9:3/4:
Judgments and Truth: Essays in Honour of Jan Woleński
It is a Preface to Volume 9:3/4 that has brought a renewed focus to the role of truth conceptions in frameworks of semantics and logic. Jan Woleński is known due to his works on epistemological aspects of logic and his systematization of semantic truth theory. He became the successor and the worthy continuer of prominent Polish logicians: Alfred Tarski and Kazimierz Ajdukiewicz. This volume is collected on the 80th anniversary of Woleński’s birth and draws together new research papers devoted to judgments and truth. These papers take measure of the scope and impact of Woleński's views on truth conceptions, and present new contributions to the field of philosophy and logic.
Proof vs Truth in Mathematics
Two crucial concepts of the methodology and philosophy of mathematics
are considered: proof and truth. We distinguish between informal proofs
constructed by mathematicians in their research practice and formal
proofs as defined in the foundations of mathematics (in
metamathematics). Their role, features and interconnections are
discussed. They are confronted with the concept of truth in mathematics.
Relations between proofs and truth are analysed.
The Mystery of the Fifth Logical Notion
(Alice in the Wonderful Land of Logical Notions)
We discuss a theory presented in a posthumous paper by Alfred Tarski
entitled “What are logical notions?”. Although the theory of these logical
notions is something outside of the main stream of logic, not presented in
logic textbooks, it is a very interesting theory and can easily be
understood by anybody, especially studying the simplest case of the four
basic logical notions. This is what we are doing here, as well as
introducing a challenging fifth logical notion. We first recall the context
and origin of what are here called Tarski-Lindenbaum logical notions. In
the second part, we present these notions in the simple case of a binary
relation. In the third part, we examine in which sense these are
considered as logical notions contrasting them with an example of a nonlogical
relation. In the fourth part, we discuss the formulations of the four
logical notions in natural language and in first-order logic without
equality, emphasizing the fact that two of the four logical notions cannot
be expressed in this formal language. In the fifth part, we discuss the
relations between these notions using the theory of the square of
opposition. In the sixth part, we introduce the notion of variety
corresponding to all non-logical notions and we argue that it can be
considered as a logical notion because it is invariant, always referring to
the same class of structures. In the seventh part, we present an enigma: is
variety formalizable in first-order logic without equality? There follow
recollections concerning Jan Woleński. This paper is dedicated to his 80th
birthday. We end with the bibliography, giving some precise references
for those wanting to know more about the topic.
Idea of Artificial Intelligence
Artificial Intelligence, both as a hope of making substantial progress, and a fear of the unknow n and unimaginable, has its roots in hum an dreams. These dreams are materialized by means of rational intellectual efforts. We see the beginnings of such a process in Lullus’s fancies. Many scholars and enthusiasts participated in the development of Lullus’s art, ars com binatoria. Amongst them, Athanasius Kircher distinguished himself. Gottfried Leibniz ended the period in whic h the idea of artificial intelligence was shap ed, and started the new period, in which artificial intelligence could be conside red part of science, by today’s standards.
Conjunctive and Disjunctive Limits: Abstract
Logics and Modal Operators
Departing from basic concepts in abstract logics, this paper introduces two
concepts: conjunctive and disjunctive limits. These notions are used to
formalize levels of modal operators.
A Judgmental Reconstruction of Some of
Professor Woleński’s Logical and Philosophical Writings
Roman Suszko said that “Obviously, any multiplication of logical values is a mad
idea and, in fact, Łukasiewicz did not actualize it.” The aim of the present paper is
to qualify this ‘obvious’ statement through a number of logical and philosophical
writings by Professor Jan Woleński, all focusing on the nature of truth-values and
their multiple uses in philosophy. It results in a reconstruction of such an abstract
object, doing justice to what Suszko held a ‘mad’ project within a generalized
logic of judgments. Four main issues raised by Woleński will be considered to test
the insightfulness of such generalized truth-values, namely: the principle of
bivalence, the logic of scepticism, the coherence theory of truth, and nothingness.
Reism, Concretism and Schopenhauer Diagrams
Reism or concretism are the labels for a position in ontology and semantics that
is represented by various philosophers. As Kazimierz Ajdukiewicz and Jan
Woleński have shown, there are two dimensions with which the abstract
expression of reism can be made concrete: The ontological dimension of reism
says that only things exist; the semantic dimension of reism says that all
concepts must be reduced to concrete terms in order to be meaningful. In this
paper we argue for the following two theses: (1) Arthur Schopenhauer has
advocated a reistic philosophy of language which says that all concepts must
ultimately be based on concrete intuition in order to be meaningful. (2) In his
semantics, Schopenhauer developed a theory of logic diagrams that can be
interpreted by modern means in order to concretize the abstract position of
reism. Thus we are not only enhancing Jan Woleński’s list of well-known
reists, but we are also adding a diagrammatic dimension to concretism,
represented by Schopenhauer.
Deontic Relationship in the Context
of Jan Woleński’s Metaethical Naturalism
In this paper, we indicate how Jan Woleński’s non-linguistic concept of the norm
allows us to clarify the deontic relationship between sentences and the given
normative system. A relationship of this kind constitutes a component of the
metalogic of relating deontic logic, which subjects the logical value of the deontic
sentence to the logical value of the constituent sentence and its relationship with a
given normative system in the accessible possible worlds.
A Note on Intended and Standard Models
This note discusses some problems concerning intended, standard, and nonstandard
models of mathematical theories. We pay attention to the role of
extremal axioms in attempts at a unique characterization of the intended
models. We recall also Jan Woleński’s views on these issues.
About Some New Methods of Analytical Philosophy.
Formalization, De-formalization and Topological Hermeneutics
In this article I want to continue the characteristics of philosophical methods
specific to analytical philosophy, which were and are important for Professor
Jan Woleński. So I refer to his work on the methods of analytical philosophy,
but I also point out a few new methods that have grown up in the climate of
studies of philosophers, especially analytical ontologists. I will therefore
describe the following methods: generalization, specialization, formalization,
de-formalization and topological hermeneutics. Instead of the term “method” I
use interchangeably the terms “operation” or “procedure”. I will show that each
of these operations makes an important contribution to ontological
investigations, and, in particular, to formal ontology.
Anti-foundationalist Philosophy of Mathematics
and Mathematical Proofs
The Euclidean ideal of mathematics as well as all the foundational schools in
the philosophy of mathematics have been contested by the new approach,
called the “maverick” trend in the philosophy of mathematics. Several points
made by its main representatives are mentioned – from the revisability of
actual proofs to the stress on real mathematical practice as opposed to its
idealized reconstruction. Main features of real proofs are then mentioned; for
example, whether they are convincing, understandable, and/or explanatory.
Therefore, the new approach questions Hilbert’s Thesis, according to which a
correct mathematical proof is in principle reducible to a formal proof, based on
explicit axioms and logic.
Necessity and Determinism in Robert Grosseteste’s De libero arbitrio
In this paper, the theory of necessity proposed by Robert Grosseteste is
presented. After showing the wide range of various kinds of
determination discussed by him (connected with: (1) one’s knowledge
about the future, (2) predestination, (3) fate, (4) grace, (5) sin and
temptation), a different context of Grosseteste’s use of the notion of
necessity is analyzed (within logical and metaphysical approaches). At
the heart of his theory lie: the definition of necessity, which is that
something lacks the capacity (posse) for its opposite, and the distinction
between two perspectives within which we can consider necessity: (1)
the one according to which the truthfulness of a dictum determines that it
cannot be the opposite, (2) a pre- or atemporal one, as if something had
not yet begun. On these grounds, Robert explains that God’s omniscience
is compatible with contingency, including human free decisions. Robert’s
theory is still relevant and useful in contemporary debates, as it can
provide strong arguments and enrich discussions, thanks to the twoperspectives
approach, which generates nine kinds of positions on the
spectrum of determinism and indeterminism.
Logical Consequence Operators and Etatism
In the paper, there is presented the theory of logical consequence operators
indexed with taboo functions. It describes the mechanisms of logical inference
in the environment of forbidden sentences. This kind of processes take place in
ideological discourses within which their participants create various narrative
worlds (mental worlds). A peculiar feature of ideological discourses is their
association with taboo structures of deduction which penalize speech acts. The
development of discourse involves, among others, transforming its deduction
structure towards the proliferation of consequence operators and modifying
penalty functions. The presented theory enables to define various processes of
these transformations in the precise way. It may be used in analyses of conflicts
between competing elm experts acting within a discourse.
The Normative Permission and Legal Utterances
The author proves that rejecting the existence of permissive norms and limitation
of norms to prohibitions and commands alone is possible only with reducing the
idea of a function. The essence of the function is then the ability of the expression
to generate independently the universal norm formation. Such manipulation is easy
on the level of logical analysis, but proves risky from other points of view. If we
want the deontic logic, which we construct, to consider the fact that permission is
pragmatically necessary for the law-maker to convey his normative preferences,
we must solve the consequences of the adopted structure of the function of norms,
which originate on the socio-linguistic level. It appears, however, that due to a lack
of a pragmatic theory useful for lawyers, there is no proof that the pragmatically
strong permission can be expressed by means of a lot of prohibitions and
commands (dos and don’ts). Besides, reducing permissions only to the language of
legal rules is an obligation to accept the structure of an act of communication,
which can find its full motivation in the Husserl’s structure of the direct cognition.
