Journal of Philosophical Logic

, Volume 32, Issue 6, pp 613–665 | Cite as

False Though Partly True – An Experiment in Logic

  • Lloyd Humberstone


We explore in an experimental spirit the prospects for extending classical propositional logic with a new operator P intended to be interpreted when prefixed to a formula as saying that formula in question is at least partly true. The paradigm case of something which is, in the sense envisaged, false though still “partly” true is a conjunction one of whose conjuncts is false while the other is true. Ideally, we should like such a logic to extend classical logic – or any fragment thereof under consideration – conservatively, to be closed under uniform substitution (of arbitrary formulas for sentence letters or propositional variables), and to allow the substitutivity of provably equivalent formulas salva provabilitate. To varying degrees, we experience some difficulties only with this last (‘congruentiality’) desideratum in the two four-valued logics we end up giving our most extended consideration to.

absorption laws congruentiality consequence relations distributive bisemilattices many-valued logic matrix methodology partly true 


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  1. 1.
    Anderson, A. R. and Belnap, N. D.: Entailment: The Logic of Relevance and Necessity, Vol. I, Princeton University Press, Princeton, NJ, 1975.Google Scholar
  2. 2.
    Blamey, S.: Partial logic, in D. M. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic III: Alternatives to Classical Logic, Reidel, Dordrecht, 1986, pp. 1–70.Google Scholar
  3. 3.
    Bradley, F. H.: The Principles of Logic, Vol. I, 2nd edn, Oxford University Press, 1992 (First edition 1883).Google Scholar
  4. 4.
    Brink, C.: Power structures, Algebra Universalis 39 (1992), 177–216.Google Scholar
  5. 5.
    Brzozowski, J. A.: De Morgan bisemilattices, in 30th IEEE International Symposium on Multiple-Valued Logic, IEEE Computer Society, Los Alamitos, CA, 2000, pp. 173–178.Google Scholar
  6. 6.
    Cooper, W. S.: The propositional logic of ordinary discourse, Inquiry 11 (1968), 295–320.Google Scholar
  7. 7.
    Cresswell, M. J.: Classical intensional logics, Theoria 36 (1970), 347–372.Google Scholar
  8. 8.
    Demos, R.: Partly so and partly not so, Mind 68 (1959), 51–56.Google Scholar
  9. 9.
    Fuhrmann, A.: When hyperpropositions meet..., J. Philos. Logic 28 (1999), 559–574.Google Scholar
  10. 10.
    Gautam, N. D.: The validity of equations of complex algebras, Math. Logik und Grundlagenforsch. 3 (1957), 117–124.Google Scholar
  11. 11.
    Goldblatt, R.: Varieties of complex algebras, Ann. Pure Appl. Logic 44 (1989), 173–242.Google Scholar
  12. 12.
    Humberstone, L.: Logical subtraction: Problems and prospects, Talk given at a University College (London) philosophy colloquium, 1981.Google Scholar
  13. 13.
    Humberstone, L.: Singulary extensional connectives: A aloser look, J. Philos. Logic 26 (1997), 341–356.Google Scholar
  14. 14.
    Humberstone, L.: Contra-classical logics, Australasian Journal of Philosophy 78 (2000), 437–474.Google Scholar
  15. 15.
    Humberstone, L.: Parts and partitions, Theoria 66 (2000), 41–82.Google Scholar
  16. 16.
    Lewis, D.: General semantics, in D. Davidson and G. Harman (eds.), Semantics of Natural Language, Reidel, Dordrecht, 1972, pp. 169–218.Google Scholar
  17. 17.
    Lewis, D.: Statements partly about observation, Philosophical Papers 17 (1988), 1–31.Google Scholar
  18. 18.
    Miller, D.: Popper's qualitative theory of verisimilitude, British J. Philos. Sci. 25 (1974), 166–177.Google Scholar
  19. 19.
    Miller, D.: Verisimilitude redeflated, British J. Philos. Sci. 27 (1976), 363–402.Google Scholar
  20. 20.
    Oddie, G.: Likeness to Truth, Reidel, Dordrecht, 1986.Google Scholar
  21. 21.
    Rautenberg, W.: A calculus for the common rules of ∧ and ∨, Studia Logica 48 (1989), 531–537.Google Scholar
  22. 22.
    Rautenberg, W.: Axiomatization of semigroup consequences, Arch. Math. Logic 29 (1989), 111–123.Google Scholar
  23. 23.
    Rautenberg, W.: Common logic of 2-valued semigroup connectives, Z. Math. Logik Grundlag. Math. 37 (1991), 187–192.Google Scholar
  24. 24.
    Rescher, N.: Many-Valued Logic, McGraw-Hill, New York, 1969.Google Scholar
  25. 25.
    Romanowska, A.: On distributivity of bisemilattices with one distributive law, in B. Csákány et al. (eds.), Colloquia Mathematica Societatis János Bolyai 29: Universal Algebra, Esztergom (Hungary), 1977, pp. 653–661.Google Scholar
  26. 26.
    Sainsbury, R. M.: Degrees of truth and degrees of truth, Philosophical Papers 15 (1986), 97–106.Google Scholar
  27. 27.
    Scott, D. S.: Completeness and axiomatizability in many-valued logic, in L. Henkin et al. (eds.), Procs. of the Tarski Symposium, Amer. Math. Soc., Providence, RI, 1974, pp. 188–197.Google Scholar
  28. 28.
    Segerberg, K.: Classical Propositional Operators, Clarendon Press, Oxford, 1982.Google Scholar
  29. 29.
    Shafaat, A.: On varieties closed under the construction of power algebras, Bull. Australian Math. Soc. 11 (1974), 213–218.Google Scholar
  30. 30.
    Shoesmith, D. J. and Smiley, T. J.: Multiple-Conclusion Logic, Cambridge University Press, Cambridge, 1978.Google Scholar
  31. 31.
    Stalnaker, R. C.: Complex predicates, The Monist 60 (1977), 327–339.Google Scholar
  32. 32.
    Tichý, P.: On Popper's definitions of verisimilitude, British J. Philos. Sci. 27 (1974), 155–160.Google Scholar
  33. 33.
    van Fraassen, B.: Facts and tautological entailments, J. Philos. 66 (1969), 477–487.Google Scholar
  34. 34.
    Wollheim, R.: F. H. Bradley, Penguin Books, Harmondsworth, Middlesex, 1959.Google Scholar

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© Kluwer Academic Publishers 2003

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  • Lloyd Humberstone

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