Skip to main content
SpringerLink
Log in
Book cover

The Road to Universal Logic pp 213–242Cite as

Birkhäuser

Two, Many, and Differently Many

  • Chapter
  • First Online:

Part of the book series: Studies in Universal Logic ((SUL))

Abstract

This paper is a modest contribution to a universal logic approach to many-valued semantic systems. The main focus is on the relation between such systems and two-valued ones. The matter is discussed for usual many-valued semantic systems. These turn out to exist for more logics than expected. A new type of many-valued semantics is devised and its use has been illustrated. The discussion that involves truth functionality and the syntactic rendering of truth-values leads to philosophical conclusions.

The author is indebted to Joke Meheus for comments on a former draft.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Throughout this chapter, “semantics of a logic \(\mathbf{L}\)” should be read as “characteristic \(\mathbf{L}\)-semantics.”

  2. 2.

    The pseudolanguage schema \(\mathcal{L}_{\mathcal{O}}\) is not a language schema whenever its set of symbols is nondenumerable. The resulting style of semantics – examples follow in the text – offers a means to quantify over nondenumerable sets.

  3. 3.

    This move is independent of the reference to the pseudolanguage schema \(\mathcal{L}_{\mathcal{O}}\). To combine the move with a different semantic style, restrict it to \(v\colon\mathcal{W}\to\{0,1\}\).

  4. 4.

    The restriction in C2 ensures that \(\langle D,v\rangle\) is only a \(\mathbf{CL}\)-model if every element of D is named by a constant or pseudoconstant. In C3, \(\wp(D^{r})\) is the power set of the rth Cartesian product of D.

  5. 5.

    The indeterministic propositional semantics was first formulated in [8]; the deterministic predicative semantics in [10].

  6. 6.

    Technically speaking, a logic \(\mathbf{L}\) is paraconsistent iff \(A,\neg A\vdash_{\mathbf{L}}B\) does not hold generally. Interesting discussions of the underlying philosophical questions are available, for example, by Béziau [18, 19].

  7. 7.

    Still, indeterministic semantic systems, have been around at least since the 1970s and led to interesting studies, for example, in work by Arnon Avron and associates [5, 6, 7].

  8. 8.

    \(\mathbf{CLuNs}\) is apparently the most popular paraconsistent logic. It is known under a mutiplicity of names. Further useful references to studies of \(\mathbf{CLuNs}\) and of its fragments are [16, 2, 22, 24, 25, 26, 27, 28, 29, 3, 33, 34, 37, 4, 8]. Proofs of some results are in [13, 16].

  9. 9.

    Another version of the approach, requiring only a single clause, is illustrated in a paper under review [39].

  10. 10.

    In (iii), a is the alphabetically first individual constant, which is used here as a metalinguistic name of itself.

  11. 11.

    Carnap [20, 21] clearly saw the linguistic relativity of the semantic enterprise. Apparently, many have forgotten his insight and seem to presume that they can talk about states of the world in an absolute way.

  12. 12.

    I do not intend to refer, for example, to a worlds semantics but rather to a nontruth-functional semantics such as the ones from Sect. 9.2, the indeterministic as well as the deterministic ones.

  13. 13.

    There is no need to assign a three-valued assignment value to all members of \(\mathcal{W}_{\mathcal{O}}\).

  14. 14.

    The three functions determine for which r-tuples the predicate is true, inconsistent, and false respectively.

  15. 15.

    Similarly for the valuation values of the three-valued \(\mathbf{LP}\)-semantics [34] and for the valuation values of four-valued semantics for relevant logics [1].

  16. 16.

    As was already pointed out by Viktor Kraft [31], nothing warrants that the syntactically atomic sentences of a language are also epistemologically atomic.

  17. 17.

    In the text I use the name M for both models although they are not only different but even different in kind. Where it matters, I shall obviously introduce different names.

  18. 18.

    If B does not have the form \(\neg C\), then \(v(B)\) does not play any role within the \(\mathbf{CLuN}\)-semantics. So one might just as well decide not to store the value of \(v(B)\) in \(V_{M}(A)\) for such \(B\in\mathrm{fsup}(A)\). While the disadvantage of the approach followed in the text is that some digits of \(V_{M}(A)\) are irrelevant, the advantage is that the approach is more general, as is the case for the assignment function of the two-valued semantics itself. That the advantage outweighs the disadvantage will be shown in Sect. 9.5.

  19. 19.

    These are actually language schemas. Still \(\mathcal{L}_{\mathcal{O}}\) need to be different from \(\mathcal{L}\) in order to allow for models that are not ω-complete.

  20. 20.

    So, if \(L_{A}=\langle B_{1},B_{2},\ldots\rangle\) and \(V_{M}(A)=\langle 1011\ldots\rangle\), then \(V_{M}(A)\) contains the information that in the corresponding two-valued model \(M^{\prime}\) holds: \(v_{M^{\prime}}(A)=1\), \(v(B_{1})=0\), \(v(B_{2})=1\), \(v(B_{3})=1\), and so on.

  21. 21.

    So, if \(L_{A}=\langle B_{1},B_{2},\ldots\rangle\) and \(V_{M}(A)=\langle 1110\rangle\), then \(V_{M}(A)\) contains the information that in the corresponding two-valued model \(M^{\prime}\) holds: \(v_{M^{\prime}}(A)=1\), \(v(B_{1})=1\), \(v(B_{2})=1\), \(v(B_{3})=0\), and \(v(B_{i})=1\) whenever i > 3; similarly, \(V_{M}(A)=\langle 0\rangle\) then contains the information that in the corresponding two-valued model \(M^{\prime}\) holds: \(v_{M^{\prime}}(A)=0\) and \(v(B_{i})=1\) for all \(i\in\{1,2,\ldots\}\).

  22. 22.

    Some values will be absent for some logics; \(\mathbf{CLuNs}\), for example, does not allow for gaps.

  23. 23.

    I use the same notation, \(v_{M}(A)\), for the valuation function in all three kinds of semantics and I shall do so for all logics. The matter is always disambiguated by the context.

  24. 24.

    The counterfactual and causal phraseology can obviously be rephrased extensionally (in terms of all models that have certain properties).

  25. 25.

    This weak characterization is preferable in order to avoid prejudged narrowing of the domain. It is equivalent to the characterization offered by Béziau [18].

  26. 26.

    The matter is handled by an adaptive strategy – see the referred survey papers.

  27. 27.

    If this sounds puzzling, please realize that \(\{q,\neg q,\neg\neg q,\neg\neg\neg q\}\) is more inconsistent than \(\{q,\neg q,\neg\neg\neg q\}\) and that \(\mathbf{CLuN}\) does not validate \(A\supset\neg\neg A\).

  28. 28.

    In some exceptional cases, one wants an adaptive logic that is a flip–flop.

  29. 29.

    The formula \((r\wedge q)\wedge\neg(r\wedge q)\) is not an abnormality but it is a \({\mathbf{CLuNs}^{\mathit{m}}}\)-consequence of \(\Gamma_{1}\). Indeed, \(q\wedge\neg q\) is a \(\mathbf{CLuNs}\)-consequence (and hence a \({\mathbf{CLuNs}^{\mathit{m}}}\)-consequence) of \(\Gamma_{1}\), r is a \({\mathbf{CLuNs}^{\mathit{m}}}\)-consequence of \(\Gamma_{1}\), and \(q\wedge\neg q,r\vdash_{\mathbf{CLuNs}}(r\wedge q)\wedge\neg(r\wedge q)\).

  30. 30.

    Still and to the best of my knowledge, paraconsistent logics that do not allow for other gluts or gaps and were proposed to serve a sensible purpose agree with \(\mathbf{CL}\) in classifying Γ as inconsistent.

  31. 31.

    The ‘‘checked’’ symbols are metalinguistic names for certain symbols of the language \(\mathcal{L}\) of logic \(\mathbf{L}\). If the \(\wedge\) is a classical conjunction in \(\mathbf{L}\) and \(\mathord{\sim}\) is a classical negation in \(\mathbf{L}\), then the formula in the text stands for \(\mathord{\sim}A\wedge\mathord{\sim}\neg A\).

  32. 32.

    Note that \(M\Vdash p\wedge\neg p\) will do just as good.

  33. 33.

    The main antecedent warrants that there are \(\mathbf{L}\)-models M and \(M^{\prime}\) of intuitively abnormal Γ such that \(\{A\in\Omega\mid M\Vdash A\}\subset\{A\in\Omega\mid M^{\prime}\Vdash A\}\). That some \(\mathbf{L}\)-models of intuitively abnormal Γ are not minimally abnormal \(\mathbf{L}\)-models of Γ entails that \({\mathbf{L}^{\mathit{m}}}\) is not a flip–flop.

  34. 34.

    The main antecedent warrants that every minimally abnormal \(\mathbf{L}\)-model M of an intuitively normal Γ is such that \(\{A\in\Omega\mid M\Vdash A\}=\emptyset\). This entails that, for all intuitively normal Γ, \(\Gamma\vdash_{\mathbf{L}^{\mathit{m}}}A\) iff \(\Gamma\vdash_{\mathbf{CL}}A\). So \({\mathbf{L}^{\mathit{m}}}\) is not a wimp.

  35. 35.

    In connection with the flip–flop problem, the result may easily be generalized to, for example, modal logics. There are indeed adaptive logics in which abnormalities have the form \(\Diamond A\wedge\neg A\) or the form \(\Diamond A\wedge\Diamond\neg A\). This, however, should not be elaborated here.

  36. 36.

    A conjunct of an inconsistency may be an inconsistency itself, as is the case for \((p\wedge\neg p)\wedge\neg(p\wedge\neg p)\). Even then the complex inconsistency is independent of the less complex one.

References

  1. Anderson, A.R., Belnap Jr., N.D., Dunn, J.M.: Entailment. The Logic of Relevance and Necessity, vol. 2. Princeton University Press, Princeton (1992)

    MATH  Google Scholar 

  2. Arieli, O., Avron, A., Zamansky, A.: Maximal and premaximal paraconsistency in the framework of three-valued semantics. Stud. Log. 97, 31–60 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Avron, A.: On an implication connective of RM. Notre Dame J. Form. Log. 27, 201–209 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. Avron, A.: Natural 3-valued logics—Characterization and proof theory. J. Symb. Log. 56 276–294 (1991)

    Google Scholar 

  5. Avron, A.: Non-deterministic matrices and modular semantics of rules. In: Béziau, J.-Y. (ed.) Logica Universalis, pp. 149–167. Birkhäuser, Basel, Boston, Berlin (2005)

    Chapter  Google Scholar 

  6. Avron, A., Ben-Naim, J., Konikowska, B.: Cut-free ordinary sequent calculi for logics having generalized finite-valued semantics. Log. Univers. 1, 41–70 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Avron, A., Konikowska, B.: Multi-valued calculi for logics based on non-determinism. J. Interest Group Pure Appl. Log. 10, 365–387 (2005)

    MathSciNet  Google Scholar 

  8. Batens, D.: Paraconsistent extensional propositional logics. Log. Anal. 90–91, 195–234 (1980)

    Google Scholar 

  9. Batens, D.: A bridge between two-valued and many-valued semantic systems: n-tuple semantics. Proceedings of the 12th International Symposium on Multiple-Valued Logic. IEEE: Los Angeles, 318–322 (1982)

    Google Scholar 

  10. Batens, D.: Inconsistency-adaptive logics. In: Orlowska, E. (ed.) Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pp. 445–472. Physica Verlag (Springer), Heidelberg, New York (1999)

    Google Scholar 

  11. Batens, D.: A general characterization of adaptive logics. Log. Anal. 173–175, 45–68 (2001), appeared 2003

    Google Scholar 

  12. Batens, D.: A universal logic approach to adaptive logics. Log. Univers. 1, 221–242 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Batens, D.: Adaptive Logics and Dynamic Proofs. Mastering the Dynamics of Reasoning, forthcoming (201x)

    Google Scholar 

  14. Batens, D.: Tutorial on inconsistency-adaptive logics. In Béziau, J.-Y., Chakraborty, M., Dutta, S. (eds.) New Directions in Paraconsistent Logic, in print. Springer (2015)

    Google Scholar 

  15. Batens, D.: Spoiled for choice? J. Log. Comput., in print. doi:10.1093/logcom/ext019 (1913)

    Google Scholar 

  16. Batens, D., De Clercq, K.: A rich paraconsistent extension of full positive logic. Log. Anal. 185–188, 227–257 (2004), appeared 2005

    Google Scholar 

  17. Batens, D., Mortensen, C., Priest, G., Van Bendegem, J.P. (eds.) Frontiers of Paraconsistent Logic. Research Studies Press, Baldock, UK (2000)

    MATH  Google Scholar 

  18. Béziau, J.-Y.: What is paraconsistent logic? In: Batens et al. [17], pp. 95–111.

    Google Scholar 

  19. Béziau, J.-Y.: Are paraconsistent negations negations? In: Carnielli, W.A., Loffredo D’Ottaviano, I.M. (eds.) Paraconsistency. The Logical Way to the Inconsistent, pp. 465–486. Marcel Dekker, New York (2002)

    Google Scholar 

  20. Carnap, R.: Meaning and Necessity. University of Chicago Press, Chicago (1947)

    MATH  Google Scholar 

  21. Carnap, R.: Logical Foundations of Probability. University of Chicago Press, Chicago (1950)

    MATH  Google Scholar 

  22. Carnielli, W.A., Marcos, J., de Amo, S.: Formal inconsistency and evolutionary databases. Log. Log. Philos. 8, 115–152 (2001), appeared 2002

    Google Scholar 

  23. da Costa, N.C.: On the theory of inconsistent formal systems. Notre Dame J. Form. Log. 15, 497–510 (1974)

    Article  MATH  Google Scholar 

  24. D’Ottaviano, I.M.L.: Sobre uma Teoria de Modelos Trivalente (in Portuguese). PhD thesis, State University of Campinas (Brazil) (1982)

    Google Scholar 

  25. D’Ottaviano, I.M.L.: The completeness and compactness of a three-valued first-order logic. Proceedings of the 5th Latin American Symposium on Mathematical Logic, pp. 77–94. Revista Colombiana de Matemáticas, 1–2 (1985)

    Google Scholar 

  26. D’Ottaviano, I.M.L.: The model extension theorems for J3-theories. In: Di Prisco, C.A. (ed.) Methods in Mathematical Logic: Proceedings of the 6th Latin American Symposium on Mathematical Logic, Lecture Notes in Mathematics 1130, pp. 157–173. Springer (1985)

    Chapter  Google Scholar 

  27. D’Ottaviano, I.M.L.: Definability and quantifier elimination for J3-theories. Stud. Log. 46(1), 37–54 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  28. D’Ottaviano, I.M.L., Epstein, R.L.: A paraconsistent many-valued propositional logic: J3. Rep. Math. Log. 22, 89–103 (1988)

    MATH  MathSciNet  Google Scholar 

  29. Esser. O.: A strong model of paraconsistent logic. Notre Dame J. Form. Log. 44, 149–156 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kleene, S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1952)

    MATH  Google Scholar 

  31. Kraft, V.: Der Wiener Kreis. Springer, Wien (1950)

    Book  Google Scholar 

  32. Lev, I.: Preferential systems for plausible non-classical reasoning. Master’s thesis, Department of Computer Science, Tel-Aviv University (2000), unpublished M.A. dissertation

    Google Scholar 

  33. Priest, G.: The logic of paradox. J. Philos. Log. 8, 219–241 (1979)

    MATH  MathSciNet  Google Scholar 

  34. Priest, G.: In Contradiction. A Study of the Transconsistent, 2nd expanded edn. (1st edn. 1987). Oxford University Press, Oxford (2006)

    Google Scholar 

  35. Rosser, J.B., Turquette, J.B.: Axiom schemes for m-valued propositional calculi. J. Symb. Log. 10, 61–82 (1945)

    Article  MATH  MathSciNet  Google Scholar 

  36. Schütte, K.: Beweistheorie. Springer, Berlin (1960)

    MATH  Google Scholar 

  37. Smirnova, E.D.: An approach to the justification of semantics of paraconsistent logics. In: Batens, D. et al. [17], pp. 255–262

    Google Scholar 

  38. Suszko, R.: The Fregean axiom and Polish mathematical logic in the 1920s. Stud. Log. 36, 377–380 (1977)

    Article  MathSciNet  Google Scholar 

  39. Verdée, P., Batens, D.: Nice embedding in classical logic, under review. Stud. Log., in print (2015)

    Google Scholar 

  40. von Wright, G.H.: Truth-logics. Acta Philos. Fennica 60, 71–91 (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Logic and Philosophy of Science, Ghent University, Ghent, Belgium

    Diderik Batens

Authors
  1. Diderik Batens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diderik Batens .

Editor information

Editors and Affiliations

  1. City University of New York, New York, New York, USA

    Arnold Koslow

  2. Federal University of Santa Catarina, Florianópolis, Santa Catarina, Brazil

    Arthur Buchsbaum

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Batens, D. (2015). Two, Many, and Differently Many. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-15368-1_9

Download citation

Publish with us

Policies and ethics

Search

Navigation