https://doi.org/10.1007/s00153-018-0626-3
Mathematical Logic
Sequent calculus for classical logic probabilized
Marija Boriˇci´c1
Received: 8 December 2017 / Accepted: 25 April 2018 / Published online: 7 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treat-ment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence prob-ability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ’probabilized sequent’ ba with the intended meaning that “the probability of truthfulness of belongs to the interval [a, b]”. This method makes it possible to define a system of derivations based on ’axioms’ of the form i baii i, obtained as a result of empirical research, and then infer conclusions of the
form ba. We discuss the consistency, define the models, and prove the soundness and completeness for the defined probabilized sequent calculus.
Keywords Consistency· Sequent calculus · Probability · Soundness · Completeness Mathematics Subject Classification 03B48· 03B50 · 03B05
1 Introduction
Wagner’s paper on probabilistic versions of modus ponens and modus tollens infer-ence rules (see Wagner [38]) immediately influenced this work. Our first step was to
The results contained in this paper were partially presented at the European Summer Meetings of Association for Symbolic Logic, Logic Colloquium 2012, held in Manchester on July 12–18, 2012, and Logic Colloquium 2014, held in Vienna on July 14–19, 2014. This research was supported in part by the Ministry of Education, Science and Technology of Serbia, Grant number 174026.
B
Marija Boriˇci´cgeneralize Wagner’s results in order to get a corresponding probabilization of hypo-thetical syllogism rule (see Boriˇci´c [4]), and the next one was the probabilization of a complete inference rules system of classical propositional calculus (see Boriˇci´c [6]). The main result of this paper can be considered a sound and complete prob-abilization LKprob of Gentzen’s sequent calculus LK for classical propositional logic.
The presence of the notion of a sentence’s truthfulness probability is evident from the very beginning of modern logic development [3,36,37]. It should be noted that Keynes [17] was one of the first authors who explicitly considered probabilities of propositions, but, unfortunately, his work did not have any significant influence on the development of this area. Probability functions were definitely introduced in modern logic in the works of Carnap and Popper (see Carnap [8], Popper [31], Leblanc and van Fraassen [18], and Leblanc [19,20]) and established a new logical discipline— probability logic. On the other hand, the sequent calculus was originally introduced by Gentzen (see Gentzen [12] or Takeuti [34]), with the intention of incorporating the deducibility relation into the object language of a logical system. The notion of sequent, through the elegant Gentzen’s formulations of classical LK and intuitionistic LJ logic, with the cut-elimination theorem, founded the proof theory. Bearing in mind that each sequent , where and are finite sequences (words) (possibly empty) of formulae, is interpreted as a formula → consisting of conjunction of formulae appearing in and disjunction of formulae appearing in , the sequent calculus can be considered a generalized calculus for Horn clauses (see Horn [15]) playing an important role in logic programming.
A perfect modern introduction to the theory of probabilized inference rules is given by Frisch and Haddawy (see Frisch and Haddawy [11]) and Wagner (see Wagner [38]) connecting their work with philosophical background of earlier authors and attempts, such as Adams, Hailperin, Lewis, Nilsson and Suppes (see Adams [1,2], Hailperin [13,14], Lewis [21,22], Nilsson [25,26] and Suppes [33]). Let us also mention here a particularly inspirational, extensive and valuable work by Fagin, Halpern and Megiddo (see Fagin et al. [10]). Finally, let us stress here that the following sources directly influ-enced our work: Frisch’s and Haddawy’s (see Frisch and Haddawy [11]) and Wagner’s (see Wagner [38]) investigations of conditional probability’s deductive properties, as well as Ognjanovi´c’s and Raškovi´c’s (see Ognjanovi´c and Raškovi´c [27] and Raškovi´c [32]) development of semantics for logic with probability operators over formulae, including an extensive overview of probability logics by Ognjanovi´c et al. [28,29]. Unlike them, we consider simple probability’s deductive properties in the context of a kind of labelled sequent calculus (see Szabo [35]). The models we define have a lot in common with Carnap’s (see Carnap [8]), Popper’s (see Popper [31]) and Leblanc’s (see Leblanc [19,20]) approach. Let us emphasize that, although Poper and Carnap do not belong to the same school of thought, in this field they have absolutely common ideas and equivalent approaches (see Leblanc and van Fraassen [18]).
connection between sequent calculi and probability, except our paper [7] concerning the sequents with high probabilities, where the basic form of probability intervals is [1−ε, 1], for a sufficiently small positive real number ε, according to Suppes’ approach [33,38], while, in this paper we deal with intervals of the form[a, b] ⊆ [0, 1].
When the propositional language is extended by some probability operators, e.g. Pr
(r ∈ I , where I is a finite subset of reals [0, 1] containing 0 and 1), then, usually, the iterations of these operators are allowed. In this case there is a possibility to express not only the statement that “the probability of A is greater than or equal to r ” as “PrA”,
but also to express the statement that “the probability of PrA is greater than or equal to s” as “PsPrA” (see Ognjanovi´c and Raškovi´c [27], Ikodinovi´c [16] and Ognjanovi´c
et al. [28]). Such an approach is absolutely in the spirit of the logical tradition of treating the unary logical operators as in modal logic. But on the other hand, our basic information regarding the tradition of probability theory is connected with one level only: if A is an event, then P(A) ≥ r means that “the probability of event A is greater than or equal to r ” and P{P(A) ≥ r} ≥ s, although a meaningful statement, does not play an important role in the context of probability theory, requiring the extension of probability space by an inductive definition claiming that if A is an event, then {P(A) ≥ r} is an event as well.
In this paper we try to combine, in the framework of the unique system, the two above—mentioned concepts: sentence probability, and the deduction relation formal-ized in a sequent calculus. In our approach to ‘probabilformal-ized sequents’ we enlarge the notion of sequent with the aim of describing formally the calculus based on the sequents of the form ba with the intended meaning of “the probability of derivability of belongs to the set [a, b] ∩ I ”, where I is a finite subset of reals [0, 1] (closed under addition), or, bearing in mind an expected kind of completeness theorem, that “the probability of truthfulness of the formula → equals c ∈ [a, b]∩ I ”. For simplicity’s sake, henceforth, we will use[a, b] instead of [a, b]∩ I . Let us emphasize that the aim of this paper is not an orthodox proof theoretical treatment of probability logics, although we believe that this would make sense in systems with sequents of high probabilities, including the possibility of obtaining a kind of the cut-elimination theorem.
We define a Gentzen-type system LKprob, a modification of Gentzen’s original sequent calculus LK for classical propositional logic, enabling one to work with the-ories based on facts of the formi baii iwhich are obtained as a result of empirical
research. Namely, our system of probabilized sequents defines the derivation rules for inferring the conclusions of the form ba , from a finite list of hypotheses i baii i. Also, LKprob can be considered a program producing the set of all
deductive consequences LKprob(σ1, . . . , σn) of a finite list (σ1, . . . , σn), where σi (1 ≤ i ≤ n) is a sequent of the form i baii i (1 ≤ i ≤ n), treated as
addi-tional axioms of LKprob. Namely, for an input(σ1, . . . , σn), by using LKprob, we
infer an output LKprob(σ1, . . . , σn). Our aim was to develop probabilities over the
possible to obtain a probabilized version LJprob of intuitionistic sequent calculus LJ in a similar way.
Compared with anthological paper by Fagin, Halpern and Megiddo (see Fagin et al. [10]) where consideration is clearly divided into three parts: propositional reasoning, reasoning about linear inequalities, and reasoning about probabilities, our aim is just to deal with the third part—probabilistic reasoning.
Dominant type of semantics for (classical) probability logic in the literature is the Kripke’s possible worlds semantics (see Fagin et al. [10], Nilsson [25,26], Ognjanovi´c and Raškovi´c [27], Ikodinovi´c [16], Ognjanovi´c et al. [28] and Raškovi´c [32]), but here, unexpectedly, we obtain a simpler type of semantics. Our approach is fully justified by the corresponding soundness and completeness results presented in this paper.
The paper is organized as follows. In the first part, a system LKprob of probabi-lized sequents is introduced. The second part deals with models and inference rules justification, proving, in fact, the soundness of our system with respect to the given semantics, with particular attention paid to the probabilistic version of the cut rule. The final paragraphs are concerned with consistency, soundness and completeness for the probabilized sequent calculus.
2 A system of probabilized sequents
The sequent , as introduced by Gentzen, consists of two finite (possibly empty) sequences (or words) of formulae—the antecedent, and —the consequent, with the main interpretation as →, where denotes the conjunction of all formulae appearing in, and denotes the disjunction of all formulae appearing in; particularly, if or is an empty sequence, then is interpreted as, as¬, and can be understood as a pure contradiction. By extending Gentzen’s calculus for classical propositional logic LK, we introduce a sequent system LKprob over the set of propositional formulae as follows. For any sequent of LK we suppose that, for any a, b ∈ I , ba is a sequent of LKprob, where the essential
meaning of b
a is that “there exists c ∈ [a, b] such that the probability of
is equal to c”. In cases when a = 1 or b = 0, for ba and a ≤ b we use the following abbreviations 1 and 0, respectively, and in the case of a > b we treat ∅ as pure contradiction.
The axioms of LKprob are the following three sequents: 1
0 0 A1 A for any words and , and any formula A.
The structural rules of LKprob are as follows:
permutation: AB b a B A b a (P b a) b aAB b aB A (b a P) contraction: AA b a A b a (C b a) b a A A b a A (b aC)
for any a, b ∈ I , the cut rule: b
a A A dc min(b+d,1)
max(0,a+c−1)
(cut[a,b][c,d])
for any a, b, c, d ∈ I , and the following specific structural rules:
weakening: b a dc A A min(1,b+1−c) max(a,1−d) (W b a) b a dc A min(1,b+d) max(a,c) A (b a W) for any a, b, c, d ∈ I , monotonicity: b a d c (M ↑) ba dc min(b,d) max(a,c) (M ↓)
for any[a, b] ⊆ [c, d], and any a ≤ b and c ≤ d, respectively, for (M ↑) and (M ↓), and the following specific rule regarding additivity:
A B1 ba A dc B
min(1,b+d) min(1,a+c) A B
(ADD)
These rules, and some rules that follow, can be immediately justified by using the well known Boole’s and Bonferroni’s inequalities: fromba A anddc B, we can infer
The following rule, regarding inconsistency: ∅ ∅ (⊥)
is directly inspired by Frisch and Haddawy (see Frisch and Haddawy [11]), by giving more elegant derivation system and status of consistency.
Let us note that the cut rule applied on the sequents ba A and A dc which probability intervals[a, b] and [c, d] are not in the neighborhood of 1 (or 0), gives a conclusion minmax(b+d,1)(0,a+c−1) with a very wide interval. For instance, from 12 1 2 A and A 12 1 2 we infer 1
0, i.e. a trivially provable sequent—an axiom. Note also that the additivity rule(ADD) makes no sense in case a + c > 1. Namely, from1a A and A B 1, by the cut rule we infer B 1a, i.e.
1−a
0 B. But, if 1
c B, then it must be that c < 1 − a, otherwise ∅ B. For this reason we believe
that there is more logical sense to consider a subsystem of LKprob concerning the probability intervals of the form[1 − ε, 1] for sequents provability, provided ε > 0 is sufficiently small real number (see Boriˇci´c [7]). Such approach was firstly developed by Suppes (see Suppes [33] and Wagner [38]).
The logical rules of LKprob are as follows: b a A ¬A b a (¬ b a) A b a b a¬A (b a¬) AB b a A ∧ B b a (∧ b a) b aA dcB min(b,d) max(0,a+c−1)A∧B( b a∧) A b a B dc A ∨ B min(b,d) max(0,a+c−1) (∨ b a) b a A B b a A∨ B (b a∨) b a A B dc A → B min(b,d) max(0,a+c−1) (→b a) A b a B b a A→ B (b a→)
The proofs in LKprob (or in LKprob(σ1, . . . , σn)) have a tree form, its initial
nodes consisting of axioms instances and each ramifying in such a way that the proof tree is made in accordance with some inference rule of LKprob. The sequent ba is provable in LKprob if there is a proof tree in LKprob ending with the sequent b
a. Particularly, we can consider the fact that there is the best an anytime proof for the sequent ba in LKprob if [a, b] is the tightest interval, such that ba is provable in LKprob i.e. [a, b] = {[x, y]| y
x is provable in LKprob}.
Obviously, in case that the tightest interval for a sequent is the empty set, i.e. that the sequent ∅ is LKprob(σ1, . . . , σn)—provable, we have an unfavorable
situation—an inconsistency of the system.
for any words and , where I = {c1, c2, . . . cm}, as an alternative to the axiom 1
0 . This kind of rules are typical for natural deduction systems (see Gentzen [12] or Prawitz [30]), but not for sequent calculi. The double lines appearing in the rule ( ∅) denote the form of derivations such that each appearance of [ c
c ]
means that it is possible to cancel cc in the set of hypotheses of derivation. More accurately, in this case the set of hypotheses is obtained by striking out some (or none, or all) occurrences, if any, of the sequent cc on the top of a derivation tree. This rule provides that, for each sequent , the probabilized sequent 10 can be treated as an axiom of LKprob, with the meaning that each probabilized sequent c
c holds, for some c ∈ [0, 1]. The rule ( ∅), accompanied with (⊥), should
make it possible to treat consistency of LKprob in a traditional way, meaning that the following two forms of inconsistency are equivalent: (a) there exists a formula A, such that both A and¬A are provable, and (b) every formula is provable.
In further text we present some examples which illustrate proofs in our system. The first two examples involve formula interpretations, while the others are more general. Example 1 Let the formulas A, B, C, D and E have the following interpretation: A—the person is a female, B—has a Bachelor’s degree, C—has Master’s degree or doctorate, D—has a high salary, E−owns at least one property. The results of a questionnaire are: (i) the probability of having a high salary, if you are a female who has a Bachelor’s degree is at least 0.873, (ii) the probability of owning at least one property, if you are a female who has a Bachelor’s degree is 0.794, and (iii) the probability of not having master’s or doctoral degree if you are a female is at least 0.951. That means that in our system LKprob, with I = {10−3k|k = 0, 1, . . . , 103}, the additional axioms are of the form (i) A B 10.873 D, (ii) A B 00.794.794 E and (iii)
A10.951¬C. Using the following proof
A B10.873 D A B00.794.794 E A B00.794.667 D∧ E ( ∧) A10.951¬C AC 10.951 (¬ ) 10 D∧ E AC 10.951 D∧ E ( W) A(B ∨ C) 0.794 0.618 D∧ E (∨ ) we can conclude that, if you are a female with Bachelor’s, master’s or doctoral degree, then the probability of having a high salary and owning at least one property belongs to the interval[0.618, 0.794].
Note that our system is particularly useful in situations when only some final results are available (hypothesis (i), (ii) and (iii)), not the hole database containing the ques-tionnaire responses.
in probability of patient having both allergies (D∧ E) or a cold (F) given that he is coughing ( A) and, sneezing or having a dermal rash (B ∨ C). Using our system LKprob, we calculate it as follows:
B00.95.95 D A B1 0.95D F (W , W) A00.97.97F G BG00.90.90E A B1 0.87F E (cut) A B10.82(D ∧ E)F (∧ )AC0.92 0.92(D ∧ E)F A(B ∨ C) 1 0.74(D ∧ E)F ( ∨)
meaning that the probability of patient having both allergies or a cold given that he is coughing and, sneezing or having a dermal rash belongs to the interval[0.74, 1]. Example 3 This example illustrates how the additivity rule can raise the probability. Let the additional axioms of LKprob, with I = {10−1k|k = 0, 1, . . . , 10}, be 00.4.4 A, 0.2
0.2 B, 0.3
0.3C, A B1, A∨ BC 1and A∨ B 0.9
0.9. We prove that the probability of C is in the interval [0.8, 1]: A B100.4.4 A 00.2.2B 0.6 0.6A∨ B (ADD)( ∨) 00.3.3C A∨ BC 1 0.9 0.9 A∨ BC (ADD)( ∨) A ∨ B 0.9 0.9 1 0.8C (cut) Example 4 This example illustrates how the rules( ∧), (¬ ) and (→) are used. Let the additional axioms of LKprob, with I = {10−2k|k = 0, 1, . . . , 102}, be 0.97 0.97 A, 0.90 0.90C and 0.93 0.93B. Then we have 0.97 0.97 A 0.90 0.90C 1 0.90C ( W) 1 0.87 A∧ C ( ∧) 0.93 0.93 B 0.93 0.93 B ( P) ¬B 0.93 0.93 (¬ ) (A ∧ C → ¬B) 1 0.80 (→) concluding that the probability of(A ∧C → ¬B) is in the interval [0.8, 1].
Let us give a few auxiliary statements regarding LKprob—provability:
Lemma 1 (a) (Probability of a complementary event) If the sequentba A is provable in LKprob, then11−a−b¬A is provable in LKprob; if the sequent A bais provable in LKprob, then¬A 11−a−bis provable in LKprob.
(b) (Additivity) If the sequents0 A∧ B, ba A anddc B are provable in LKprob, thenminmin(1,b+d)(1,a+c) A∨ B is provable in LKprob.
(c) If is provable in LK, then 1 is provable in LKprob.
Proof (a) This follows from the rules(¬ ba) and (ba ¬) and the weakening rule (W b
a).
(b) This follows from the additivity rule(ADD).
(c) By induction on the length of the proof for 1 in LKprob. (d) Immediately by the cut rule.
(e) Fromcc A and A1 B, by (d), we have1c B. From A 1−c
1−cand B 1 A, by (cut), we have B 11−c, i.e.c0 B, by (a). By monotonicity(M ↓), from 1c B and
c
0 B, we infercc B, i.e. c= d. ∇
Part (e) of the previous lemma justifies the principle that equiprovable sequents are
equiprobable.
Also, we obtain:
Corollary 2 (a) (Subadditivity) If the sequents0 A∧B, 1a A and1c B are provable in LKprob, then1min(1,a+c) A∨ B is provable in LKprob.
(b) (Superadditivity) If the sequentsb0 A andd0 B are provable in LKprob, then min(1,b+d)
0 A∨ B is provable in LKprob.
3 Models for probabilized sequents
Now we describe models for LKprob (see Boriˇci [5]). Let Seq be the set of all unlabelled sequents, i.e. of sequents of the form , and I a finite subset of reals [0, 1] closed under addition and containing 0 and 1. Then a mapping p : Seq → I will be a model, if it satisfies the following conditions:
(i) p(A A) = 1, for any formula A;
(ii) if p(AB ) = 1, then p( AB) = p( A) + p( B), for any formulae A and B; (iii) if sequents and are equivalent in LK, in sense that there are proofs for both sequents → → and → → in LK, then p( ) = p( ).
We have to emphasize that the above axioms roughly correspond to the Carnap’s and Popper’s sentence probability axioms (see Carnap [8] and Popper [31]), variations of which were considered by Leblanc and van Fraassen (see Leblanc and van Fraassen [18], and Leblanc [19,20]).
Let us note also that in case when the above condition (ii) is replaced by the following one: (iv) if A B is provable in LK, then p( AB) = p( A) + p( B), for any formulae A and B, the condition (iii) would be redundant in this definition, as consequence of (i) and (iv). For the reasons of better intuitiveness, we prefer the definition of model as given above.
Satisfiability in a model for the probabilized sequents is defined by clause: |p ba iff a ≤ p( ) ≤ b
and we say that the probabilized sequent ba is satisfied in a model p. A sequent b
a is valid iff it is satisfied in each model, and this is denoted by | ba.
Lemma 3 For any formulas A and B, the following equalities and inequalities hold: (a) p( ¬A) = 1 − p( A);
(b) p( AB) = p( A) + p( B) − p( A ∧ B); (c) p( AB) ≥ p( A);
(d) p(A B) ≤ p(A ) + p( B); (e) p(A A) = p(A ) + p( A);
(f) p( A) ≥ p( A ∧ B).
Proof (a) Using the properties of p (i), (ii), and (iii), we have 1= p(A A) = p( A¬A) = p( A) + p( ¬A).
(b) As (A∧B)(¬A∧B) and B are equivalent, and p(AB¬AB ) = 1, from (ii) and (iii) , we have p( B) = p( (A∧B)(¬A∧B)) = p( (A∧B))+p( (¬A∧B)). Also, AB and A(¬A ∧ B) are equivalent, and p(A¬AB ) = 1. By using (ii), (iii) and the previous fact, we have p( AB) = p( A(¬A ∧ B)) = p( A) + p( (¬A ∧ B)) = p( A) + p( B) − p( A ∧ B).
(c) Similarly, by using p(A¬A ) = 1, (i), (ii) and (iii) , we have 1 = p(A A) = p( A¬A) = p( A) + p( ¬A), and p( AB) = p( A(¬A ∧ B)) = p( A) + p( ¬A ∧ B) ≥ p( A).
(d) Follows immediately from (b), because p( ¬A ∧ B) ≥ 0. (e) Follows from (ii), and (iii).
(f) Follows from (b), and (c).
The following lemmata argues the soundness of LKprob with respect to the given models.
Lemma 4 For any formula A and each sequent , we have: (a) | 10;
(b) |0; (c) | A 1 A.
Proof (a)|p 10 holds for each p, because p( ) ∈ [0, 1], for each p. (b) 1 = p(A A) = p( A¬A) = p( A) + p(¬A) − p( A ∧ ¬A), by Lemma3(b), concluding that p( A ∧ ¬A) = 0, wherefrom, as A ∧ ¬A and are interdeducible in LK, p() = 0, i.e. |p0holds for each p.
(c) p(A A) = 1, so |p A1 A holds for each p.
By condition (iii) of our definition, we have immediately:
Lemma 5 For any formulas A and B, and each words, , and : (a) p( AB ) = p(B A );
(b) p( AB) = p( B A); (c) p( A ) = p( AA ); (d) p( A) = p( AA); (e) p( A) = p(¬A );
We continue with semantical justification of rules of our calculus.
Lemma 6 For any formulas A and B, and each words, , and , we have: (a) if a≤ p( ) ≤ b and c ≤ p( A) ≤ d, then
max(a, 1 − d) ≤ p( A ) ≤ min(1, b + 1 − c); (b) if a≤ p( ) ≤ b and c ≤ p( A) ≤ d, then
max(a, c) ≤ p( A) ≤ min(1, b + d); (c) if a≤ p( A) ≤ b and c ≤ p( B) ≤ d, then
max(0, a + c − 1) ≤ p( A ∧ B) ≤ min(b, d); (d) if a≤ p( A ) ≤ b and c ≤ p(B ) ≤ d, then
max(0, a + c − 1) ≤ p( A ∨ B ) ≤ min(b, d); (e) if a≤ p( A) ≤ b and c ≤ p(B ) ≤ d, then
max(0, a + c − 1) ≤ p( A → B ) ≤ min(b, d). Proof In each part of this proof we essentially use facts given in Lemma3.
(a) Suppose that p( ) ∈ [a, b] and p( A) ∈ [c, d], then p( A ) = p( ¬()¬A)
= p( ¬()) + p( ¬A) − p( (¬()) ∧ ¬A) = p( ) + 1 − p( A) − p( (¬()) ∧ ¬A) i.e. p( A ) ∈ [max(a, 1 − d), min(1, b + 1 − c)], by Lemma3(f). (b) Similarly as (a).
(c) Suppose that p( A) ∈ [a, b] and p( B) ∈ [c, d]. We have that p( (A ∧ B)) = p( (A ∧ B)¬())
= p( (A ∨ ∨ ¬()) ∧ (B ∨ ∨ ¬()))
= p( A¬()) + p( B¬()) − p( AB¬()) = p( A) + p( B) − p( AB¬())
(d) Suppose that p( A ) ∈ [a, b] and p(B ) ∈ [c, d]. We have that p((A ∨ B) ) = p( (¬A ∧ ¬B)¬())
= p( (¬A ∨ ∨ ¬()) ∧ (¬B ∨ ∨ ¬()))
= p( ¬A¬()) + p( ¬B¬()) − p( ¬A¬B¬()) = p( A ) + p(B ) − p( ¬A¬B¬())
Therefore, p((A ∨ B) ) ∈ [max(0, a + c − 1), min(b, d)].
(e) Similarly, if we suppose that p( A) ∈ [a, b] and p(B ) ∈ [c, d], then p((A → B) ) = p( (A ∧ ¬B)¬())
= p( (A ∨ ∨ ¬()) ∧ (¬B ∨ ∨ ¬()))
= p( A¬()) + p( ¬B¬()) − p( A¬B¬()) = p( A) + p(B ) − p( A¬B¬())
Therefore, p((A → B) ) ∈ [max(0, a + c − 1), min(b, d)]. The following lemma deals with the cut rule and its analogues.
Lemma 7 Let p( A) = a, p( B) = b, p( C) = c, p(A B) = r and p(B C) = s, with a + r ≥ 1. Then:
(a) [14] (modus ponens probabilized)
a+ r − 1 ≤ p( B) ≤ r (b) [38] (modus tollens probabilized)
r− b ≤ p(A ) ≤ r (c) [4] (hypothetical syllogism rule probabilized)
max(1 − a, b) + max(1 − a, 1 − b, c) − 1 ≤ p(A C) ≤ 2 − a − b + c (d) [4] (hypothetical syllogism rule probabilized)
max(r − a, r + s − 1) ≤ p(A C) ≤ min(s + 1 − a, r + c) The bounds in (a), (b), (c) and (d) are the best possible.
(c) Let us consider the following derivation:
A → B
A A B C A(A → B) C(∗)
A C (∗∗)
where the step denoted by(∗) has done by the rule for ’introducing the implication in antecedent’(→) having the following form:
A B C D A(B → C) D(→)
while the step denoted by(∗∗) can be considered a particular case of the cut rule, i.e. modus ponens rule:
B AB C A C
This derivation justifies that the hypothetical syllogism rule can be inferred as a logical consequence of the modus ponens rule. Let us suppose that p( A) = a, p( B) = b, p( C) = c, p(A B) = r and p(B C) = s, and denote p(A ((A → B) → C)) = t. Then, we have:
max(1 − a, b) ≤ r ≤ min(1, 1 − a + b), (1) max(1 − b, c) ≤ s ≤ min(1, 1 − b + c) (2) and
max(1 − a, c) ≤ p(A C) ≤ min(1, 1 − a + c) (3) From
t = p(A(A → B) C) = p( ¬A ∨ ¬B ∨ C) = = p(B C) + p( ¬A) − p( ¬A ∧ (B → C)) we conclude:
max(1 − a, s) ≤ t ≤ s + 1 − a (4) On the other hand, bearing in mind Hailperin’s result (a) related to the probability analogue of modus ponens rule, i.e. its particular case denoted by(∗∗), we have
r+ t − 1 ≤ p(A C) ≤ t (5) From (2) and (4) we infer:
max(1 − a, 1 − b, c) ≤ t ≤ min(1, 1 − b + c) + 1 − a (6) and finally, from (1), (3), (5) and (6), we have:
(d) From the previous relation, immediately, we have:
max(r − a, r + s − 1) ≤ p(A C) ≤ min(s + 1 − a, r + c, 1) bearing in mind 1− a ≤ r, and 1 − a + c ≤ r + c. Note that the part (c) of our Lemma generalizes and contains as its particular subcases both results Hailperin’s modus ponens probabilized and Wagner’s modus tollens probabilized. Also, as an immediate consequence of this Lemma, bearing in mind that max(r −a, r +s−1) ≥ r +s−1 and min(s+1−a, r +c, 1) ≤ r +c ≤ r +s, we obtain the required form of the cut rule:
Corollary 8 If a≤ p( A) ≤ b and c ≤ p(A ) ≤ d, then max(0, a + c − 1) ≤ p( ) ≤ min(b + d, 1).
4 Consistent LKprob—theories
Letσi(1 ≤ i ≤ n) be a finite list of sequents of the form i abii i, for ai, bi ∈ I (1 ≤
i ≤ n). Then an LKprob—theory over σ1, . . . , σn, denoted by LKprob(σ1, . . . , σn),
presents a deductive closure of the system LKprob extended by the list of sequents σ1, . . . , σn as additional axioms. We say that a theory LKprob(σ1, . . . , σn) is inconsistent if there are two sequents b
a and dc , both provable in LKprob(σ1, . . . , σn) such that [a, b] ∩ [c, d] = ∅; otherwise, LKprob(σ1, . . . , σn)
is consistent. A sequent ba is said to be consistent with respect to LKprob(σ1, . . . , σn) if LKprob(σ1, . . . , σn, ba ) is consistent. A finite set of
sequents{τ1, . . . , τk} is consistent with respect to LKprob(σ1, . . . , σn) if the sequent τi is consistent with respect to LKprob(σ1, . . . , σn, τ1, . . . , τi−1, τi+1, . . . , τk), for
each i(1 ≤ i ≤ k). We also say that a denumerable set of sequents is consistent with respect to the theory LKprob(σ1, . . . , σn) if each of its finite subsets is consistent
with respect to LKprob(σ1, . . . , σn). By proper extension of a theory we mean any its
deductively closed proper superset. A consistent theory is called a maximal consistent theory if each of its proper extensions is inconsistent.
Note that in the case of an inconsistent theory, from ba and dc , by
inference rule(M ↓) we can infer ∅, and in the sequel by (M ↑), ba, for
any a, b ∈ I , and moreover ba, for any a, b ∈ I and any , , by (⊥). Lemma 9 Each consistent theory can be extended to a maximal consistent theory. Proof LetT be a consistent theory, and let α1, α2, . . . , αn, . . . be the sequence of all
unlabelled sequents i.e.αnisn n, and for each c ∈ I , let αc1, α2c, . . . , αnc, . . .
be the sequence of the corresponding labelled sequents, i.e.αcnisncc n. Let the
sequence(Tn) of theories be defined inductively as follows: T0 = T , and Tn+1 = Tn ∪ {αnc1}, if αnc1 is consistent with respect toTn, but if it is not consistent, then: Tn+1 = Tn ∪ {αnc2}, if αcn2 is consistent with respect to Tn, but if it is not, then
…Tn+1= Tn∪ {αncm−1}, if α cm−1
Tn∪ {αncm}, otherwise; where {c1, c2, . . . , cm} = I . Let us note that the final result of
this construction depends on the order of points c1, c2, . . . , cmof the set I . Let T= ∪n∈ωTn
Then, by induction on n we will prove thatTis a maximal consistent extension ofT . First, we prove that ifTnis consistent, thenTn+1is consistent. The only interesting
case is whenTn+1= Tn∪{αcnm}. Suppose that Tn+1is inconsistent, i.e. that the sequent αcm
n is not consistent with respect toTn. Then there exists an interval[a, b] ⊂ [0, 1]
such that cm /∈ [a, b] and nbanis provable inTn, which is impossible because the
theoryTn∪ {α cj
n } is inconsistent for each j (1 ≤ j ≤ m − 1). In order to prove that T
is a maximal consistent extension ofT we extend Tby the sequentk bak. In case
that this is a proper extension, we already have that the theoryTk+1 ⊂ Tcontains kcc kfor some c /∈ [a, b], and, consequently, this extension will be inconsistent.
We will prove that the consistency problem of an LKprob—theory can be reduced to the consistency problem of its fragment of surely provable sequents, i.e. the sequents of the form 1:
Lemma 10 LKprob(σ1, . . . , σn) is consistent iff there exists the sequent 1 which is not provable in LKprob(σ1, . . . , σn).
Proof If the theory LKprob(σ1, . . . , σn) is inconsistent, then the sequent ba is
provable for all, and all a, b ∈ I , so all sequents of the form 1 are provable. Conversely, if all sequents of the form 1 are provable, then 1is provable as well, meaning that the theory LKprob(σ1, . . . , σn) is inconsistent, bearing in mind
that0is an axiom.
It follows that the consistency problem can be treated in the traditional way: Corollary 11 LKprob(σ1, . . . , σn) is inconsistent iff there exists a formula A such that both sequents1 A and1¬A are provable in LKprob(σ1, . . . , σn).
5 Soundness and completeness
In this part we define the notion of canonical model and give soundness and complete-ness theorems for LKprob—theories.
Soundness Theorem. If an LKprob—theory has a model, then it is consistent. Proof By induction on the length of the proof for any sequent ba provable in LKprob we can prove that the system LKprob is sound; this fact follows immediately from our justification of inference rules through Lemmata 1–7. Consequently, any satisfiable set of sequents{σ1, . . . , σn} is consistent with respect to LKprob.
LKprob(σ1, . . . , σn). Then, for any X ∈ ConExt(LKprob(σ1, . . . , σn)) we define pX( ) = r, for an arbitrary r ∈ [a, b], where a ≤ max{c| 1c ∈ X} and b≥ min{c| c0 ∈ X}. Obviously, such a definition provides that the mapping pX, depending on X , always has an adequate value, meaning that pX( ) ∈ [a, b]. In
that case we have that:
Lemma 12 |pX ba iff ba ∈ X.
Proof From|pX ba, by the above definition and the monotonicity rule (M ↑),
we have 1a ∈ X. Similarly, b0 ∈ X, wherefrom, by the monotonicity rule (M ↓), we conclude ba ∈ X. Conversely, if ba ∈ X, then, as X is deductively closed and by the monotonicity rule (M ↑), 1a ∈ X, so a ≤ max{c| 1c ∈ X}. Similarly, b0 ∈ X, so b ≥ min{c|
c
0 ∈ X}.
Consequently, we have|pX ba.
For brevity, below we omit X from the denotation for pX. Then, we also have:
Lemma 13 Any canonical model is a model.
Proof Let p be any mapping defined as above on an arbitrary maximal consistent set of any LKprob—theory. Then:
(1) We have that p(A A) = 1, since A 1 A is an axiom.
(2) If p(AB ) = 1, then, by the additivity rule (ADD), we infer p( AB) = p( A) + p( B), for any formulas A and B.
(3) From the previous lemma, and Lemma1(e), we have that if the sequents and are equivalent in LK, then the sequents and have the same probabilities in the canonical model. Completeness Theorem. Each consistent LKprob—theory has a model.
Proof Bearing in mind that each consistent theory can be extended to a maximal consistent theory, and that one such theory is described exactly as the canonical model, it suffices to prove that for each consistent theory there is a world of the canonical
model satisfying it.
6 Concluding remarks
In order to respect both, the probabilistic Carnap’s, Popper’s and Leblance’s tradition, and the spirit of Gentzen’s foundations of proof theory, we introduce a sequent cal-culus LKprob enabling the expression of the following statement: “the probability of truthfulness of sequent is in the interval [a, b]”, denoted by ba, obtaining
form:i baii i (1 ≤ i ≤ n), shortly denoted by σi (1 ≤ i ≤ n). The axiom σi can
be understood as an empirical conclusion of statistical research. In this case an exten-sion LKprob(σ1, . . . , σn) of LKprob by the new empirical facts σ1, . . . , σn, from
the logical point of view, presents a theory LKprob(σ1, . . . , σn) based on LKprob. LKprob(σ1, . . . , σn) is a sequent calculus, enabling to manipulate with
probabili-ties, which needs a formal description of the appropriate syntax and semantics, and, if possible, the corresponding proof-theoretic and model-theoretic treatment. Here we determine the notion of consistency, define the corresponding models and obtain soundness and completeness results for LKprob.
Finally, let us note that the system LKprob can be used as a good basis to introduce an intuitionistic version of probabilistic sequent calculus LJprob, in the same way as Gentzen did (see Gentzen [12]), with restriction to singular form of sequents, and specific rules customized. For instance, additivity rule in LJprob may have the following form
A B1 ba A dc B
¬A¬B min(1,b+d) min(1,a+c)
(ADD)
Anyway, there remains the problem how to define the models, probably the Kripke-type models, with respect to which the system LJprob will be sound and complete.
Acknowledgements The author would like to thank the members of the Seminar for Probabilistic Logic of
Mathematical Institute of SANU, Belgrade, and particularly her teachers Z. Ognjanovi´c, Z. Markovi´c, M. Raškovi´c, N. Ikodinovi´c and A. Perovi´c, as well as the anonymous referee of the Archive for Mathematical Logic, for making very valuable suggestions and helpful comments regarding earlier versions of this paper.
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