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 providing 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

Full text

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
,ifφ
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 ,andτ
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.AtripleX
sG
,R
sG
,λ
s
,thatis,thetreeX
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
References
[1] McAllester, D., and R. Givan, Taxonomic syntax for first order inferences, Journal
of ACM 40(2):246–283, 1993.
[2] Av r o n , A., The method of hypersequents in the proof theory of propositional non-
classical logics, in W. Hodges et al. (eds.), Logic: Foundations to Applications, Oxford
Science Publications, Clarendon Press, Oxford, 1996, pp. 1–32.
[3] Buss,S.R.(ed.),Handbook of Proof Theory, Elsevier, Amsterdam, 1998.
[4] Heeffer,A.,Automated Theorem Prover for CL and CLuN Based on the Method of
Socratic Proofs, Centre for Logic and Philosophy of Science, Ghent University, 2003.
[5] Leszczy
´
nska, D., Socratic proofs for some normal modal propositional logics, Logique
et Analyse 185–188, 259–285, 2004.
[6] Leszczy
´
nska,D.,The Method of Socratic Proofs for Normal Modal Propositional
Logics, Adam Mickiewicz University Press, Pozna´n, 2007.
[7] Leszczy
´
nska-Jasion, D., The method of socratic proofs for modal propositional
logics: K5, S4.2, S4.3, S4M, S4F, S4R and G, Studia Logica 89(3):371–405, 2008.
[8] Leszczy
´
nska-Jasion, D., A loop-free decision procedure for modal propositional log-
ics K4, S4 and S5, Journal of Philosophical Logic 38(2):151–177, 2009.
[9] Leszczy
´
nska-Jasion,D.,On Determining Modal Sequent Proofs by Socratic Proofs,
Research Report, Adam Mickiewicz University, Institute of Psychology, Chair of Logic
and Cognitive Science, 2010.
[10] Negri, S., and J. von Pla to, Structural Proof Theory, Cambridge University Press,
Cambridge, 2001.
[11] Skura, T., Intuitionistic socratic procedures, Journal of Applied Non-Classical Logics
15(4):453–464, 2005.
[12] Smullyan,R.M.,First-Order Logic, Springer-Verlag, Berlin, 1968.
[13] Troelstra,A.S.,andH.Schwichtenberg, Basic Proof Theory (2nd ed.), Cam-
bridge University Press, Cambridge, 2000.
[14] Urba
´
nski,M.,Socratic Proofs for Some Temporal Logics, Research Report,Adam
Mickiewicz University, Institute of Psychology, Chair of Logic and Cognitive Science,
2005.
[15] Wi
´
sniewski, A., Erotetic implications, Journal of Philosophical Logic 23(2):174–195,
1994.
[16] Wi
´
sniewski,A.,The Posing of Questions: Logical Foundations of Erotetic Inferences,
Kluwer Academic Publishers, Dordrecht/Boston/London, 1995.
[17] Wi
´
sniewski, A., The logic of questions as a theory of erotetic arguments, Synthese
109(2):1–25, 1996.
[18] Wi
´
sniewski, A., Questions and inferences, Logique et Analyse 173–175, 5–43, 2001.
[19] Wi
´
sniewski, A., Socratic proofs, Journal of Philosophical Logic 33(3):299–326, 2004.
[20] Wi
´
sniewski,A.,G.Vanackere,andD.Leszczy
´
nska, Socratic proofs and para-
consistency: a case study, Studia Logica 80(2–3):433–468, 2005.
[21] Wi
´
sniewski, A., and V. Shangin, Socratic proofs for quantifiers, Journal of Philo-
sophical Logic 35(2):147–178, 2006.

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