INTERNAL AND EXTERNAL LOGIC

Transcription

1 Bulletin of the Section of Logic Volume 17:3/4 (1988) pp reedition 2005 [original edition pp ] V. A. Smirnov INTERNAL AND EXTERNAL LOGIC In an essential way I make use of Frege s and Vasilev s ideas. N. A. Vasilev distinguished two levels in a logic. The abstract (external) logic depends on gnoeologic assumptions while the empirical (internal) part of logic depends on ontological ones. Vasilev did not change the external logic but he did change the internal one. His system can be viewed as a non-standard syllogistics based on classical propositional logic (see [5] [6]). Vasilev s ideas become plain if we discern cleary acts of predication (the synthesis of a property with an object) and acts of assertion the relation of mental contents with the way things are. The second sourse of the idea seems to Frege s differentation of mental process (Gedanke) and assertion statement (Urteil). As a symbol of assertion for Frege serves a vertical stroke. A horizontal stroke following it is correlated with holding a proposition it makes the sign determined everywhere. In the string of signs the one following the vertical stroke does not express a proposition it describes a possible state of affairs. Getting started from those ideas I introduce several combined propositional calculi of events. In [6] I examined a case when the external logic did not change. Here we are going to examine both the case when the internal logic changes and the one when the external does. We describe a language of a combined calculi of sentences and events. Let p q... be event s variables. We assume that event s variables make terms. If a and b are terms then (a b) (a b) a are terms as well. If a is a term then Θa is a formula; if α and β are formulas then (α&β) (α β) (α β) α are formulas. The connectives: I call internal and & external ones. Let W bs a non-empty set of possible worlds. Let Π be a class of subsets of W closed under the operations of union and. < Π >

2 Internal and External Logic 171 can be a lattice a bounded lattice a distributive lattice (it is possible to examine any algebra). Events will be identified with pairs of sets of possible worlds i.e. with elements of Π Π. Let ϕ be a function assigning to events variables elements from Π Π; ϕ 1 (p) is the first part of ϕ(p) and ϕ 2 (p) the second one. The function ϕ we shall extend for all terms as follows 1). ϕ 1 (a b) = ϕ 1 (a) ϕ 1 (b) ϕ 1 (a b) = ϕ 1 (a) ϕ 1 (b) ϕ 1 ( a) = ϕ 2 (a) ϕ 2 (a b) = ϕ 2 (a) ϕ 2 (b) ϕ 2 (a b) = ϕ 2 (a) ϕ 2 (b) ϕ 2 ( a) = ϕ 1 (a). Let T be a set of subjective points of reference (in particular of time moments of propositions) and R a binary relation on T. The model structure is a system M = << T R > < W Π > d > where d is a function from T to W. Now we are going to describe the notion of substitution of a formula A in a moment t of a modal structure M relative to an assignment ϕ t = ϕ Θa t 1 (Rtt 1 d(t 1 ) ϕ 1 (a)) t = ϕ α&β t = ϕ α and t = ϕ β t = ϕ α β t = ϕ α or t = ϕ β t = ϕ α β t 1 (Rtt 1 (t 1 = ϕ α t 1 = ϕ β)) t = ϕ α t 1 (Rtt 1 t 1 = ϕ α). We can introduce the notion of falsity as well: t ϕ = Θa t 1 (Rtt 1 d(t 1 ) ϕ 2 (a)) and analogously for external connectives. The notion of falsity for classical and intuitionistic external logic is not necessary; but still we need it for the cases of de Morgan logic as well as for constructive one with strong negation. If the relation R is the relation of partial order we will obtain the case of external intuitionistic logic; if R is an identity we will obtain the classical one.

3 172 V. A. Smirnov Now we consider the case when the external logic is classical and the internal one is de Morgan lattice i.e. the system CM. In this case we need R to be the identity and < Π > a distributive lattice. Axiom schemata: W 0. Axiom schemas to classical sentential logic W 1. Θ(a b) Θa & Θb W 2. Θ(a b) Θa Θb W 3. Θ (a b) Θ a Θ b W 4. Θ (a b) Θ a & Θ b W 5. Θ a Θa. As the unique rule of inference will serve Modus Ponens. It is possible to give an elegant formulation of this system in a sequential mode. General sequences: Θa Γ Θa. The logical figures of inferences for internal connectives: Γ Θa Γ Θb Γ Θ(a b) Γ Θ a Θ b Γ Θ (a b) Γ Θa Θb Γ Θ(a b) Γ Θ a Γ Θ b Γ Θ (a b) Γ Θa Γ Θ a Θa Θb Γ Θ(a b) Γ Θ a Γ Θ b Γ Θ (a b) Γ Θa Γ Θb Γ Θ(a b) Γ Θ a Θ b Γ Θ (a b) Γ

4 Internal and External Logic 173 Θa Γ Θ a Γ. The figures for external connectives are standard as well as structural rules of inference. If we demand that ϕ 1 (p) ϕ 2 (p) = then we obtain the Hao Wan logic; in this case we add to axioms W 0-W 5 the following one: W 6. (Θa & Θ a) and to basic sequences: Θa Θ a Γ. If we assume ϕ 1 (p) ϕ 2 (o) = W then we obtain a dual Hao Wan logic; in this case we add: W 7. Θa Θ a and to basic sequences we add Γ Θa Θ a. If we take both above assumptions we obtain a Boolean algebra as an internal logic. In that case Θ a Θa so the distinction between internal and external logic is inessential. If we want to have de Morgan logic as the external logic then except of & and it is necessary to introduce external negation and to define together with the notion of truth the notion falsity: t ϕ = Θa t 1 (Rtt 1 d(t 1 ) ϕ 2 (a)). The relation R is identity so t ϕ = Θa d(t) ϕ 2 (a) t ϕ = α&β t ϕ = α or t ϕ = β t ϕ = α β t ϕ = α and t ϕ = β t ϕ = α t = ϕ α. For the negation we need to add conditions for truth: t = ϕ α t ϕ = α. In the sequential form the system MM is described as follows. Basic sequences: Γ Θa Θa Γ Θa Θa.

5 174 V. A. Smirnov The logical figures of inferences for internal connectives are the same as for the system CM and for the external connectives are the following: Γ α Γ β Γ α&β Γ α β Γ (α&β) Γ δ α β Γ δ α β Γ α Γ β Γ (α β) Γ α Γ α α β Γ α&β Γ α Γ β Γ (α&β) Γ α Γ β Γ α β Γ α β Γ (α β) Γ α Γ α Γ. The structural figures of inferences are as usual. Adding to basic sequences Θa Θ a Γ and Γ Θa Θ a we obtain correspondingly CW and CDW but if both hold then CW ; adding Θa Θa Γ and Γ Θa Θa or both we have correspondingly the external logic W DW or C. We note three questions rising from the approach presented. The first one is the problem of events identity. If a and b are terms then a = b is a formula. The formula a = b is true relative to ϕ iff ϕ 1 (a) = ϕ 1 (b). The question is: is the rule Θa Θb valid? It is valid if we assume that the a = b function d is the counterimage from T on W i.e. w t(d(t) = w). The second questions concerns iteration of the act of asserting. According to the presented approach such iteration is meaningless. Neverthe-

6 Internal and External Logic 175 less the very asserting becomes an event and talking about it is meaningful. That is why it is reasonable to enrich our language as follows. In the definition of a formula we add one more condition: if α is a formula then [α] is a term. In the sequential system CM we add the following figures of inference: Γ α Γ Θ[α] α Γ Γ Θ [α] α Γ Θ[α] Γ Γ α Θ [α] Γ. The system obtained is up to designation equivalent to von Wright system. The system mentioned will be published in Proceedings of V Soviet-Finish Logic Colloquium. The last third question concerns understanding of relations. It is natural to regard a relation as a function from Cartesian product of individual domain to an algebra of events. Then R(a 1... a n ) is not an assertion but as an expression denoting event. It is naturally neither true nor false. We can regard Θ as an operator assigning assertions to events but Θa does mean nothing. Analogously the sign of asserting identity of events = assigns to a couple of events an assertion and a = b does mean nothing. The presented way of understanding relations as images in an algebra of events was used in algebra (see [4]). In this context rises the question of distinction of internal and external quantifiers. References [1] G. Frege Begriffsschrift und Ondere Ausätze Zweite Aulage Darmrfadt [2] N. A. Vasilev Voobrazemaja logika [3] N. A. Vasilev Logika i metalogika 1913.

7 176 V. A. Smirnov [4] V. N. Salij Reschetki s edinstvennymi dopolneniami M [5] V. A. Smirnov Axiomatizacja logiceskih sistem N. A. Vasileva [in:] Sovremennaja logika i metodologia nauki M pp [6] V. A. Smirnov Logiceskije metody analiza nauc novo znania M [7] E. D. Smirnova Logiceskaja semantika i filosofskie osnovania logiki M Notes: 1) If a language contains pseudodifference and pseudocomplement then: ϕ 1 (a b) = ϕ 1 (a) ϕ 1 (b) ϕ 1 (a b) = ϕ 1 (a) ϕ 1 (b) ϕ 2 (a b) = ϕ 2 (b) ϕ 2 (a) ϕ 2 (a b) = ϕ 2 (b) ϕ 2 (a).