Logic and Its Philosophy

Preface

This book collects 20 of my papers published in the years 2011–2016. All of them, with one exception, are devoted to logic and its philosophy. The essay XX is the only exception, although it also alludes to some logical questions. However, I included considerations on Keret’s House for personal reasons. My roots are Jewish. I spent the years 1941–1944 in Warsaw, not in ghetto, but in relatively normal (if anything was normal at the time) circumstances. Although my family survived, but… (let me not finish). Hence, I am very sensitive to philosophical (and other) problems related to the Holocaust.

The rest of the book is a sequel to my Essays on Logic and its Applications in Philiosophy, published by Peter Lang in 2011. But this collection is more compact because all chapters belong to systematic philosophy (I did not include historical studies). However, I continue topics considered in the mentioned book of 2011 and use similar analytic tools taken from formal logic, especially the logical square and its generalization. Generally speaking, both collections can be viewed as contributions to so-called philosophical logic.

The papers included in this collection are reprinted here with changes introduced for avoiding repetitions. Yet, particular chapters overlap at some points; in particular, the diagrams re-appear in few places. Although I tried to unify symbolism to some extent, there are some differences caused by the fact that specific letters and signs play various roles. However, the context always explains what is going on at a given place. The numeration of formulas and statements is separate for each chapter.

The book is financially supported by the University of Information, Technology and Management in Rzeszów, Poland. I am grateful to its authorities, especially Dr. Wirgiliusz Gołąbek for this help. I would also like to thank Mr. Łukasz Gałecki for his initiative that this collection could be published by Peter Land. I am also grateful the text-editors for thei efforts toward preparing the final version of the book. All papers are reprinted with permission of particular publishers.

Jan Woleński

← 7 | 8 → ← 8 | 9 →

Source note

Semantic Loops, in Philosophy in Science. Methods and Applications, ed. by B. Brożek, W. P. Grygiel and J. Mączka, Copernicus Center Press, Kraków 2011, 256–271.

Logic as Calculus Versus Logic as Universal Medium, and Syntax Versus Semantics, Logica Universalis 6(3)(2012), 587–596.

Do We Need to Reform the Old T-Scheme, Discusiones Filozóficas (20)(13) (2012), 73–85.

Truth is Eternal if and only if it is Sempiternal, in: Studies in the Philosophy of Herbert Hochberg, ed. by E. Tegtmeier, Ontos, Frankfurt am Main 2012, 223-230.

Is Identity a Logical Constant and are there Accidental Identities?, Studia Humana 3-4(2012), studia humana.com/current-issue.html.com.

Naturalism and the Genesis of Logic, in Papers on Logic and Rationality. Festschrift in Honor of Andrzej Grzegorczyk (Studies in Logic, Grammar and Rhetoric 27(40)); together with K. Trzęsicki and S. Krajewski, University of Bialystok, Bialystok 2012, 223–240.

Some Analogies between Normative and Epistemic Discourse, in: The Many Faces of Normativity, ed, by J. Stelmach, B. Brożek and M. Hohol, Copernicus Center, Kraków 2013, 51–71.

Theology and Logic, in: Logic in Theology, ed. by B. Brożek, A. Olszewski and M. Hohol, Copernicus Center Press, Kraków 2013, 11–38.

Ens et Verum Concertuntur (Are Being and Truth Convertible)? A Contermporary Pesrpective, in: Truth, ed. by M. Dumitriu and G. Sandu, Editura Universitãtii din Bucureşti, Bucureşsti 2013, 75–83.

An Abstract Approach to Bivalance, Logic and Logical Philosophy 23(1)(2014), 3–14.

Philosophical Reflections on Logic, Proof and Truth, in: Trends in Logic XIII. Gentzen’s and Jaskowski’s Heritage. 80 Years of Natural Deduction and Sequent Calculi, ed. by A. Indrzejcak, J. Kaczmarek and M. Zawidzki, Wydawnictwo Uniwersytetu Łódzkiego, Lódź 2014, 27–39.

Constructivism and Metamathematics, in The Road to Universal Logic. Festschrift for 50th Birthday of Jean-Yves Béziau, v. I, ed. by A. Koslow and A. Buchsbaum, Birkhäuser, Basel 2015, 513-520. ← 9 | 10 →

Rule-Following and Logic, in Problems of Normativity, Rules and Rule-Following, ed. by M. Araszkiewicz, P. Banas, T. Gizbert-Studnicki and Krzysztof Płeszka, Springer, Heidelberg 2014, 395–402.

Truth-Makers and Convention T, in Philosophical Papers for Kevin Mulligan, ed. by A. Reboul, in Mind, Values, and Metaphysics. Philosophical Essays in Honor of Kevin Mulligan, v. 1, Springer, Dordrecht 2014, 79–84.

Constructivism and Metamathematics, in The Road to Universal Logic. Festschrift for 50th Birthday of Jean-Yves Béziau, v. I, ed. by A. Koslow and A. Buchsbaum, Birkhäuser, Basel 2015, 513–520.

An Analysis of Logical Determinism, in Themes from Ontology, Mind and Logic. Present and Past. Essays in Honour of Peter Simons, ed. by S. Lapointe, Brill, Leiden 2015, 423–442.

Normativity of Logic, in The Normative Mind, ed. by J. Stelmach, B. Brożek and Ł. Kwiatek, Copernicus Center 2016, 169–195.

Formal and Informal Aspects of the Semantic Theory of Truth, in: Uncovering Facts and Values. Studies in Contemporary Epistemology and Political Philosophy, ed. by A. Kuźniar and J. Odrowąż-Sypniewska, Brill/Rodopi, Leiden 2016, 56–66.

The Paradox of Analyticity and Related Issues, in Modern Logic 1850–1950. East and West, ed. by F. Abeles and M. E. Fuller, Birkhäuser, Basel 2016, 135–138.

Some Liar-Like Paradoxes, Logic-Philosophical Studies. Yearbook of St. Perterburg Logic Association (14(2016), 70–75.

Nothingness, Philosophy and Keret’s House, in: Przestrzenie pustki/Void Spaces, Fundacja Polskiej Sztuki Nowoczesnej/Foundation of Polish Modern Art., Warszawa 2013, 23–39. ← 10 | 11 →

I
Semantic Loops

Michael Heller introduced (see Heller 1999, pp. 90, 99–100) the idea of semantic loops. They are essentially involved in interactions between languages and their metalanguages (I will use the letter L as referring to a language and the symbol ML as denoting a metalanguage of L). As it is well-known, such interactions lead to semantic antinomies related to the self-referential use of expressions. The famous Liar antinomy (LA for brevity) is perhaps the most paradigmatic example. Consider the sentence:

A simple inspection shows that (λ) is true if and only if it is false (it will be demonstrated below). The situation illustrated by (λ) can be metaphorically characterized as a closed semantic loop, because we pass from truth to falsehood and back without any possibility of leaving the loop (or a circle, if you prefer this, more popular, figure of speech; see also a historico-terminological digression below). On the other hand, the language/metalanguage distinction cannot be liquidated, because we need to speak about languages and their various properties.

Although ML can be effectively reduced to L in the case of syntax (for example, via the method of Gödel numbering), this is impossible in the case of semantics; Tarski showed that doing semantics of L in ML requires that the latter is essentially richer than the former. Thus, closed semantic loops operate somehow between L and ML. As Heller indirectly (as he speaks about loops in physics) suggests, a solution of a semantic paradox assumes that loops related to such antinomies are not quite closed (for stylistic reasons, I use the labels ‘antinomy’ and ‘paradox’ as synonyms). In order to have a convenient façon de parler, I will contrast closed loops resulting in paradoxes and open loops, which save us from falling into antinomies as contradictions. The Gödelian sentence

provides an example of an open loop, because it is not paradoxical, but generates undecidable sentences or shows that arithmetic is incomplete (of course, after suitable formulation in terms of number theory and assuming its consistency); in fact, (G) is purely syntactic and does not involves semantic notions. My main task is to outline a formal approach to the problem of semantic loops. ← 11 | 12 → Yet, I will also make some general and special philosophical suggestions at the end of this paper.

Let me start with a diagnosis of LA paradox outlined by Leśniewski and Tarski (see Tarski 1933, Tarski 1935, and especially Tarski 1944, pp. 671–674). They argued (Leśniewski never published his results) that LA is generated by the following facts (my formulation differs literally from that of Tarski):

This diagnosis opens three ways out. Firstly, we can exclude self-reference from semantics. Secondly, we drop the T-scheme. Thirdly, we decide to modify logic. In each case, a solution of LA costs something in the sense that we must sacrifice self-reference or the T-scheme or classical logic. Nothing is free of charge here and eliminating semantic antinomies requires a sacrifice of something for something. For Tarski (and the same concerns Leśniewski), playing with (A) was the most natural way (or cost less than rejecting the T-scheme) or changing logic. I do not know of any solutions consisting in rejecting the T-scheme. However, this remark must be properly understood. If we exclude self-reference, some T-equivalences resulting from the T-scheme are not admissible, for instance ‘λ is true if and only if λ’, because it leads to LA (see Thomason 1986). Similarly, we block analogous statements having the form of equivalences and related to other semantic paradoxes, for instance, Grelling’s antinomy of heterological adjectives (see below).

It is important to see the role of the T-scheme in producing LA. Not every case of self-reference is harmful (in the sense of leading to an antinomy). In fact, we have ‘innocent’ self-referential sentences (for a general concise treatment of cases of this phenomenon, not only linguistic, see Smith 1986; Bartlett, Suber 1987 and Bartlett 1992 are useful collections) in which self-reference occurs. Consider, the phrase