Rasiowa-Sikorski proof system for the non-fregean sentential logic SCI Joanna Golińska-Pilarek National Institute of Telecommunications, Warsaw, J.Golinska-Pilarek@itl.waw.pl We will present complete and sound proof system for the non-fregean sentential logic SCI in the style of Rasiowa-Sikorski. It provides a natural deduction-style method of reasoning for SCI. The non-fregean logic The sentential logic SCI is obtained from the classical sentential calculus by adding a new identity connective (different from ) and axioms which say α β means α is identical to β. From the axioms for it follows that the range of sentences has at least two elements. No other special presupposition about the meaning of is identical to nor the range of sentences are assumed. Any additional conditions for the range or the nature of connectives lead to an extension of SCI. In other words, SCI seems to be as weak as it is possible. Most of known sentential calculus - classical, modal, intuitionistic - are extensions of SCI. In this section we present the basic definitions of the non-fregean sentential logic. They can be found in [1], [7] and [8], among others. The SCI-language contains the following symbols: sentential variables α k, for k ω, written also as α, β, γ,......, truth-functional connectives,, and identity connective. On SCI-language the consequence operation C in Tarski s sense generated by the Modus Ponens rule, logical axioms for truth-functional connectives and axioms for identity connective, is defined. Axioms for identity connective (A1) α α, (A2) (α β) ( α β), (A3) (α β) ( α β), (A4) [(α β) (γ δ)] [(α#γ) (β#δ)], for # {,, }.
A partial SCI-model is a structure M = (M,,,,, D), where (M,,,, ) is an arbitrary algebra of the same type as the SCIlanguage and D is a subset of M such that for all a, b M: a D a D a b D a D or b D a b D a, b D A partial SCI-model is an SCI-model whenever it satisfies: for all a, b M a b D a = b Let M be an (partial) SCI-model. A valuation in M is any homomorphism h : L M. An SCI-formula ϕ is satisfied by h in M (for short M = h ϕ) if h(ϕ) D, and a set X of SCI-formulas is satisfied by h in M (for short M = h X) if every ϕ X is satisfied by h in M. An SCI-formula ϕ is satisfiable in M if it is satisfied by some M-valuation and true in M if it is satisfied by all M-valuations. A formula ϕ is SCI-valid if it is true in all SCI-models. Rasiowa-Sikorski proof system Rasiowa-Sikorski proof systems are validity checkers in a natural deduction style. They contains axiomatic sets of formulas and rules that apply to finite sets of formulas. Rules preserve and reflect validity of sets of formulas which are their premisses and conclusions, where validity of a set means validity of the disjunction of its formulas. To prove validity of a formula we build a finitely branching tree with this formula at the root and decompose it to simpler ones by applying the rules until all the branches are with an axiomatic set or there is an infinite branch that cannot be. This technique of deduction was first formalized for first-order logic in [6]. However Rasiowa-Sikorski proof systems can be also applied to a wide variety of logics, e.g. first-order, modal or relation. They were studied for example in [3], [4], [5]. Let ϕ, ψ be SCI-formulas. Rasiowa-Sikorski proof system for SCI contains the following decomposition rules: 2
{ϕ ψ} (RS ) {ϕ, ψ} (RS ) {ϕ ψ} {ϕ} {ψ} (RS ) (RS ) { (ϕ ψ)} { ϕ} { ψ} { (ϕ ψ)} { ϕ, ψ} (RS ) { ϕ} {ϕ} and the following specific rules: (RS ) {ϕ(α)} {α β, ϕ(α)} {ϕ(β), ϕ(α)} (RS cut) {ϕ} { ϕ} β is any variable, ϕ is any SCI-formula A finite set of formulas is RS axiomatic whenever it includes a subset (AX1) or (AX2), where (AX1) (AX2) {ϕ ϕ}, {ϕ, ϕ}, where ϕ is any SCI-formula. A finite set of formulas {ϕ 1,..., ϕ n } is called an RS set whenever the disjunction of its elements is SCI-valid. It means that, (comma) is interpreted as disjunction and (branching) as conjunction. ( ) K K A rule, is RS correct whenever K is an RS set iff H 1 H 1 H 2 H and H 2 are RS sets (K is an RS set iff H is an RS set). Let ϕ be an SCI-formula. An RS proof tree for ϕ is a finitely branching tree whose nodes are sets of formulas satisfying the following conditions: the formula ϕ is at the root of this tree, each node except the root is obtained by an application of an RS rule to its predecessor node, a node does not have successors whenever it is an RS axiomatic set. 3
A branch of an RS proof tree is said to be whenever it contains a node with an RS axiomatic set of formulas. An RS proof tree is whenever all of its branches are. A formula ϕ is RS provable whenever there is a RS proof tree for ϕ. Example 1. The following tree is a RS proof tree for the axiom (A4) in the case when # = (formula to which a rule is applied is underlined): [(α β) (γ δ)] [] (RS ) [(α β) (γ δ)], [] (RS ) (α β), (γ δ), (α β), (γ δ), α β, (RS ) (α β), (γ δ), (β γ) (β δ), (RS ) (α β), (γ δ), γ δ, (β γ) (β δ), We will prove that RS system for SCI is sound and complete: (α β), (γ δ), (β δ) (β δ), (β γ) (β δ), Theorem 1 (Soundness and Completeness). A formula ϕ is SCI-valid iff there is a RS proof tree for ϕ.
References [1] Bloom S. L., Suszko R., Investigation into the sentential calculus with identity, Notre Dame Journal of Formal Logic 13/3 (1972), pp. 289-308 [2] Golińska-Pilarek J., Huuskonen T., Number of Extensions of Non-Fregean Logics, The Journal of Philosophical Logic 2005,Vol. 34, No. 2, pp. 193-206 [3] Konikowska B., Rasiowa-Sikorski Deduction Systems - Foundations and Applications, in: R. Dyckhoff (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Proceeding of the International Conference Tableaux 2000, St Andrews, Scotland, 2000 [4] Or lowska E., Relational proof system for relevant logics, Journal of Symbolic Logic 57 (1992), 1425-1440 [5] Or lowska E., Relational proof system for modal logics,in: H. Wansing (ed.), Proof Theory of Modal Logics, Kluwer, Dordrecht, 1996, pp. 55-77 [6] Rasiowa H., Sikorski R., The Mathematics of Metamathematics, Polish Science Publishers, Warsaw 1963 [7] Suszko R., Adequate models for the non-fregean sentential calculus SCI, w: Logic, language and probability. A selections of papers of the 4 th International Congress for Logic, Methodology and Philosophy, Reidel Pub. Company, Dordrecht-Holland (1973), pp. 49-54 [8] Suszko R., Abolition of the Fregean axiom, Lectures Notes in Mathematics 453 (1975), pp. 169-239, 5