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Dorota

Leszczy

´

nska-Jasion

Mariusz Urba

´

nski

Andrzej Wi

´

sniewski

Socratic Trees

Abstract. The method of Socratic proofs (SP-method) simulates the solving of logical

problem by pure questioning. An outcome of an application of the SP-method is a sequence

of questions, called a Socratic transformation. Our aim is to give a method of translation

of Socratic transformations into trees. We address this issue both conceptually and by pro-

viding certain algorithms. We show that the trees which correspond to successful Socratic

transformations—that is, to Socratic proofs—may be regarded, after a slight modiﬁcation,

as Gentzen-style proofs. Thus proof-search for some Gentzen-style calculi can be performed

by means of the SP-method. At the same time the method seems promising as a foundation

for automated deduction.

Keywords: Socratic transformations, Socratic proofs, Sequent calculi, Proof search,

Automated deduction.

Introduction

The aim of this paper is to analyze a certain application of the method of

Socratic proofs (hereafter SP-method). We will show how the outcomes of

successful applications of the method can be translated into Gentzen-style

proofs in some sequent calculi.

The SP-method enables a formal explication of the idea of solving logi-

cal problems (i.e problems of entailment/derivability/theoremhood) by pure

questioning, that is, by transforming the relevant initial question into con-

secutive questions without making any use of answers to the questions just

transformed. A transformation of this kind is called Socratic. There are suc-

cessful and unsuccessful Socratic transformations; a successful transforma-

tion ends with a question of a required ﬁnal form, roughly, a question which

can be answered in only one rational way. A successful transformation is a

Presented by Andrzej Indrzejczak; Received August 4, 2011

Studia Logica (2012)

DOI: 10.1007/s11225-012-9404-0

c

Springer Science+Business Media B.V. 2012

D. Leszczy´nska-Jasion et al.

Socratic proof. Socratic transformations are guided by erotetic rules

1

which

have only questions as premises and conclusions. These rules form the core

of erotetic calculi. Since entailment/derivability/theoremhood is always rel-

ative to a logic, one needs different erotetic calculi for different logics. So

far calculi of this kind have been developed for Classical Logic (both prop-

ositional and ﬁrst-order; [19,21]), normal modal propositional logics [5–7],

intuitionistic propositional logic [11], some paraconsistent [21] as well as

temporal propositional logics [14].

The SP-method is grounded in the logic of questions called Inferential

Erotetic Logic (IEL for short, see [15–18]). IEL deﬁnes conditions of validity

of erotetic inferences, that is, inferences whose conclusions are questions.

Erotetic rules are designed in such a way that IEL-validity is retained. On

the other hand, the underlying semantics warrants that the existence of a

successful Socratic transformation of an initial question amounts to the aﬃr-

mative answer to the question: a formula is entailed by /derivable from a

set of formulas (in the light of the just analyzed logic), or is a theorem (of

the logic).

In order to formulate an erotetic calculus for a logic one needs a lan-

guage which extends the language of the logic. In particular, such language

has questions among its well-formed expressions. It is convenient to build

the language in such a way that sequents form its atomic well-formed for-

mulas. This requires an introduction of the turnstile to the (object-level)

language; the well-formed formulas of the language of the analyzed logic

or of an associated language (e.g. labelled formulas) stay at the left and

right of the turnstile. Questions are then based on sequences of sequents.

In order to express answers one needs additional connectives which apply

to sequents. It should be stressed that a language of an erotetic calculus

is supplemented with its own semantics by means of which basic semantic

relations between well-formed expressions (questions included!) are deﬁned.

For details see [6,19–21].

Formally, Socratic transformations are sequences of questions. Given that

questions are based on sequences of sequents, erotetic calculi are akin to hy-

persequent calculi (although with some peculiarities: there are no primary

structural rules and “questions-hypersequents” are understood “conjunc-

tively”, see [19, p. 321, footnote 15], compare also [2]). In this paper we

will show how to transform Socratic transformations, that is, sequences of

questions, into trees which will be called Socratic trees. On the one hand,

1

“Erotetic” comes from Greek “erotema” which means “question”. The logic of ques-

tions is sometimes called erotetic logic.

Socratic Trees

this brings us closer to the proof format which is characteristic for tableau

methods. On the other hand, since the nodes of Socratic trees are (anno-

tated) sequents, the trees we shall deﬁne may be considered, after some

reformulation(s), as derivations in a traditional sequent calculus (we exam-

ine this matter in Section 3). It has been already observed (cf.[19]) that

Socratic proofs can be transformed into Gentzen-style proofs.

2

In this paper

we analyze the matter in detail. We address the issue not only conceptually,

but we also provide (in the Appendix) algorithms which translate Socratic

transformations into the corresponding trees.

Since Socratic transformations produce, via the algorithms, derivations

in the corresponding sequent calculi, proof-search can be performed at the

level of erotetic calculi. In general, the SP-method seems promising as a foun-

dation for automated deduction, avoiding well-known limitations of resolu-

tion-based techniques. There are some remarkable implementations of the

SP-method for quite sophisticated propositional cases in Prolog.

3

As a proof

of completeness of the ﬁrst-order version of the method (cf.[21]) involves

a model-elimination technique which is not based on backtracking, it is an

interesting perspective to consider the SP-method as a basis for a ﬁrst-order

theorem prover. Translation into trees allows for search for more eﬃcient

proof strategies without calling for resolution.

Let us also add here that the term “Socratic proof” is sometimes used

(cf.[1]) for a proof in the form of a series of statements, each of which obvi-

ously follows from earlier statement(s) of the proof, and the last of which is

a desired fact or theorem. However, the idea underlying Socratic proofs in

our sense is different (and probably more “Socratic”): proofs are sequences

of questions, not statements, and what is obvious is not a transition from

one to another but the answer to the last question of the sequence.

1. Socratic Transformations

We shall show here how the SP-method works on the simple example of

Classical Propositional Calculus (hereafter: CPC), and in Section 3.2 we will

2

Moreover, a simulation of the Smullyan-type Analytic Tableau Method by the SP-

method has also been presented there, (see [19, pp. 313–318] and [12]). It should be noted,

however, that since the nodes of the trees considered here are labeled with sequents ,the

trees strongly resemble Hintikka-type tableaux rather than Smullyan-type tableaux.

3

Cf.[4]. The idea of the SP-method was discussed already in 2003, and thus this Prolog

program was written before the ﬁrst paper concerning the method of Socratic proofs was

published.

D. Leszczy´nska-Jasion et al.

brieﬂy discuss the case of First-Order Logic (FOL). For a detailed account of

the cases presented in this paper see [19,21], for more sophisticated cases see

[5–7,11,14,20], and for examples of possible applications of the SP-method

in the area of (the metatheory of) modal logics see [8,9].

1.1. Language

Let us start with a language of CPC with ¬ (negation), → (implication),

∧ (conjunction) and ∨ (disjunction) as primitive connectives and let us

extend the language, int. al., with a question-forming operator ? and the

sign . The resulting language L

∗

has two disjoint categories of meaningful

expressions: declarative well-formed formulas (hereafter: d-wﬀs) and ques-

tions. Questions of L

∗

are based on sequences of atomic d-wﬀs, that is,

expressions of the form:

S A (1)

where S is a ﬁnite sequence (possibly empty) of CPC-formulas, and A is a

CPC-formula (this latter concept is deﬁned in the standard manner). Note

that atomic d-wﬀs of L

∗

are (single-conclusioned) sequents. In what follows

we will refer to atomic d-wﬀs of L

∗

simply as to sequents, yet always having

in mind that only sequents with single CPC-formulas in the succedent are

taken into consideration. We use Greek lower case letters φ, ψ, χ, ω (pos-

sibly with subscripts) as metavariables for sequents, and Greek upper case

letters Φ, Ψ, Γ as variables for sequences of sequents.

A question of the language L

∗

is an expression of the form:

?(Φ) (2)

where Φ is a non-empty ﬁnite sequence of sequents; the terms of this

sequence are called constituents of the question, and we say that the question

is based on the sequence.

Some notational conventions will be useful. The following:

S

T

stands for the concatenation of sequences S and T of CPC-formulas. By

S

A

we refer to the concatenation of S and the one-term sequence A, where

A is a CPC-wﬀ. The concatenation of sequences Φ and Ψ of sequents is

referred to as:

Φ; Ψ

Socratic Trees

whereas the inscription:

Φ; φ

denotes the concatenation of a sequence of sequents Φ and the one-term

sequence φ, where φ is a sequent. Of course, the inscription:

Φ; φ;Ψ

refers to the concatenation of Φ; φ and a sequence of sequents Ψ. Any of S,

T ,Φ,andΨcanbeempty.

Thus when Φ = φ

1

,...,φ

n

, the corresponding question can be written

as:

?(φ

1

; ...; φ

n

)

and we will proceed that way. If Φ = φ, then we write the question as:

?(φ)

and we say that the question is based on a single-conclusioned sequent.

A question of the form: ? (S

1

A

1

; ...; S

n

A

n

) can read: “Is it the

case that: A

1

is CPC-entailed by S

1

and ... and A

n

is CPC-entailed by

S

n

?”; due to the completeness of CPC, “CPC-entailed by” can be replaced

by “CPC-derivable from”. (Of course, by entailment by/derivability from a

sequence we mean entailment by/derivability from the set of all the terms

of the sequence.) When n = 1, the question pertains to the claim of a single

sequent.

1.2. Socratic Proofs

In a Socratic transformation one transforms a question into a question. Here

is the list of erotetic rules that govern the relevant transformations of ques-

tions of L

∗

:

L

α

:

?(Φ;S

α

T C;Ψ)

?(Φ;S

α

1

α

2

T C;Ψ)

R

α

:

?(Φ;S α;Ψ)

?(Φ;S α

1

; S α

2

;Ψ)

L

β

:

?(Φ;S

β

T C;Ψ)

?(Φ;S

β

1

T C; S

β

2

T C;Ψ)

R

β

:

?(Φ;S β;Ψ)

?(Φ;S

β

∗

1

β

2

;Ψ)

L

¬¬

:

?(Φ;S

¬¬A

T C;Ψ)

?(Φ;S

A

T C;Ψ)

R

¬¬

:

?(Φ;S ¬¬A;Ψ)

?(Φ;S A;Ψ)

The above rules constitute calculus of questions called E

∗

.

The letters “L”and“R” indicate that the appropriate rule “operates”

on the left or right side of the turnstile . The second part of the rule’s

name indicates the form of a formula acted upon, e.g. rule R

α

operates on

D. Leszczy´nska-Jasion et al.

Table 1. α, β-notation

αα

1

α

2

ββ

1

β

2

β

∗

1

A ∧ BAB¬(A ∧ B) ¬A ¬BA

¬(A ∨ B) ¬A ¬BA∨ BAB¬A

¬(A → B) A ¬BA→ B ¬AB A

an α-formula occurring on the right side of the turnstile. We have used the

α, β–notation for brevity (this is explained in Table 1, see also [12]). β

∗

1

(as

deﬁned in Table 1) may be called the complement of β

1

.

We shall call rules R

α

and L

β

branching rules, as the resulting “ques-

tion-conclusion” has more constituents than the “question-premise”. Con-

sequently, we will call the remaining erotetic rules: L

α

, R

β

, L

¬¬

and R

¬¬

non-branching rules.

4

It is easily visible that the rules of E

∗

are designed in such a way that

each constituent of the “question-conlusion” is CPC-valid if and only if each

constituent of the “question-premise” is CPC-valid. On the other hand, it

can be shown that each application of a rule of E

∗

retains validity (in the

sense of IEL) of the corresponding erotetic inference. For a detailed pre-

sentation of the relevant languages and their semantics, as well as for the

justiﬁcation of the above claims see [19].

The concept of Socratic transformation is given by the following defini-

tion:

Definition 1. A sequence s

1

,s

2

,... of questions is a Socratic transfor-

mation of a question ?(S A) via the rules of an erotetic calculus E

∗

iﬀ

the following conditions hold:

(i) s

1

=?(S A);

(ii) s

i

, where i>1, results from s

i−1

by an application of an erotetic rule

of E

∗

.

Here is an example of a Socratic transformation of question ? ((p ∨ p)

∨ (q ∨ q) p ∨ q), where p, q are propositional variables (this question can

read: “Is it the case that p ∨ q is CPC-entailed by (p ∨ p) ∨ (q ∨ q)?”); for

4

We consider here only binary branching rules, as E

∗

is based on CPC; adjustments to

erotetic calculi with n-ary, n>2, branching rules (like calculi based on many-valued log-

ics) are rather straightforward. It should be noted, however, that some other notions need

then to be adjusted as well (like the ones introduced by Definitions 4 and 5; cf. Section 2).

Socratic Trees

further reference, we will call this transformation s. We highlight the con-

stituents which a given rule acts upon and, for clarity, put the name of the

rule to the right:

Example 1.

?((p ∨ p) ∨ (q ∨ q) p ∨ q ) L

β

?(p ∨ p p ∨ q ; q ∨ q p ∨ q) L

β

?(p p ∨ q ; p p ∨ q ; q ∨ q p ∨ q) R

β

?(p, ¬p q ; p p ∨ q ; q ∨ q p ∨ q) R

β

?(p, ¬p q ; p, ¬p q ; q ∨ q p ∨ q ) L

β

?(p, ¬p q ; p, ¬p q ; q p ∨ q ; q p ∨ q) R

β

?(p, ¬p q ; p, ¬p q ; q, ¬p q ; q p ∨ q ) R

β

?(p, ¬p q ; p, ¬p q ; q, ¬p q ; q, ¬p q)

The last question of the above sequence has an interesting property: the

aﬃrmative answer to it is, in a sense, evident, as all the constituents of this

question express some basic facts about (CPC) entailment. Thus, the answer

to the ﬁrst question of the sequence is also aﬃrmative: it is true that p ∨ q

is entailed by (p ∨ p) ∨ (q ∨ q), and the sequence of Example 1 is not just

a transformation: it is a proof.

Definition 2. A ﬁnite Socratic transformation Q

1

,...,Q

n

of question

?(S A) via the rules of E

∗

is a Socratic proof of sequent S A in the

calculus E

∗

iﬀ for each constituent φ of Q

n

:

(a) φ is of the form T

B

U B,or

(b) φ is of the form T

B

U

¬B

W C,or

(c) φ is of the form T

¬B

U

B

W C.

The existence of a Socratic proof of S A amounts to CPC-entail-

ment/derivability of A by/from S. Moreover, if A is CPC-entailed/derivable

by/from S, then there exists a Socratic proof of S A. For details, see [19].

2. Socratic Transformations and Trees

Socratic transformations are deﬁned as sequences of questions, and so are

Socratic proofs. This is a departure from the most popular proof format,

that is, proof trees. The aim of this section is to deﬁne a translation of a

Socratic transformation/proof into the corresponding tree, whose nodes are

annotated sequents.

D. Leszczy´nska-Jasion et al.

Before we deﬁne the notion of an annotated sequent let us recall that

questions of the considered language L

∗

are based on (ﬁnite) sequences of

sequents. Thus each sequent which occurs in a question performs the role of

a term of the sequence on which the question is based; since sequences with

repetitions are allowed, it may happen that a given sequent is both i-th and

j-th term of the corresponding sequence, where i = j, that is, it is both i-th

and j-th constituent of the question.

Until the end of this paper we assume that s = s

1

,s

2

,... is an arbitrary

but ﬁxed Socratic transformation of a question based on a single sequent,

that is, a Socratic transformation of a question of the form ? (S A). An

ordered triple n, i, φ where sequent φ is i-th constituent of n-th question

s

n

of s will be called an annotated sequent of Socratic transformation s.For

simplicity, an annotated sequent n, i, φ will be sometimes designated by φ

n

i

.

We will use , σ, τ, possibly with subscripts, for annotated sequents. We

will also need a function assigning sequents to annotated sequents of Socratic

transformation s; this is set by: g

s

(φ

n

i

)=φ.

Suppose that question s

n+1

of s results from question s

n

by erotetic rule

r. The constituent of question s

n

whose schema is distinguished in the “ques-

tion-premise” of r—with respect to which the rule is applied—will be called

the active constituent of question s

n

or the active sequent of this question.

Moreover, we will say that the constituent or sequent (the constituents or

sequents) of question s

n+1

whose schema is distinguished in the “question-

conclusion” of r results from the active constituent/sequent of s

n

,orthat

it is constituent-conclusion of question s

n+1

or its sequent-conclusion.Obvi-

ously, each question of s has exactly one active constituent (or, if it is the

last question of s—it has none) and at most two constituents-conclusions;

these constituents-conclusions are “new” with respect to the previous ques-

tion. We say that sequent ψ results in s from sequent φ by erotetic rule r

if for some term s

n

of s,theterms

n+1

results from s

n

by r, φ is the active

constituent of s

n

and ψ is a sequent-conclusion of s

n+1

.

We will also say that annotated sequent φ

n

i

is active in s

n

if i-th con-

stituent of question s

n

is active in this question; and, analogously, we will

say that annotated sequent φ

n+1

k

results from annotated sequent ψ

n

i

if k-th

constituent of question s

n+1

results from i-th constituent of question s

n

.

The rules of erotetic calculi operate on questions which are based on

sequences of sequents. However, a rule acts upon only one constituent of a

question leaving the remaining sequents unchanged. From the point of view

of the logic of questions this is a desired eﬀect, since an application of an

erotetic rule amounts to a reformulation of the problem expressed by the

Socratic Trees

“question-premise”.

5

However, from the present point of view, the unchanged

sequents that are “left” in the resulting question are redundant and thus we

will aim at leaving them out of the tree (cf. Definition 4). Moreover, since in a

Socratic transformation sequents are organized “horizontally”—in sequences

on which the questions are based—we need to change the organization into

a “vertical” one. Thus we will deﬁne a relation, P

s

, that links two annotated

sequents of Socratic transformation s whenever one of these annotated se-

quents is active and the other is its sequent-conclusion. Moreover, if after

an application of a rule a sequent is rewritten in the next question (that is,

a rule has been applied with respect to some other—active—constituent),

then the relation P

s

links the annotated sequent and its annotated repetition

(cf. Definition 5). Next, we will consider the transitive closure of relation

P

s

, as we aim at a tree. And ﬁnally, in order to get rid of the redundant

repetitions of sequents we will restrict the transitive closure of P

s

to the set

X

s

(cf. Definition 6).

Now let us proceed with the trees. If R is a transitive and asymmetric

relation on X,thenwesaythatx is an immediate R-predecessor of y or

that y is an immediate R-successor of x iﬀ xRy and ¬∃

z∈X

(xRz ∧ zRy).

Moreover,

Definition 3. An ordered pair X, R is atreewithoriginx iﬀ R is a tran-

sitive and asymmetric relation on X, x is the least element in X, R and

each y ∈ X \{x} has exactly one immediate R-predecessor. If X, R is a

tree, then we call y ∈ X a node of the tree. Moreover, we say that a tree

X, R is ﬁnite iﬀ X is ﬁnite. If X, R is a ﬁnite tree, then we say that its

node y ∈ X is a leaf of the tree iﬀ y has no immediate R-successors in X.

Let us recall that s is an arbitrary but ﬁxed Socratic transformation.

Definition 4 speciﬁes the set of nodes of our tree.

Definition 4. By X

s

we mean the smallest set of annotated sequents of s

such that the following conditions hold:

1. φ

1

1

∈ X

s

2. Let ψ

n

i

be the active annotated sequent in question s

n

,

(a) if s

n+1

results from s

n

by a non-branching rule, then χ

n+1

i

∈ X

s

,

(b) if s

n+1

results from s

n

by a branching rule, then (ω

1

)

n+1

i

∈ X

s

and

(ω

2

)

n+1

i+1

∈ X

s

.

5

For more information on this point see [19].

D. Leszczy´nska-Jasion et al.

According to Definition 4 the set X

s

contains the annotated sequent φ

1

1

(it will perform the role of the origin of the tree) and the annotated se-

quents-conclusions of the questions of s. Thus we pick from every question

different from the ﬁrst one only these sequents that are “new” with respect

to the previous question and we leave the remaining redundant sequents out.

We will use the Socratic transformation s of Example 1 as an illustration.

The active constituents are highlighted as before, whereas the constituents-

conclusions (and the only constituent of the ﬁrst question) are framed. The

set X

s

is given below.

s

1

=?((p ∨ p) ∨ (q ∨ q) p ∨ q ) L

β

s

2

=?(p ∨ p p ∨ q ; q ∨ q p ∨ q ) L

β

s

3

=?(p p ∨ q ; p p ∨ q ; q ∨ q p ∨ q ) R

β

s

4

=?(p, ¬p q ; p p ∨ q ; q ∨ q p ∨ q ) R

β

s

5

=?(p, ¬p q ; p, ¬p q ; q ∨ q p ∨ q ) L

β

s

6

=?(p, ¬p q ; p, ¬p q ; q p ∨ q ; q p ∨ q ) R

β

s

7

=?(p, ¬p q ; p, ¬p q ; q, ¬p q ; q p ∨ q ) R

β

s

8

=?(p, ¬p q ; p, ¬p q ; q, ¬p q ; q, ¬p q )

X

s

= { (p ∨ p) ∨ (q ∨ q) p ∨ q

1

1

,p∨ p p ∨ q

2

1

,

q ∨ q p ∨ q

2

2

,p p ∨ q

3

1

,p p ∨ q

3

2

,p,¬p q

4

1

,p,¬p q

5

2

,

q p ∨ q

6

3

,q p ∨ q

6

4

,q,¬p q

7

3

,q,¬p q

8

4

}

As we can see, the ﬁrst question excepted, in each case we simply pick

as the members of X

s

the (annotated) constituents that are “new” with

respect to the previous question.

In order to deﬁne a relation R

s

on the set X

s

we need a certain auxiliary

concept.

Definition 5. Let Y

s

be the set of all annotated sequents of Socratic trans-

formation s.ByP

s

we mean the smallest binary relation on Y

s

such that for

each n ≥ 1:

1. If s

n+1

=?(ψ

1

; ...; ψ

j

) results from s

n

=?(φ

1

; ...; φ

j

) by a non-branch-

ing rule, then for each i ≤ j the following pair of annotated sequents of

s : n, i, φ

i

, n +1,i,ψ

i

belongs to P

s

, that is,

(φ

i

)

n

i

, (ψ

i

)

n+1

i

∈P

s

.

Socratic Trees

Figure 1. Illustration of Definition 5

2. If s

n+1

=?(ψ

1

; ...; ψ

j+1

) results from s

n

=?(φ

1

; ...; φ

j

) by a branching

rule, and k-th constituent of s

n

is active in s

n

,then:

(a) for each i ≤ k, (φ

i

)

n

i

, (ψ

i

)

n+1

i

∈P

s

,

(b) for each i : k ≤ i ≤ j, (φ

i

)

n

i

, (ψ

i+1

)

n+1

i+1

∈P

s

.

It follows from Definition 5 that if , σ∈P

s

, then either and σ repre-

sent the same (differently annotated) sequent or is active in some question

of s and σ results from . For further reference let us also observe that the

following holds:

Corollary 1. If , σ∈P

s

and τ,σ∈P

s

,then = τ.

Again, we will use the Socratic transformation s of Example 1 as an

illustration (cf.Figure1). The arrows show which annotated sequents stay

in relation P

s

to each other. The elements of X

s

are framed.

Finally, let us introduce:

D. Leszczy´nska-Jasion et al.

Figure 2. Illustration of Definition 6

Definition 6. Let P

tr

s

be the transitive closure of P

s

.ThenbyR

s

we mean

relation P

tr

s

restricted to set X

s

,thatis, σ∈R

s

iﬀ both , σ∈P

tr

s

and , σ ∈ X

s

.

If we apply Definition 6 to our previous example, we will receive the

structure presented in Figure 2. Only the elements of X

s

are displayed, the

lines represent relation R

s

.

For further reference let us note:

Corollary 2. Let s be a Socratic transformation via the rules of E

∗

.If

annotated sequent φ

n

i

is an immediate R

s

-successor of ψ

m

k

,thensequent

g

s

(φ

n

i

) results in s from sequent g

s

(ψ

m

k

) by a rule of E

∗

.

By the very definition of P

s

,ifφ

n

i

, ψ

m

j

∈P

s

,thenm = n + 1, hence,

obviously, m>n.AsP

tr

s

is the transitive closure of P

s

, it follows that if

φ

n

i

,ψ

m

j

∈P

tr

s

,thenm>n, and similarly in the case of R

s

. This is enough

to show that R

s

is asymmetric. Moreover, as a restriction of transitive rela-

tion P

tr

s

, relation R

s

is also transitive. Thus:

Corollary 3. R

s

is an asymmetric and transitive relation on set X

s

.

Moreover, it is easy to check that, by the definition of P

s

, the following

holds:

Corollary 4. For each τ ∈ X

s

other than φ

1

1

there are τ

1

,...,τ

n

∈ X

s

,

where n>1, such that τ

1

= φ

1

1

, τ

n

= τ ,andτ

i

,τ

i+1

∈P

s

for

i =1,...,n− 1.

By Tr(s) we shall mean the structure X

s

,R

s

. We may ﬁnally prove:

Theorem 1. If s = s

1

,s

2

,... is a Socratic transformation of a question

based on single sequent φ via the rules of E

∗

,thenTr(s) is a tree with

origin φ

1

1

.

Socratic Trees

Proof. By Corollary 3, R

s

is a transitive and asymmetric relation on set

X

s

. By Corollary 4 and Definition 6, φ

1

1

is the least element in Tr (s). Finally,

from Corollary 4 together with Corollary 1 it follows that each element of

X

s

other than φ

1

1

has exactly one immediate R

s

-predecessor.

We have shown that the structure Tr(s) is a tree. We shall call Tr(s)the

Socratic tree determined by a Socratic transformation s of a question based

on a single sequent. Definitions 4, 5 and 6 give us a recipe how to construct

Tr(s) for a given Socratic transformation s. (The corresponding algorithm

is presented in the Appendix.)

Since Socratic proofs are ﬁnite Socratic transformations, we also have:

Theorem 2. Let s = s

1

,s

2

,...,s

n

be a Socratic proof of sequent ψ in the

calculus E

∗

. The structure Tr(s)=X

s

,R

s

is a tree with the origin ψ

1

1

and

such that if φ

m

k

is a leaf of Tr(s),thensequentg

s

(φ

m

k

) is of one of the forms

(a), (b) or (c) speciﬁed in Definition 2. Moreover, each node of the tree

other than its origin is an annotated sequent-conclusion of a question of s.

Proof. What needs to be proved is that the leaves are actually of the indi-

cated forms. Let φ

m

k

be a leaf of Tr(s), that is, an annotated sequent which

has no immediate R

s

-successors. If m = n, then sequent g

s

(φ

m

k

)=φ occurs

in the last question of s, and thus it must be of one of the forms (a), (b), (c)

speciﬁed in Definition 2, since s is a Socratic proof. Suppose that m = n − 1.

The leaf φ

m

k

is not an immediate R

s

-predecessor of any other node of Tr(s),

but it still must stand in relation P

s

to some χ

n

i

, where sequent g

s

(χ

n

i

)

occurs in the n-th question of s. However, χ

n

i

/∈ X

s

, for otherwise we would

have had φ

m

k

,χ

n

i

∈R

s

,andthenφ

m

k

would not be a leaf of Tr(s). But the

fact that χ

n

i

/∈ X

s

implies that χ

n

i

is not an annotated sequent-conclusion,

that is, g

s

(φ

m

k

) is simply the same sequent as g

s

(χ

n

i

). Obviously, sequents

occurring in s

n

are of the speciﬁed forms (a), (b) or (c), and thus g

s

(φ

m

k

)is

in one of these forms as well.

The reasoning is analogous if m<n− 1 and we leave it to the reader.

Let us ﬁnish this section with some examples.

Example 2. Here is a Socratic proof of the sequent (p → (q → r)) →

((p → q) → (p → r)) :

?( (p → (q → r)) → ((p → q) → (p → r))) R

β

?(p → (q → r) (p → q) → (p → r)) R

β

?(p → (q → r),p→ q p → r) R

β

?(p → (q → r),p→ q, p r) L

β

?(¬p, p → q,p r ; q → r, p → q, p r) L

β

D. Leszczy´nska-Jasion et al.

?(¬p, p → q,p r ; ¬q, p → q,p r ; r, p → q,p r) L

β

?(¬p, p → q,p r ; ¬q, ¬p, p r ; ¬q, q, p r ; r, p → q, p r)

And here is the Socratic tree determined by this Socratic proof:

(p → (q → r)) → ((p → q) → (p → r))

1

1

p → (q → r) (p → q) → (p → r)

2

1

p → (q → r),p→ q p → r

3

1

p → (q → r),p→ q, p r

4

1

¬p, p → q, p r

5

1

q → r, p → q, p r

5

2

¬q, p → q,p r

6

2

¬q, ¬p, p r

7

2

¬q, q, p r

7

3

r, p → q,p r

6

3

Example 3. This time we consider a Socratic transformation which is not

a proof:

?( ((p → q) ∧ q) → p) R

β

?((p → q) ∧ q p) L

α

?(p → q, q p) L

β

?(¬p, q p ; q,p q)

The Socratic tree determined by the above Socratic transformation can be

displayed as follows:

(p → q) ∧ (q → p)

1

1

(p → q) ∧ q p

2

1

p → q,q p

3

1

¬p, q p

4

1

q, q q

4

2

3. Socratic Transformations and Gentzen-style Proofs

It is known that Socratic proofs can be transformed into Gentzen-style

proofs in a “parallel” calculus of sequents (cf.[19]and[21]). In the following

Socratic Trees

subsections we present the rules of such parallel calculi of sequents per-

taining to CPC and to FOL. Our aim is to show that each Socratic tree

determined by a Socratic proof of a sequent may be further considered as a

Gentzen-style proof of the sequent in the relevant calculus.

Traditionally, by a proof of a sequent in a Gentzen system one means a

ﬁnite labeled tree regulated by the rules where the leaves are labeled with

axioms.

6

What is meant by “regulation” is that each node-label is con-

nected with the label(s) of the immediate successor(s) node(s) (if there are

any) according to one of the rules, that is, the node-label falls under the

schema of the conclusion of the rule, whereas the immediate successor(s)

label(s) falls (fall) under the schema of its premise(s).

Obviously, labels are what counts, the nature of the nodes is inessential.

However we may—and we will—assume that if the nodes of a Socratic tree

are objects of the form n, i, φ, then the pair n, i constitutes a node of

the parallel Gentzen-style tree and the sequent φ constitutes its label. More-

over, in the Appendix (see page 21) we show how one can further use this

feature of our construction of a tree in order to simplify the algorithm of

tree construction.

3.1. Calculus G

∗

We present the Gentzen-style counterpart of the erotetic calculus E

∗

. The

calculus, called G

∗

, constitutes a formalization of CPC; it has the following

rules:

G

∗

L

α

:

S

α

1

α

2

T C

S

α

T C

G

∗

R

α

:

S α

1

S α

2

S α

G

∗

L

β

:

S

β

1

T CS

β

2

T C

S

β

T C

G

∗

R

β

:

S

β

∗

1

β

2

S β

G

∗

L

¬¬

:

S

A

T B

S

¬¬A

T B

G

∗

R

¬¬

:

S A

S ¬¬A

Obviously, the rules operate on sequents of the form (1). Axioms of the cal-

culus are of the forms (a), (b) and (c) speciﬁed in Definition 2. There are no

primary structural rules. According to our previous remarks, by a proof of

sequent S A in G

∗

we mean a ﬁnite tree labeled with sequents, regulated

by the rules of G

∗

and such that the origin is labeled with sequent S A

and the leaves are labeled with axioms of the calculus.

Now we show that after a certain reformulation of its structure a Socratic

tree Tr(s)=X

s

,R

s

produces a proof in G

∗

. For suppose that X

s

,R

s

is

6

Cf. for example [13, p. 60].

D. Leszczy´nska-Jasion et al.

the Socratic tree determined by a Socratic transformation s. Then we put:

X

sG

— the set of pairs of numerals which are parts of the annotated se-

quents in X

s

, R

sG

—the relation extracted from relation R

s

in a similar way;

formally:

Definition 7. Let X

s

,R

s

be the Socratic tree determined by a Socratic

transformation s. Then set X

sG

and relation R

sG

on this set are deﬁned as

follows:

1. n, i∈X

sG

iﬀ for some sequent φ, n, i, φ∈X

s

,

2. n

1

,i

1

, n

2

,i

2

∈ R

sG

iﬀ for some sequents φ

1

,φ

2

, n

1

,i

1

,φ

1

,

n

2

,i

2

,φ

2

∈ R

s

.

Obviously, the structure X

sG

,R

sG

is a tree, since X

s

,R

s

is a tree.

Moreover, we will assume that the tree is labeled, namely the labelling func-

tion λ

s

is such that if i, j, φ∈X

s

, then sequent φ is the label λ

s

(i, j)of

node i, j.AtripleX

sG

,R

sG

,λ

s

,thatis,thetreeX

sG

,R

sG

together

with its labelling function λ

s

, will be called the labeled tree determined by a

Socratic transformation s.

Now let us observe that if the ﬁrst question of s is of the form ?(S A),

then the node 1, 1 of the tree is labeled with sequent S A;andifs hap-

pens to be a Socratic proof, then the leaves of Tr(s) are annotated axioms of

G

∗

and therefore the leaves of X

sG

,R

sG

are labeled with axioms as well.

Thus we arrive at:

Theorem 3. If X

s

,R

s

is the Socratic tree determined by a Socratic proof of

sequent S A, then the labeled tree X

sG

,R

sG

,λ

s

is a proof of S A in G

∗

.

Proof. The structure X

sG

,R

sG

,λ

s

is a ﬁnite tree labeled with sequents.

Moreover, Corollary 2 together with the structure of the rules of E

∗

and G

∗

warrants that the tree is regulated by the rules of G

∗

.

If s is a Socratic proof of sequent S A in E

∗

, then the labeled tree

X

sG

,R

sG

,λ

s

will be called the proof of S A in G

∗

based upon s.We

will denote it by G(s).

Example 4. Here is the proof of sequent (p ∨ p) ∨ (q ∨ q) p ∨ q based

upon s (cf. Example 1).

p, ¬p q

p p ∨ q

p, ¬p q

p p ∨ q

p ∨ p p ∨ q

q, ¬p q

q p ∨ q

q, ¬p q

q p ∨ q

q ∨ q p ∨ q

(p ∨ p) ∨ (q ∨ q) p ∨ q

Socratic Trees

Since calculus E

∗

is sound and complete (cf.[19]) we have the following

result:

Theorem 4. Let A be a formula of CPC and let S be a ﬁnite sequence of

such formulas. A is CPC-entailed by the set of terms of S ( CPC-derivable

from this set) iﬀ sequent S A has a proof in G

∗

.

Proof. The ﬁrst implication (completness of G

∗

) follows directly from com-

pleteness of E

∗

and from the previous theorem.

For soundness of G

∗

suppose that S A has a proof in G

∗

.IfS A is

an axiom, then, obviously, A is CPC-entailed by the set of terms of S.And

for the other cases we need to show that the rules of G

∗

“transmit CPC-

entailment” in the following sense: if for each premise S

i

A

i

of a rule of G

∗

it holds that A

i

is CPC-entailed by the set of terms of S

i

, then, similarly,

for the conclusion S

0

A

0

of the rule it also holds that A

0

is CPC-entailed

by the set of terms of S

0

. Equivallently, one may show that the rules of E

∗

have an analogous property of transmission of CPC-entailment from bottom

to top, that is, from a conclusion (or conclusions) to a premise; and this

result may be found in [19].

3.2. First Order Logic

In [21] the authors introduce an erotetic calculus E

PQ

pertaining to FOL, as

well as its Gentzen-style counterpart with the following properties: (a) only

single-conclusioned sequents are operated on; (b) the rules are semantically

invertible; and (c) there are no primary structural rules. Nevertheless, the

calculus is sound and complete. We will call this calculus G

PQ

. We present

both of the calculi in the following subsections.

3.2.1. Calculus E

PQ

. First, let us brieﬂy describe some syntactic notions.

By L we mean the language of Classical Predicate Calculus with individual

parameters, but without function symbols and identity. By a term of L we

mean an individual variable or a parameter. As before, we enrich this lan-

guage with a question-forming operator “?” and the sequent sign “”. All

the notions of the enriched language (e.g. that of a d-wﬀ and of a question)

are deﬁned according to the pattern presented in subsection 1.1; in particu-

lar, sequents of the language are of the form (1), where S is a ﬁnite sequence

of sentences of L and A is also a sentence. Moreover, a pure sentence is a

sentence (of L) with no individual parameters, and a sequent is called pure

if it contains only pure sentences.

Below the reader will ﬁnd the quantiﬁcational rules of the erotetic cal-

culus (the other rules cover the cases of propositional connectives and thus

D. Leszczy´nska-Jasion et al.

Table 2. κ and κ

∗

κ κ

∗

¬∀x

i

A ∃¬x

i

A

¬∃x

i

A ∀¬x

i

A

∀x

i

A,wherex

i

is not free in AA

∃x

i

A,wherex

i

is not free in AA

are analogous to the rules of E

∗

) and an example of a Socratic proof in this

calculus. The notions of a Socratic transformation and of a Socratic proof of

a pure sequent in E

PQ

are introduced as in definitions 1 and 2. For details

we refer the reader to [21].

L

∀

:

?(Φ; S

∀x

i

A

T B;Ψ)

?(Φ; S

∀x

i

A

A(x

i

/τ)

T B;Ψ)

R

∀

:

?(Φ; S ∀x

i

A;Ψ)

?(Φ; S A(x

i

/τ); Ψ)

provided that x

i

is free in A, provided that x

i

is free in A,

τ is any parameter and τ is a parameter which

does not occur in S nor in A

L

∃

:

?(Φ; S

∃x

i

A

T B;Ψ)

?(Φ; S

A(x

i

/τ)

T B;Ψ)

R

∃

:

?(Φ; S ∃x

i

A;Ψ)

?(Φ; S

∀x

i

¬A A(x

i

/τ); Ψ)

provided that x

i

is free in A, provided that x

i

is free in A,

and τ is a parameter which τ is any parameter

does not occur in S, A, T, B

L

κ

:

?(Φ; S

κ

T C;Ψ)

?(Φ; S

κ

∗

T C;Ψ)

R

κ

:

?(Φ; S κ;Ψ)

?(Φ; S κ

∗

;Ψ)

Obviously, the quantiﬁcational rules of E

PQ

are non-branching. Rules L

κ

and R

κ

cover the cases of quantiﬁers in the scope of negation and dummy

quantiﬁcation according to Table 2.

Example 5. A Socratic proof of sequent ∃xP (x) ∨∃xQ(x) →∃x(P (x) ∨

Q(x)) :

? ( ∃xP (x) ∨∃xQ(x) →∃x(P (x) ∨ Q(x)) ) R

β

?(∃xP (x) ∨∃xQ(x) ∃x(P (x) ∨ Q(x)) ) L

β

?(∃xP (x) ∃x(P (x) ∨ Q(x)) ; ∃xQ(x) ∃x(P (x) ∨ Q(x)) ) L

∃

?(P (a) ∃x(P (x) ∨ Q(x)) ; ∃xQ(x) ∃x(P (x) ∨ Q(x)) ) R

∃

?(P (a), ∀x¬(P (x) ∨ Q(x)) P (a) ∨ Q(a); ∃xQ(x) ∃x(P (x) ∨ Q(x)) ) R

β

?(P (a), ∀x¬(P (x) ∨ Q(x)), ¬P (a) Q(a); ∃xQ(x) ∃x(P (x) ∨ Q(x)) ) L

∃

?(P (a), ∀x¬(P (x) ∨ Q(x)), ¬P (a) Q(a); Q(a) ∃x(P (x) ∨ Q(x)) ) R

∃

?(P (a), ∀x¬(P (x) ∨ Q(x)), ¬P (a) Q(a); Q(a), ∀x¬(P (x) ∨ Q(x)) P (a) ∨ Q(a)) R

β

?(P (a), ∀x¬(P (x) ∨ Q(x)), ¬P (a) Q(a); Q(a), ∀x¬(P (x) ∨ Q(x)), ¬P (a) Q(a))

Calculus E

PQ

pertains to the Pure Calculus of Quantiﬁers in the follow-

ing sense:

Socratic Trees

Theorem 5. Let S A be a pure sequent. S A is provable in E

PQ

iﬀ

S A is valid.

7

The reader will ﬁnd the proof in [21] (it is worth noting that the authors

give a direct proof of the completeness part).

Now we may easily adjust the concepts from Section 2 to erotetic calculus

E

PQ

.Ifs is a Socratic transformation of (a question based on) a single-conc-

lusioned sequent via the rules of E

PQ

, then the structure X

s

,R

s

, where

X

s

and R

s

are understood according to Definitions 4, 5 and 6, will be called

the Socratic tree determined by Socratic transformation s.

3.2.2. Calculus G

PQ

. The rules of calculus G

PQ

for connectives are anal-

ogous to those of G

∗

, and the quantiﬁcational rules have the following form

(κ

∗

and κ are deﬁned as in Table 2):

L

+

∀

:

S

∀x

i

A

A(x

i

/τ)

T B

S

∀x

i

A

T B

R

+

∀

:

S A(x

i

/τ)

S ∀x

i

A

provided that x

i

is free in A, provided that x

i

is free in A,

τ is any parameter and τ is a parameter which

does not occur in S nor in A

L

+

∃

:

S

A(x

i

/τ)

T B

S

∃x

i

A

T B

R

+

∃

:

S

∀x

i

¬A A(x

i

/τ)

S ∃x

i

A

provided that x

i

is free in A, provided that x

i

is free in A,

and τ is a parameter which τ is any parameter

does not occur in S, A, T, B

L

+

κ

:

S

κ

∗

T C

S

κ

T C

R

+

κ

:

S κ

∗

S κ

As before, axioms of the calculus are of the forms speciﬁed in Defini-

tion 2. Also the notion of a proof of a sequent in the calculus is understood

as before.

In Figure 3 we present the Socratic tree determined by the above Socratic

proof (cf. Example 5) and the proof in calculus G

PQ

based upon it.

As in the propositional case we arrive at:

Theorem 6. If s is a Socratic proof of sequent S A in E

PQ

, then the tree

G(s) baseduponitisaproofofS A in G

PQ

.

7

The concept of validity of a sequent is deﬁned as follows: for each model M of L,for

each M -valuation v: if every element of S is satisﬁed by v in M ,thenA is satisﬁed by v

in M. For details see [21].

D. Leszczy´nska-Jasion et al.

Figure 3. Continuation of Example 5

Since calculus E

PQ

is sound and complete, we also have:

Theorem 7. Let S A be a pure sequent. S A is valid iﬀ S A has a

proof in calculus G

PQ

.

At the end of this section let us emphasize that sequent calculi G

∗

and

G

PQ

have the following properties: all the rules of the calculi are context-

sharing and invertible, there are no structural rules and the sequents are

single-conclusioned (single-succedent). None of the sequent calculi for clas-

sical logic which are traditionally considered in structural proof theory (cf.

[10,13]) has this collection of proof-theoretical features.

4. Final Remarks

As Buss [3, p. 18] correctly points out, there are two properties desirable

for a proof system amenable to computerized proof search. In such a proof

system (a) proof search is eﬃcient and does not require too many ‘arbitrary’

choices, and (b) proof lengths are not excessively long.

On the one hand, in the case of SP-method proof search is eﬃcient indeed.

First, there are no primary structural rules. Second, due to the “immedi-

ate-consequence” mode of rules’ application the only choice to be made at

Socratic Trees

a certain stage of a transformation concerns the sequent to be picked up as

an active one. Even this is suppressed in the case of r-minimal transforma-

tions (see the Appendix).

On the other hand, it is rather typical for tableau methods that they

avoid “dummy” occurences of wﬀs to be present within a transformation:

only active and leaf nodes are displayed, or, equivalently, only the nodes

obtained by an application of some rule. The same holds in the case of

Socratic trees (cf. Theorem 2). As a consequence, the number of non-leaf

nodes determines the number of rules’ application in a given Socratic trans-

formation and thus its computational complexity. This is important for the

SP-method itself. But one can also say that Socratic trees measure compu-

tational complexity of the corresponding Gentzen-style proofs.

There is one more advantage of Socratic trees worth to be mentioned,

although probably a side one: they are great didactic tool. It stems from

teaching experience of the second author that when it comes to manual cal-

culations, then operating on single-conclusioned sequents, lack of primary

structural rules and immediate-consequence style of rules give an advantage

to SP-method over virtually any other sequent-based proof setting. Socratic

trees, in turn, oﬀer even more user-friendly framework by avoiding tedious

rewriting of non-active sequents. In case of sequent calculus the tree-format

provides an element of geometrical intuition (as it does in case of natural

deduction; cf. [13, p. 34]).

One ﬁnal remark is in order. We considered here only the cases of CPC

and FOL. But a translation of a Socratic proof into a Socratic tree is also an

important part of the procedure of determining proofs in standard sequent

calculi for modal logics (cf.[9]) which opens another interesting perspective

for research in the ﬁeld of automated deduction in modal logics. On the

other hand, it is possible to transform Socratic trees into proofs in some

other proof formats, like Smullyan’s analytic tableaux or Rasiowa-Sikorski

diagrams of formulas. This issues are still under investigation, but it is clear

at the present moment that Socratic trees may occur to be a very universal

tool in proof-theory.

Appendix

It is easy to construct the set X

s

of the Socratic tree determined by a Socratic

transformation s—namely, it is sufﬁcient to compare the constituents of the

consecutive questions and to pick these constituent(s) that occur as “new”.

Such new (annotated) constituents together with the only constituent of the

D. Leszczy´nska-Jasion et al.

ﬁrst question form the required set. However, extracting relation R

s

from a

Socratic transformation seems more complicated. One possibility is to follow

definitions 4, 5 and 6, the result, however, occurs quite complex (see Algo-

rithm 1 below), as it requires generation of three relations: P

s

, P

tr

s

and R

s

,

instead of the (solely) last one. (Recall that R

s

is a restriction of P

tr

s

and

P

tr

s

is a transitive closure of P

s

.) Algorithm 1 uses four procedures presented

below. It may be used to generate Socratic trees for both CPC and FOL—

in the second case a Socratic transformation determining the tree must be

ﬁnite.

Data: ﬁnite Socratic transformation s = s

1

,...,s

n

of question ?(S A)

Result: the set X

s

of nodes of Tr(s) and the relations P

s

, P

tr

s

, R

s

begin

X

s

←{1, 1,S A}

P

s

←∅

P

tr

s

←∅

R

s

←∅

n ← the number of questions of s

if n>1 then

for i=2 to n do

k ← the number of constituents of question s

i

j ← 1

while j-th constituent of s

i−1

and j-th constituent of s

i

is the same

sequent do

Relations1(s, i, j, P

s

, P

tr

s

)

j ← j +1

end

if s

i−1

and s

i

have the same number of constituents then

Relations2(s, i, j, X

s

, P

s

, P

tr

s

, R

s

)

for l=j+1 to k do

Relations1(s, i, l, P

s

, P

tr

s

)

end

else

Relations3(s, i, j, X

s

, P

s

, P

tr

s

, R

s

)

for l=j+1 to k-1 do

Relations4(s, i, l, P

s

, P

tr

s

)

end

end

end

end

end

Algorithm 1: Socratic tree determined by s

Socratic Trees

Input: s—Socratic transformation, i—indicates the question of s, j—indicates the

constituent of question s

i

, P

s

, P

tr

s

Result: consecutive pairs are “added” to relations P

s

and P

tr

s

φ ← j-th constituent of s

i−1

ψ ← j-th constituent of s

i

P

s

← P

s

∪{i − 1,j,φ, i, j, ψ}

foreach pair , σ∈P

tr

s

do

if σ = i − 1,j,φ then

P

tr

s

← P

tr

s

∪{, i, j, ψ}

end

end

P

tr

s

← P

tr

s

∪{i − 1,j,φ, i, j, ψ}

Procedure Relations1(s, i, j, P

s

, P

tr

s

)

Input: s—Socratic transformation, i—indicates the question of s, j—indicates the

constituent of s

i

, X

s

, P

s

, P

tr

s

, R

s

Result: consecutive annotated sequents/their pairs are “added” to set X

s

/relations

P

s

, P

tr

s

and R

s

φ ← j-th constituent of s

i−1

ψ ← j-th constituent of s

i

X

s

← X

s

∪{i, j, ψ}

P

s

← P

s

∪{i − 1,j,φ , i, j, ψ}

foreach pair , σ∈P

tr

s

do

if σ = i − 1,j,φ then

P

tr

s

← P

tr

s

∪{, i, j, ψ}

if ∈ X

s

then

R

s

← R

s

∪{, i, j, ψ}

end

end

end

P

tr

s

← P

tr

s

∪{i − 1,j,φ , i, j, ψ}

if i − 1,j,φ∈X

s

then

R

s

← R

s

∪{i − 1,j,φ , i, j, ψ}

end

Procedure Relations2(s, i, j, X

s

, P

s

, P

tr

s

, R

s

)

As we have mentioned in Section 3 the fact that the nodes of a Socratic

tree are annotated sequents may be used to significantly simplify the algo-

rithm of tree construction (see Algorithm 3). However, the simpliﬁed algo-

rithm works properly provided that the tree is determined by a Socratic

transformation of a special sort, which will be called r-minimal (see Algo-

rithm 2). We will now present and discuss some additional advantages of

such r-minimal Socratic transformations.

D. Leszczy´nska-Jasion et al.

Input: s—Socratic transformation, i—indicates the question of s, j—indicates the

constituent of s

i

, X

s

, P

s

, P

tr

s

, R

s

Result: consecutive annotated sequents/their pairs are “added” to set X

s

/relations

P

s

, P

tr

s

and R

s

φ ← j-th constituent of s

i−1

ψ

1

← j-th constituent of s

i

ψ

2

← (j + 1)-st constituent of s

i

X

s

← X

s

∪{i, j, ψ

1

, i, j +1,ψ

2

}

P

s

← P

s

∪{i − 1,j,φ , i, j, ψ

1

, i − 1,j,φ , i, j +1,ψ

2

}

foreach pair , σ∈P

tr

s

do

if σ = i − 1,j,φ then

P

tr

s

← P

tr

s

∪{, i, j, ψ

1

, , i, j +1,ψ

2

}

if ∈ X

s

then

R

s

← R

s

∪{, i, j, ψ

1

, , i, j +1,ψ

2

}

end

end

end

P

tr

s

← P

tr

s

∪{i − 1,j,φ, i, j, ψ

1

, i − 1,j,φ, i, j +1,ψ

2

}

if i − 1,j,φ∈X

s

then

R

s

← R

s

∪{i − 1,j,φ , i, j, ψ

1

, i − 1,j,φ , i, j +1,ψ

2

}

end

Procedure Relations3(s, i, j, X

s

, P

s

, P

tr

s

, R

s

)

Input: s—Socratic transformation, i—indicates the question of s, j—indicates the

constituent of question s

i

, P

s

, P

tr

s

Result: consecutive pairs are “added” to relations P

s

and P

tr

s

φ ← l-th constituent of s

i−1

ψ ← (l + 1)-st constituent of s

i

P

s

← P

s

∪{i − 1,l,φ , i, l +1,ψ}

foreach pair , σ∈P

tr

s

do

if σ = i − 1,l,φ then

P

tr

s

← P

tr

s

∪{, i, l +1,ψ}

end

end

P

tr

s

← P

tr

s

∪{i − 1,l,φ , i, l +1,ψ}

Procedure Relations4(s, i, l, P

s

, P

tr

s

)

We will use B-rule (NB-rule) for branching (non-branching) rules and

we will say that a sequent is a B-sequent (an NB-sequent) if a B-rule (an

NB-rule) is applicable with respect to it. A sequent S A is called atomic

if A and each element of S is either a propositional variable or negation of

a propositional variable. A sequent is called open if it is of neither of the

forms speciﬁed in Definition 2.

Algorithm 2 allows to bound the complexity of Socratic transformations

and thus it bounds also the complexity of Socratic trees and Gentzen-style

Socratic Trees

Data: question ?(S A)

Result: r-minimal Socratic transformation of question ?(S A) (a Socratic proof of

S A , if the sequent is provable)

begin

Q ←?(S A)

while there is no open and atomic constituent of Q AND there is an open

constituent of Q do

while there is an open NB-constituent of Q do

φ ← the rightmost open NB-constituent of Q

apply an NB-rule with respect to φ (if more than one NB-rule is

applicable, the choice is optional)

Q ← the last question obtained so far

end

if there is an open B-constituent of Q then

φ ← the rightmost open B-constituent of Q

apply a B-rule with respect to φ (if more than one B-rule is applicable,

the choice is optional)

Q ← the last question obtained so far

end

end

if there is an open constituent of Q then

return “sequent S A is not provable”

else

return “sequent S A is provable”

end

end

Algorithm 2: r-minimal Socratic transformation

proofs based on them. The algorithm runs according to the following princi-

ples: ﬁrst, the work is ﬁnished if the last question of the constructed Socratic

transformation has an open and atomic constituent, that is, a constituent

which is of neither of the required forms but yet no rule is applicable with

respect to it—in this case the initial sequent is not provable and further

continuation of the Socratic transformation (if possible at all) is pointless;

second, the rules are never applied with respect to sequents which are not

open—again, there is no point to do this if we aim at a proof; and third, a

branching rule is never applied with respect to an NB-sequent.

The letter “r” indicates another important feature of the algorithm—the

rules are applied with respect to the rightmost constituents. Obviously, it is

possible to give an algorithm for constructing an l-minimal Socratic trans-

formation, however, the feature of starting from the right side is crucial for

Algorithm 3 to work properly.

The last algorithm generates the set of nodes, X

s

, of a Socratic tree and—

omitting the construction of relations P

s

and P

tr

s

—a relation symbolized by

D. Leszczy´nska-Jasion et al.

Data: ﬁnite r-minimal Socratic transformation s = s

1

,...,s

n

of question S A

Result: the set X

s

of nodes of Tr(s)andrelationR

s

begin

X

s

←{1, 1,S A}

R

s

←∅

n ← the number of questions of s

if n>1 then

for i=2 to n do

j ← 1

while j-th constituent of s

i−1

and j-th constituent of s

i

is the same

sequent do

j ← j +1

end

if s

i−1

and s

i

have the same number of constituents then

φ ← j-th constituent of s

i

X

s

← X

s

∪{i, j, φ}

k ← i − 1

while for no sequent ψ: k, j, ψ∈X

s

do

k ← k − 1

end

ψ ← the sequent such that k, j,ψ∈X

s

R

s

← R

s

∪{k, j,ψ, i, j, φ}

else

φ ← j-th constituent of s

i

ψ ← (j + 1)-st constituent of s

i

X

s

← X

s

∪{i, j, φ, i, j +1,ψ}

k ← i − 1

while for no sequent χ: k, j, χ∈X

s

do

k ← k − 1

end

χ ← the sequent such that k,j, χ∈X

s

R

s

← R

s

∪{k, j,χ, i, j, φ, k, j,χ, i, j +1,ψ}

end

end

end

end

Algorithm 3: relation R

s

of tree Tr(s)

R

s

, which is the relation of the immediate predecessor of relation R

s

, that is,

, σ∈R

s

iﬀ is the immediate R

s

-predecessor of σ. Generating relation

R

s

is obviously enough to construct the tree.

Algorithm 3 allows for a simpliﬁed construction of a Socratic tree in the

case of CPC. The case of FOL is obviously different, since—for well-know

reasons—algorithms like Algorithm 2 do not work for FOL in general. It is

an interesting question whether there are heuristics that could be used to

overcome this problem in the framework of Socratic trees, however, we are

not going to adress this issue in this paper.

Socratic Trees

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D. Leszczy´nska-Jasion et al.

Dorot a Leszczy

´

nska-Jasion, Mariusz Urba

´

nski and Andrzej Wi

´

sniewski

Chair of Logic and Cognitive Science

Adam Mickiewicz University

60-586 Pozna´n, Poland

Dorota.Leszczynska@amu.edu.pl

Mariusz Urba

´

nski

Mariusz.Urbanski@amu.edu.pl

Andrzej Wi

´

sniewski

Andrzej.Wisniewski@amu.edu.pl