PLAUSIBLE REASONING EXPRESSED BY p-CONSEQUENCE

In this paper we present a formal way of describing plausible (or non-deductive) reasoning. Ajdukiewicz’s distinction between deductive and non-deductive reasoning [1] provides our theoretical framework. Our formal approach is given by so called operation of p-consequence, which has been described earlier e.g. in [2]. At every stage of our work we try to show that Ajdukiewicz’s framework is relevant for our investigations. In the last paragraph axiomatization of ”plausible” counterpart of Lukasiewicz’s many valued logics is given. 1. Assume that a propositional language L = (L, f1 . . . , fn) is given. For the purposes of this paper it is assumed that L is fixed. Definition 1. By p-consequence operation on the language L we understand any operation that fulfills the following two conditions (for every X,Y ⊆ L): (i) X ⊆ Z(X) (reflexivity) (ii) X ⊆ Y implies Z(X) ⊆ Z(Y ) (monotonicity). It is well known that if we add to the definition of p-consequence an idempotency condition: Z(Z(X)) ⊆ Z(X) then we obtain an ordinary 162 Szymon Frankowski notion of logical consequence. The latter is a formal counterpart of deductive reasoning that is specific for formal sciences. In fact, idempotency of operation can be expressed by the statement conclusions of conclusions are conclusions, as well. In other words, immediate consequences of given premisses have the same (true) value as their premisses (see [1], §40), so two-steps reasoning has the same quality. In plausible reasoning things look different – if a sentence B is a conclusion of the set {A1, A2, . . . , An} then its value can be less then the value of the worst of premisses ([1], §44). So, if we add B to {A1, A2, . . . , An}, we automatically diminish the value of the set of premisses. Definition 2. By p-inference (for the language L) we shall mean finite and non empty sequence (〈α1, x1〉, 〈α2, x2〉, . . . , 〈αk, xk〉), where αi is a formula and xi ∈ {∗, 1} for i = 1, 2, . . . , k. A symbol xi occurring in a pair, indicates a degree of certainty of formula αi. Thus, if a pair is of the form 〈β, 1〉, then the formula is understood as “absolutely” true or “accepted with a maximal degree”. When a formula γ is followed by the symbol *, then one can read this as “γ is plausible” or “γ is not rejected”. In p-inference (〈α1, x1〉, 〈α2, x2〉, . . . , 〈αk, xk〉) a subsequence (〈α1, x1〉, 〈α2, x2〉, . . . , 〈αk−1, xk−1〉) is a string of premisses and the last pair 〈αk, xx〉 (we remind that k ≥ 1) is a conclusion. p-rule is simply any set of pinferences. Now, we can put the definition of p-proof : Definition 3. By p-proof of formula α from the set of formulas X, based on the set of p-rules R we understand a sequence (a1, . . . , am) ∈ (L× {∗, 1}) such that: 1 am = (α, x) for some x ∈ {∗, 1}; 2 for every i ∈ {1, . . . , k}: 2.1 ai = 〈αi, 1〉 and αi ∈ X or 2.2 (b1, . . . , bk, ai) ∈ ⋃ R for some b1, . . . , bk ∈ {a1, . . . , ai−1}. The condition 2.1. in the above definition means that a formula taken from the set of initial premisses is treated as proven with maximal degree of certainty. Plausible Reasoning Expressed by p-Consequence 163 We put X ‖–R α iff there exists p-proof of α from the set X based on R. We say that p-consequence Z is finitary iff Z(X) = ⋃ {Z(Xf ) : Xf is finite subset of X}. Theorem 1. (See [2]) For every finitary p-consequence Z there exists a set of p-rules R for which X ‖–R α iff α ∈ Z(X) for every X ⊆ L, α ∈ L holds. It is easy to see that the opposite statement holds true, i.e. every operation ZR defined by α ∈ ZR(X) iff X ‖–R α is a finitary p-consequence. When we have X ‖–R α it is not possible to recognize whether α has been proved with 1 or with ∗. Similarly, like in the case of operation of p-consequence we lost an information about a strength of deductivity of α, that is we do not know if α is deductive or plausible conclusion from X. So, let us introduce the other relation: Definition 4. X =⇒R 〈α, x〉 iff there exists p-proof (a1, . . . , ak, 〈α, x〉) from the set X based on R. We leave without proof the following characterization of =⇒ relation: Theorem 2. For every families of p-rules R, S, sets of formulas X, Y : i) α ∈ X ⇒ (X =⇒R 〈α, 1〉) ii) (X ⊆ Y & X =⇒R a) ⇒ (Y =⇒R a) iii) (X =⇒R 〈β1, 1〉, . . . , 〈βk, 1〉) & {β1, . . . , βk} =⇒R a) ⇒ (X =⇒R a) iv) X =⇒R a iff (Xf =⇒R a) for some Xf ∈ Fin(X) v) R ⊆ S & (X =⇒R a) ⇒ (X =⇒S a). What is important point iii) is a counterpart of idempotency condition. In the case of =⇒ relation there does not occur any loss of information, which makes iii) holds true. Definition 5. p-rule r is derivable from the set of p-rules R (symb. r ∈ Der(R)) iff for every (a1, . . . , an) ∈ r there exist: p-inference (b1, . . . , bm), such that: 164 Szymon Frankowski (a1, . . . , an−1) = (b1, . . . , bk), (k < m), an = bm and for every i ∈ {k + 1, . . . ,m} the following holds: there exists (c1, . . . , cj) ∈ ⋃ R such that {c1, . . . , cj−1} ⊆ {b1, . . . , bi−1} and cj = bi. p-rule r is weakly derivable from the set R (symb. r ∈ D̂er(R)) iff for every (a1, . . . , an) ∈ r, X ⊆ L, condition X =⇒R a1, . . . , an−1 implies X =⇒R an. Fact 1. i) R ⊆ Der(R) ⊆ D̂er(R). ii) =⇒R==⇒Der(R)==⇒D̂erR iii) R ⊆ S ⇒ Der(R) ⊆ Der(S) & D̂er(R) ⊆ D̂er(S) iv) Der(Der(R)) ⊆ Der(R) & D̂er(D̂er(R)) ⊆ D̂er(R). It can be shown that the inclusions from i) can be strict.