the three wise men puzzle. In this proposal the connection between meta-level and object level theory is a semantic one: determining the truth of a formula involving PR in the ML can be performed by “semantic attachment”, i.e. performing a proof in the OL. This semantic connection can be exploited to simplify proofs and only when complete information on the object level theory is available; in principle however, all the reasoning can be performed at the meta-level using the “simulated” version of provability. The major drawback of this approach is the lack of expressiveness. Since there is a different truth and provability predicate for each level, sentences that do not belong specifically to one level, such as “the agent has a false belief”, are not naturally expressed. In Weyrauch’s FOL (Weyrauch, 1980) the beliefs of each agent is represented as a pair of OL/ML theories connected by “reflection principles”; the reflection rules are expressed as follows: (in T) (in META) (in META) (in T)
A Theorem(‘A’)
Theorem(‘A’) A
1989), each agent is represented as a structure which includes three object level theories, for representing the knowledge of the other two agents and himself, plus a meta-level theory where all the reasoning is done. This is necessary in order to maintain separate between the object level facts which represent the beliefs of each agent, including the agent itself. An evolution of the ideas of FOL is the work on multilanguage systems (Giunchiglia and Serafini, 1990). A multi-language system includes a set of distinct languages each associated with it’s own theory. These theories are not independent of one another but connected by “bridge” rules, whose premises and conclusions belong to different languages. For example reflection rules, which are the most typical example of bridge rules, could be expressed as follows:
(reflect up)
(reflect down)
without dependencies (reflect up)
(reflect down)
where META is the meta-theory of T, including as theorems all the facts that are assumptions in T, and an axiomatization of provability. The rules can be phrased as follows, in our notation: _T A _MT PR(‘A’)
(reflect up)
_MT PR(‘A’) _T A
(reflect down)
MΤ is the meta theory of T.These rules require that A has been derived with no dependencies (A is a theorem of Τ) in order to be able to reflect it up. In FOL, theories are kept separate, no amalgamation is attempted, and therefore it is an obvious fact that no theorems across theories are generated. The approach to nested beliefs is not elegant. For example, in the solution based on FOL to the three wise men puzzle proposed in (Aiello, Nardi and Schaerf,
where represents the formula A in the language i. This approach has the flavour of general framework more then of a specific theory: no commitment is made to any particular formulation of the reflection rules, nor to the fact that the languages are necessarily distinct. This makes it difficult to make a comparison with other proposals. All we can say is that we seem to remain in the realm of layered approaches with the shortcomings in expressiveness we have discussed above.
4 AMALGAMATION OF LANGUAGE AND META-LANGUAGE In the amalgamated approach we have just one theory including both the object level and meta-level theories, connected by inference rules. The interesting case is when the meta-language is the same as the object language (M = L); we will write both L and M as meaning derivability in the amalgamated theory. The amalgam includes an axiomatization of the provability predicate PR. With these assumptions, the inference rules for BowenKowalski’s amalgamation become: Γ_A _ PR(‘Γ’, ‘A’)
(1 – reflect up)
4
(1 – reflect down)
using a set of assumptions can be reflected down in the context of the same assumptions.
We will refer globally to these rules as RR11. If PR(‘Γ’, ‘A’) is a theorem of the extended calculus, then A can be proved from Γ and vice versa. Note that these rules are derivable (provided that a “faithful” axiomatization of provability is included) and therefore no new theorems are generated by virtue of these new rules. Therefore the amalgamation is a conservative extension and provability coincides with provability in the predicate calculus.
Moreover, with (1–reflect up) we can deduce the theorem PR (Γ’, ‘A’) only if there is a valid deduction of A from Γ alone, while with (2–reflect up), if there is a deduction of A from a set of assumptions Γ1 ∪ Γ2 we can deduce PR(‘Γ2’, ‘A’) in the context of Γ1; it is like performing a kind of partial closure on a subset of the assumptions used to deduce A and discharging them, while remaining in the context of the rest of the assumptions.
In Omega (Attardi and Simi, 1984) the reflection rules are stated as follows (here (‘s’ in vp) means that ‘s’ can be derived from the set of assumptions ‘vp’):
Note that. with RR2, deduction in the extended calculus is no more classical deduction as in Bowen-Kowalski.
_ PR(‘Γ’, ‘A’) Γ_A
(‘s’ in ‘vp’) vp _ s
(reflect down)
We show that the extension with RR2 is non conservative, by proving a formula across object and meta-level. A property of deduction that we will use is the following:
vp _ s ('s' in 'vp' )
That is, if (‘s’ in ‘vp’) can be derived, then s can be derived from the assumptions vp, and vice versa, if there is a proof of s using a set of assumptions vp, then also (‘s’ in ‘vp’) can be proved. In the notation of this paper, and making explicit the dependencies, the above rules can be phrased as: Γ1 _ PR(‘Γ2’, ‘A’) Γ1 ∪ Γ2 _ A Γ1 ∪ Γ2 _ A Γ1 _ PR(‘Γ2’, ‘A’)
If Γ
(reflect up)
A then Γ ∪ Γ1
A
(monotonicity)
which is obvious from the classical definition of derivability. We will write RR1 and RR2 to indicate explicitly which version of the reflection rules is used in a derivation. Lemma 1: Proof:
RR2
PR(‘A’, ‘B’)
(2–reflect down)
PR(‘A’, ‘B’) ⇒ (A ⇒ B)
RR2
PR(‘A’, ‘B’)
PR(‘A’, ‘B’) ∪ A PR(‘A’, ‘B’) (2–reflect up) RR2
RR2
RR2
B
(A ⇒ B)
PR(‘A’, ‘B’) ⇒ (A ⇒ B)
(definition of
)
(2–reflect down) (⇒ introduction) (⇒ introduction)
■
We will refer globally to these rules as RR2. There is a significant difference between the two formulations of reflection. Rules RR1 are a special case of rules RR2 where Γ1 is empty. In fact, with (1- reflect down), the only facts that can be reflected down are the theorems of the amalgamated calculus, that is formulas that are derivable using no proper axioms nor assumptions. The assertions about PR made at the metalevel have no influence on the object level. According to rule (2–reflect down) instead, facts that are deduced
If (2–reflect down) is available, the converse of the implication can also be proved: Lemma 2: Proof:
RR2
(A ⇒ B) ∪ A (A ⇒ B) RR2
RR2
(A ⇒ B) ⇒ PR(‘A’, ‘B’)
RR2
B
PR(‘A’, ‘B’)
(A ⇒ B) ⇒ PR(‘A’, ‘B’)
(Modus Ponens) (2–reflect up) (⇒ introduction)
■ 1In (Bowen and Kowalski, 1982) the formulation of these ru-
les includes explicitly the axiomatization of the provability relation, denoted as Pr.
If the amalgam is achieved using the inference rules RR1, the proofs of the two lemmas cannot go through. The crucial difference is that in order to apply (1–reflect
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down) to PR(‘A’, ‘B’) this formula must be provable without assumptions, which is not the case here. For a similar reason (1–reflect up) cannot be used in the proof of Lemma 2. More general forms of Lemma 1 and Lemma 2 are also derivable:
Γ1
then, using Lemma 1 and Modus Ponens Γ1
Γ1 ∪ Γ2
where the notation ∧Γ should be interpreted as the conjunction of the statements in Γ. Putting the two lemmas together we get: (∧Γ ⇒ B) ⇔ PR(‘Γ’, ‘B’)
This result is problematic. In fact undesirable properties of PR will follow, such as: ¬ PR(‘Γ’, ‘A’) ⇒ PR(‘Γ’, ‘¬A’) PR(‘Γ’, ‘A ∨ B’) ⇒ PR(‘Γ’, ‘A’) ∨ PR(‘Γ’, ‘B’) (PR(‘Γ’, ‘A’) ⇒ PR(‘Γ’, ‘B’)) ⇒ PR(‘Γ’, ‘A ⇒ B’) A ⇒ PR({}, ‘A’) The last property above was called visibility in (Attardi and Simi, 1984) and was used for a very simple and natural solution to the three wise men puzzle. Visibility is however too strong and sanctions counterintuitive arguments. In the solution of the puzzle we used visibility to model the idea that any agent is able to reproduce the reasoning of another one who shares the same assumptions. There exists a solution to the puzzle, as simple and natural, where (1–reflect up) is used instead of visibility. More seriously, we can prove that the amalgam using RR2 is inconsistent. For this we refer to the discussion in section 7. Lemma 1 can be seen as an alternative formulation of (2reflect down). In fact not only Lemma 1 is derivable using (2-reflect down) but the converse is also true. Lemma 3: The following axiom and inference rule are equivalent: RR2
PR(‘Γ’, ‘A’) ⇒ (∧Γ ⇒ A)
Γ1 _ PR(‘Γ2’, ‘A’) Γ1 ∪ Γ2 _ A
A
■
(∧Γ ⇒ B) ⇒ PR(‘Γ’, ‘B’)
RR2
(∧Γ2 ⇒A)
finally, by Modus Ponens,
PR(‘Γ’, ‘B’) ⇒ (∧Γ ⇒ B)
Theorem 1:
PR(‘Γ2’, ‘A’)
(Lemma 1) (2–reflect down)
Proof: We have to show that (2-reflect down) can be derived from Lemma 1. Assume
As discussed above, Theorem 1 is not derivable with RR1 reflection rules. It turns out however that, for any pair of formulas A and B such that A RR1 B, and in particular for those formulas which are provable without assumptions, i.e. theorems of the calculus, the proof can be carried out also with RR1 reflection rules. That is, one can prove at least all the theorems of the form RR1
PR(‘Γ’, ‘B’) ⇔ (∧Γ ⇒ B)
where Γ RR1
RR1
B. For example:
PR({}, ‘A ⇒ A’) ⇔ (A ⇒ A)
The Bowen-Kowalski amalgamation is consistent if and only if the extended calculus, including the axiomatization of PR but not the reflection rules, is consistent. Even if a full account of consistency was not provided in (Bowen and Kowalski, 1982) the restriction on Theorem 1 that Γ RR1 B is sufficient to avoid paradoxes and guarantees consistency. This is a consequence of the results in section 7 where we prove the consistency of a richer logic.
5 A CONSISTENT NON CONSERVATIVE EXTENSION A consistent non conservative extension can be obtained with the following formulation of the inference rules (RR3): Γ1 _ PR(‘Γ2’, ‘A’) Γ1 ∪ Γ2 _ A Γ _PC A _ PR(‘Γ’, ‘A’)
(3-reflect down) (3–reflect up)
therefore weakening the rule (2–reflect up), which turned out to be problematic while retaining the strong version of reflect down. Note that the precondition in (3–reflect up) is provability in the predicate calculus. More precisely PC means “derivable using any inference rule except reflect down and reflect up”. This restriction is necessary for consistency (see section 7). Let us call this logic RT.
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⇒ PR(‘ΓV’, ‘Γ ⇒ ¬A’)
The extension is still non conservative since one can prove Lemma 1. However Theorem 1 does not hold nor all its unpleasant consequences. In section 7 we prove that the proposed extension is consistent. We have several reasons to believe that Lemma 1 is reasonable and desirable. First of all it can be intuitively justified. In case PR is used to represent beliefs, i.e. PR(‘Γ’, ‘A’) is read as “A is one of the beliefs of an agent whose assumptions are Γ”, then Lemma 1 properly models the following property of beliefs: “if an agent holds belief A, then either some of the agent’s assumptions are false or belief A is true”.
⇒ (ΓV ⇒ (Γ ⇒ ¬A)) ⇒ (Γ ⇒ ¬A) Similarly the answer of a Doo would be false: Says(Doo, ‘Says(Voo, A)’) ⇒ PR(‘ΓD’, ‘¬Says(Voo, A)’) ⇒ PR(‘ΓD’, ‘¬PR(Γ, A)’) ⇒ PR(‘ΓD’, ‘PR(Γ, ¬A)’) ⇒ PR(‘ΓD’, ‘Γ ⇒ ¬A’) ⇒ (ΓD ⇒ (Γ ⇒ ¬A))
In the following section we present an example where the lemma is used to draw conclusions based on an agent’s reliability or lack of reliability. Lemma 1 can also be regarded as a two argument counterpart of axiom T of modal logics, i.e. ( A ⇒⊇A). However, while Lemma 1 is perfectly reasonable when PR is interpreted as belief, axiom T is not, and in fact is usually not included in modal logics for belief. Finally, we can observe that Lemma 1 looks right when PR is given an intuitionistic reading: in fact if the intuitionistic implication holds then the material implication holds as well but not vice versa.
6 AN EXAMPLE Using PR, we can provide a formalization of the following puzzle: The members of the tribe of Voo always speak the truth, while the members of the tribe of Doo always speak the contrary. An explorer meets a Voo and a Doo, and has to figure out which is which. By denoting with Γ the common knowledge that both Voos and Doos have, we can write: Says(Voo, x) ⇒ PR(‘ΓV’, x) Says(Doo, x) ⇒ PR(‘ΓD’, ¬x) where ΓV = Γ ∪ {Says(Doo, x) ⇔ PR(Γ, ¬x)} ΓD = Γ ∪ {Says(Voo, x) ⇔ PR(Γ, x)} If the explorer asks to one native what the other would answer to a yes/no question, the answer A he would get from a Voo would be false; in fact Says(Voo, ‘Says(Doo, A)’) ⇒ PR(‘ΓV’,‘Says(Doo, A)’) ⇒ PR(‘ΓV’, ‘PR(Γ, ¬A)’)
⇒ (Γ ⇒ ¬A) The step in the third line of the second part is justified by the choice of a yes/no question, for whose answer A it is either PR(Γ, A) or PR(Γ, ¬A). Such is the case for instance for the answer “I am a Voo”. Therefore a suitable question to solve the puzzle could be: “What would your friend answer to the question: «Are you a Voo?»”.
7 THE PROBLEM OF CONSISTENCY Many authors have discussed the implications of Montague’s negative results about inconsistency of amalgamated theories which have enough technical machinery to represent their own syntax2 and strong linking rules. In these theories a self referential formula R can be constructed such that: Lemma 4:
R ⇔ PR(‘¬ R’)
where PR(‘A’) is an abbreviation for PR({},‘A’). In order to show the incompleteness of the amalgam, a similar self referential formula was constructed in (Bowen and Kowalski, 1982). Theorem 2. (Montague) Let T be a theory such that: (i)
PR(‘A’) ⇒ A
(ii)
PR(‘PR(A) ⇒ A’)
(iii)
PR(‘A’), if
(iv) If
PC
A
PR(‘A ⇒ B’) and
PR(‘A’) then
PR(‘B’)
(v) T includes enough machinery to represent its own syntax 2 This is a minimal requirement that cannot be dispensed with
in any reasonable theory (see for example the arguments in (Perlis, 1985).
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then T is inconsistent.
IP
T(‘A ⇒ B’) ⇒ (T(‘A’) ⇒ T(‘B’))
Proof:
T
T(‘A’) ⇒ A
BAR
∀xT(‘A’) ⇒ T(‘∀xA’)
LNEC
_PC A _ T(‘A’)
PC
(R ⇒ ¬R) ⇒ ¬R
PC (PR(‘¬ R’) ⇒ ¬R) ⇒ ¬R
(tautology of PC) (Lemma 4)
PR(‘(PR(¬R) ⇒ ¬R) ⇒ ¬R’)
(iii)
PR(‘PR(¬R) ⇒ ¬R’)
(ii)
PR(‘¬R’)
(iv)
τ(A) = A, if A does not contain PR
(i)
τ(PR(‘A’, ‘B’)) = T(‘τ(A) ⇒ τ(B)’)
¬R R
(Lemma 4)
The translation between RT and TLN is defined recursively as follows:
τ(A ∧ B) = τ(A) ∧ τ(B)
■
τ(¬A) = ¬τ(A)
Corollary. The amalgamated logic with the inference rules RR2 is inconsistent.
τ(∀xA) = ∀xτ(A)
The theorem PR(‘A’) ⇒ A together with the inference rule _A _ PR(‘A’) also lead to inconsistency. In fact it is easy to show that conditions (i)-(v) of Theorem 2, can be derived (Montague, 1963). The restrictions introduced by RR3 reflection rules are enough to prevent application of Theorem 2. In fact Montague and Kaplan have shown that if any of the hypothesis (i)–(v) is dropped, Theorem 2 does not hold any longer. In our case, condition (i) follows from Lemma 1, condition (iii) is the weak version of reflect up, and condition (iv) is derivable with any version of the reflection rules. Condition (ii) however can be proved with rule (2–reflect up) but not with the weaker version (3–reflect up). Condition (ii) is not true for some sentences and therefore cannot be assumed as an axiom if we want to preserve consistency. This leads to the following: Theorem 3. The amalgamated logic RT is consistent.
We must show that all axioms and inference rules of RT hold in TLN. Considering 3–reflect up, we must show that: Γ _ PC A _ T('∧Γ ⇒ A' ) From the premise we obtain PC (∧Γ ⇒ A) by implication introduction, and hence we obtain the conclusion by means of LNEC. For (3-reflect down), which is equivalent to Lemma 1, we need to show: T(‘∧Γ ⇒ B’) ⇒ (∧Γ ⇒ B) which is just an instance of axiom T. All the axioms of RT which correspond to the axiomatization of provability, hold in TLN. Consider for instance the meta-level counterpart of Modus Ponens. We need to show that: T(‘∧Γ ⇒ A’) ∧ T(‘∧Γ ⇒ (A ⇒ B)’) ⇒ T(‘∧Γ ⇒ B’) Since PC
(∧Γ ⇒ (A ⇒ B)) ⇒ ((∧Γ ⇒ A) ⇒ (∧Γ ⇒ B))
from LNEC we obtain Proof: To prove consistency, we show that the logic RT can be translated into the logic TLN, which is a subset of the logic S4LN (Davies, 1990) proved consistent from results in (Turner, 1989). As a consequence RT is consistent. The logic TLN is a modal logic of truth, whose language includes the truth predicate T and the falsity predicate F. The axiomatization of TLN consists of standard axioms for first-order logic with equality and the following axioms for truth:
T(‘(∧Γ ⇒ (A ⇒B)) ⇒ ((⇒Γ ⇒ A) ⇒ (∧Γ ⇒ B))’) and then, using K T(‘(∧Γ ⇒ A) ⇒ (∧Γ ⇒ B)’) Finally, using K again, T(‘∧Γ ⇒ B’) Analogously, for any other inference rule and axiom it can be shown that their translation is provable in RT. ■
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The following theorem establishes correspondence between the two logics.
a
tighter
Theorem 4. RT and TLN are equivalent Proof: TLN can be translated in RT by means of the following mapping: ρ(T(‘A’)) = PR({}, ‘ρ(A)’) = PR(‘ρ(A)’)
because from the truth of PR(‘Γ’, ‘A’) one would be able A and therefore the truth of to infer that Γ PR({}, PR(‘Γ’, ‘A’)’), by reflecting up twice. This is however not a reasonable principle for belief, since it would amount to the following: “if an agent believes A then any other agent believes that the first agent believes A”.
IP’
PR(‘A ⇒B’) ⇒ (PR(‘A’) ⇒ PR(‘B’))
T’
PR(‘A’) ⇒ A
Davies proves consistent a stronger logic of syntactic modality which includes positive introspection. Based on this result and on the correspondence we have established, it would be safe to add positive introspection as an axiom to logic RT.
BAR’
∀xPR(‘A’) ⇒ PR(‘∀xA’)
8 RELATED WORK
LNEC’
_PC A _ PR(‘A’)
The translated axioms and inference rules are:
We can observe that IP’, T’ and LNEC’ are instances of the meta axiom for Modus Ponens, Lemma 1 and reflect up respectively. BAR’ can be derived using the object level rule for implication elimination and the meta-level axiom for implication introduction. ■ Our goal is to find a consistent amalgamated logic to serve as the basis for a syntactic treatment of modalities. Therefore we are looking for the most powerful consistent logic which is compatible with the interpretation of PR as belief. We have shown that the RR3 formulation of reflection rules results in a logic which is equivalent to the syntactical counterpart of the modal logic T. Modal logics for beliefs however usually include a positive introspection axiom (called S4 ), i.e. PR(‘Γ’, ‘A’) ⇒ PR(‘Γ’, ‘PR(Γ, A)’) This is widely accepted as a valid property of belief: “if one believes something, he believes that he believes it”. Positive introspection is not however a theorem of RT. In fact the truth of PR(‘Γ’, ‘A’) does not imply the existence of a proof; we can derive only a weaker form of positive introspection, namely If Γ
PC A
then PR({}, ‘PR(Γ, A)’)
This, at first sight, looks like a disadvantage of our logic over other logics which could, in an intuitionistic style, define PR as “derivable”. But in such logics, a stronger theorem is derivable, that is3 PR(‘Γ’, ‘A’) ⇒ PR({}, ‘PR(Γ, A)’)
3 This is not the case however for the Bowen-Kowalski’s
amalgamation.
Turner (Turner, 1989) shows that a number of choices are available to us for a syntactic treatment of truth and modalities. There are, after all, ways to circumvent, naturally enough, Montague’s negative results. Based on the work by Turner, Davies (Davies, 1990) proposes a first order theory of reasoning agents where truth, belief and knowledge are treated as syntactic predicates and which compares favourably with other similar proposals found in the literature. Our starting point has been an analysis of reflection rules and a focus on the provability relation. The logic RT, which is the result of our investigation, is very similar to one of the logics proposed by Turner and Davies and in fact we could establish a formal correspondence and base our proof of consistency on their results. We believe that a correct axiomatization of the provability relation can be the starting point also for a formal treatment of truth and other modalities; these, in the work of Turner and Davies, are introduced with separate axiom systems and then integrated. NProlog ((Gabbay and Reyle, 1985) and (Gabbay, 1985)) introduces hypothetical implications (→) in clauses and queries of logic programs. The goal P ? (A → B) is taken to be equivalent to the goal P + A ? B, that is B is a consequence of P extended with A. A theoretical motivation for introducing hypothetical implication is to give a definition to the provability predicate, in the context of an amalgamation of language and meta-language in the style of Bowen-Kowalski. In NProlog this is called NDemo and is defined according to the following: NDemo(‘P’, ‘A’) = Suspend and (∧P → A) and Restore where P represents a set of clauses (the program), A a formula, and ∧ P the conjunction of all the clauses of P. According to this definition, provability of A from P
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amounts to trying to prove A after asserting the conjunction of the clauses in P. The use of Suspend is “to get rid of the context” which could be used to derive A, and which could produce a positive answer even in the case that A is not derivable from P alone. Restore is for restoring the context after the evaluation of the goal. The effect of the right hand side of the definition is therefore equivalent to the existence of a proof (in logic programming) of A from P.
Omega proposal results in the logic RT, which is equivalent to a consistent logic of syntactic modality. In RT, a counterpart of axiom T can be derived which proves useful in some applications.
A definition without Suspend and Restore would have the effect of the (2-reflect up) reflection rule. In fact if NDemo(‘P’, ‘A’) is defined as (∧P → A), NDemo(‘P’, ‘A’) could be proved by asserting ∧P in the current database and proving A in the context of the extended database.
Acknowledgements
With the correction of Suspend and Restore however, NDemo succeeds only if the goal of proving A from ∧P alone succeeds, which is similar to the effect of (1reflect up). We have seen that in the Bowen-Kowalski amalgamation, assertions (non logical axioms) concerning PR at the meta-level do not have influence on the object level, due to the restricted form of the rule (1-reflect down). The situation with NProlog is different. According to the above definition, the assertion NDemo(‘P’, ‘A’) amounts to imposing that the goal P ? A be successful. What if this is not the case? To conclude, we believe that the only reasonable effect that an assertion PR(‘Γ’, ‘A’) can have on the object level is the one that our Lemma 1 sanctions, that is, the weak semantic constraint (as opposed to “syntactic constraint” concerning derivation) that when Γ is true then A is true.
9 CONCLUSIONS We argued about the advantages of an amalgamated approach and about the utility of reflection rules which result in a non conservative extension. We have presented an analysis of reflection rules and discussed the implications of seemingly slight variations in their formulation, specifically in the context of an amalgamated approach. In the Bowen and Kowalski amalgamation, the interpretation that can be given to PR is intuitionistic: PR(‘Γ’, ‘A’) means Γ A. At the other extreme the Omega approach, which results in PR being equivalent to the classical material implication, is shown to lead to inconsistency. We have shown that a version of the reflection rules which lays in between the Bowen-Kowalski and the
We think that logic RT is a better candidate to serve as the basis for a theory of viewpoints, where belief is treated as provability in a viewpoint. This aspect will be developed in future work.
We are grateful to Daniele Nardi for expressing doubts about the correctness of the proof of the three wise men in (Attardi and Simi, 1984) and to Piero Bonatti for helpful hints. We owe to one of the referees the suggestion that the different formulations of the reflection rules correspond to modal and intuitionistic logics and for bringing to our attention the work on NProlog. References L. C. Aiello, D. Nardi, M. Schaerf (1989) “Reasoning about Knowledge and Ignorance”, Proc. of Int'l Conf. on Fifth Generation Computer Systems, 618–627. G. Attardi, M. Simi (1984) “Metalanguage and Reasoning across Viewpoints”, in T. O'Shea (ed.), ECAI–84: Advances in Artificial Intelligence, Proc. of 6th European Conference on Artificial Intelligence. Amsterdam, Elsevier Science Publishers. K.A. Bowen, R.A. Kowalski (1982) “Amalgamating Language and Metalanguage In Logic Programming”, in K. Clark and S. Tarnlund (eds), Logic Programming , 153–172. Academic Press. N. Davies (1990) “Towards a First Order Theory of Reasoning Agents”, Proc. of 9th European Conference on Artificial Intelligence, Stockholm. D. M. Gabbay, U. Reyle (1985) “N-PROLOG: an Extension of PROLOG with hypothetical implications. I”, The Journal of Logic Programming, 1; 319-355. D. M. Gabbay, (1985) “N-PROLOG: an Extension of PROLOG with hypothetical implications. II. Logical Foundations, and Negation as Failure”, The Journal of Logic Programming, 4; 251-283. F. Giunchiglia, L. Serafini, (1990) “Multilanguage First Order Theories of Propositional Attitudes”, IRST Technical Report #9001-02, Trento. F. Giunchiglia, A. Smaill (1989) “Reflection in Constructive and Non-Constructive Automated Reasoning”, in Lloyd J. (ed.), Proc. Workshop on Meta-
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Programming in Logic Programming, Cambridge, MA: MIT Press. K. Konolige (1982) “A First-order Formalization of Knowledge and Action for a Multi-agent Planning System”, Machine Intelligence 10. J. McCarthy (1979) “First Order Theories of Individual Concepts and Propositions”, Machine Intelligence 9, 129–147. R. Montague (1963) “Syntactical Treatment of Modalities, with Corollaries on Reflexion Principles and Finite Axiomatizability”, Acta Philosoph. Fennica, (16), 153–167. R. C. Moore (1977) “Reasoning about Knowledge and Action”, Proc. of IJCAI–77, 223–227. Cambridge, MA: Morgan Kaufmann. D. Perlis (1985) “Languages with Self-Reference I: Foundations”, Artificial Intelligence, (25), 301–322. D. Perlis (1988) “Languages with Self–Reference II: Artificial Knowledge, Belief and Modality”, Intelligence, (34), 2, 179–212. J. R. Shoenfield (1967) Mathematical Logic. Reading, MA: Addison Wesley. R. Turner (1989) Truth and Modality for Knowledge Representation. London, Pitman Press. R. W. Weyhrauch (1980) “Prolegomena to a Theory of Mechanized Formal Reasoning”, Artificial Intelligence, (13), 133–170.
