Negation

Negation is a connective that occurs in virtually all known natural languages. Apart from being universal, it is also a cause of disputes between experts in various domains interested in human communicative skills. Can we identify negation with a unary connective formalized by classical logic? Positive answer to this question would end most of these discussions. Nevertheless, a lot of researches – especially from the last century – suggest that ordinary approach to negation is too shallow. In other words, the perspective designated by stoics and Gottlob Frege distorts our thinking about negation.

Taking into account, for example, pragmatic perspective, it would expose many problems with this connective. First of all, negation in natural languages may have a great deal of non-equivalent forms. Consequently, the role of negation in some language may not rely only on a denial of a given statement. Despite of what many thinkers from Plato to contemporary ones claim, negative statements seem to be derivative in their psychological, grammatical, epistemological and ontological aspects. Denials provide less information than affirmations, they are distinguished in language more clearly and require other epistemological strategies than synthesis. Moreover, one might believe that negation is an essential part of a meaning and, consequently, propositional attitude associated with denials are irreducible to simple disclamation of some assertion. In broader perspective one may ask whether states of affairs corresponding to negative statements exist.

The problems pointed above are not exhaustive. An argumentative debate on negation should include comments of philosophers, psychologists, linguists, logicians, mathematicians, and computer scientists. The main goal of the conference is to allow such discussion.

The conference will be held in Polish and English.

Marie Duzi

Presuppositions and two kinds of negation

In this talk I deal with sentences that come with a presupposition that is entailed by the positive as well as negated form of a given sentence. However, there are two kinds of negation, namely narrow-scope and wide-scope negation. I am going to prove that while the former is presupposition-preserving, the latter is presupposition-denying. Thus, the main contribution of this paper is the proof that these two kinds of negation are not equivalent. This issue has much in common with the difference between topic and focus articulation within a sentence. Whereas articulating the topic of a sentence activates a presupposition, articulating the focus frequently yields merely an entailment. My background theory is Transparent Intensional Logic (TIL). TIL is an expressive logic apt for the analysis of sentences with presuppositions, because in TIL we work with partial functions, in particular with propositions with truth-value gaps. Moreover, procedural semantics of TIL makes it possible to define a general analytic schema of sentences associated with presuppositions, which is another novel contribution of this paper.

Pavel Materna

Logical analysis of empirical expressions. What is wrong with empiricism

The following well-known problem motivated my handling more general problems: As we surely know our pupils and even students are confronted with much more troubles when learning mathematics (and even physics) than when they learn empirical sciences like biology, mineralogy etc. There are many factors that can at least partially explain this phenomenon, I would however mention one factor that is not too frequently adduced: mathematics, logic and much of physics use concepts that are abstract while the empirical sciences seem to support understanding by using expressions concerning (denoting? expressing?) concrete objects.

Therefore the first topic to be explained (or explicated) is: Abstract vs. concrete. The second point will consist of applying the first point to explanation of the troubles with learning mathematics. The third point will ask Logical Analysis of Natural Language how to tell abstract expressions from the concrete ones.

The fourth point will confront the conception described in the foregoing point with the conceptions trying to abandon the distinction between the analytic and empirical expressions. Here it will be shown that empiricism representing this latter conception deprives semantics as applied to Natural language of important features of expressivity.

David Charles McCarty

Logical rules and the meanings of the connectives

In the talk, I plan to answer three important questions concerning logical relations between intuitionistic and classical logic and mathematics - and their respective negations. The first question is, "What is a truth-value, anyway?" Second, "Can every model of classical set theory be extended to a model of a reasonable, strictly intuitionistic theory?" (This is an old question, and I repeat an old, but interesting answer.) And, third, "Can the syntactic rules of deduction in either classical or intuitionistic formal logic determine the meanings of the respective connectives signs?"

Adam Olszewski

Negation in theological language

In the first part of my project I want to present some general considerations regarding what negation essentially is from the philosophical perspective. In the second part I indicate five different kinds of negation, functioning in other disciplines too, how they work in the theological investigations. I base these considerations on concrete examples of the functioning of negation.

Bartłmiej Skowron

Negating as turning upside down

I suggest comparing the phenomenon of negation to the dual categories in category theory. For each category, if one reverses the direction of the arrow and compositions, one gets the opposite category to it, as if it were its own "negation”. This type of formal operation sometimes has no clear meaning, but every category has its opposite category. Moreover, the phenomenon of duality, in general, has a clear form in category theory: If the sentence holds for all categories, the dual sentence also holds for all categories.

Dual to each other are: epimorphism and monomorphism, an initial object and a terminal object, invertibility of an arrow, the product and coproduct, etc. The opposite categories and the phenomenon of duality can be thought of as turning the world upside-down. It turns out that this approach has a number of interesting and unexpected consequences - in logic the mechanism of duality can give the idea of using the reverse of proofs: figuratively and not strictly speaking, in the dual world one does not prove theorems from assumptions, but one deduces assumptions from theorems.