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Intensional models for first degree formulas1

Published online by Cambridge University Press:  12 March 2014

Nuel D. Belnap Jr
Nuel D. Belnap Jr*
Affiliation:
University of Pittsburgh
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Extract

In Anderson and Belnap [8] there was developed a semantics for first degree entailments (fde), i.e., entailments A → B between formulas A and B involving only truth-functions (defined in terms of “or” and “not”) and quantifiers. The key ideas were (i) the notion of a frame ⟨P, FIP, I⟩, where P is a set of (intensional) propositions closed under negation and multiple disjunction, I is a domain of individuals, and FIP is the set of functions from I into P; and (ii) the semantic relation of cons (consequence), as obtaining between a set of propositions taken conjunctively, and a set taken disjunctively; and (iii) the notion of an atomic frame, i.e., a frame generated by a set of propositions X closed under negation, such that for any disjoint subclasses Y and Z of X, Y does not bear cons to Z.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 32 , Issue 1 , June 1967 , pp. 1 - 22
Copyright
Copyright © Association for Symbolic Logic 1967

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Footnotes

1

This research was supported in part by National Science Foundation Grant GS-190 (History & Philosophy of Science). I wish to thank J. Barwise for his considerable assistance in the early stages of this research, and P. Woodruff and M. Dunn for reading later drafts.

References

[1]Anderson, A. R., Completeness theorems for the systems E of entailment and EQ of entailment with quantification, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 6 (1959), pp. 201–216.CrossRefGoogle Scholar
[2]Anderson, A. R., Some open problems concerning the system E of entailment, Acta Philosophica Fennica, fasc. 16, Helsinki, 1963.Google Scholar
[3]Anderson, A. R. and Belnap, N. D. Jr., A modification of Ackermanr's ‘rigorous implication’, [abstract] this Journal, vol. 23 (1958), pp. 457–458.Google Scholar
[4]Anderson, A. R. and Belnap, N. D. Jr., Enthymemes, The journal of philosophy, vol. 58 (1961), pp. 713–723.CrossRefGoogle Scholar
[5]Anderson, A. R. and Belnap, N. D. Jr., The pure calculus of entailment, this Journal, vol. 27 (1961a), pp. 19–52.Google Scholar
[6]Anderson, A. R. and Belnap, N. D. Jr., Tautological entailments, Philosophical studies, vol. 13 (1961b), pp. 9–24.CrossRefGoogle Scholar
[7]Anderson, A. R. and Belnap, N. D. Jr., Entailment with negation, Zeitschrift für mathematische Logik uud Grundlagen der Mathematik, vol. 11 (1965) pp. 277–289.Google Scholar
[8]Anderson, A. R. and Belnap, N. D. Jr., First degree entailments, Mathematische Annalen, vol. 149 (1963), pp. 302–319.CrossRefGoogle Scholar
[9]Belnap, N. D. Jr., A formal analysis of entailment, Technical Report No. 7, Contract No. SAR/Nonr-609(16), Office of Naval Research (Group Psychology Branch), New Haven (1960).CrossRefGoogle Scholar
[10]Belnap, N. D. Jr., Entailment and relevance, this Journal, vol. 25 (1960a), pp. 144–146.Google Scholar
[11]Belnap, N. D. Jr., First degree formulas, [abstract] this Journal, vol. 25 (1960b), pp. 388–389.Google Scholar
[12]Belnap, N. D. Jr., EQ and the first order functional calculus, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 6 (1960c), pp. 217–218.CrossRefGoogle Scholar
[13]Belnap, N. D. Jr., and Spencer, J. H., Intensionally complemented distributive lattices. Portugaliae Mathematica, forthcoming.Google Scholar
[14]Church, A., The weak theory of implication, Kontrolliertes Denken, Munich, 1951.Google Scholar
[15]Curry, H. B., Foundations of mathematical logic, McGraw-Hill, New York, 1963.Google Scholar
[16]Shaw-Kwei, Moh, The deduction theorems and two new logical systems, Methodos, vol. 2 (1950), pp. 56–75.Google Scholar
[17]Prawitz, D., Normal deductions, presented to the Association for Symbolic Logic, Hotel New Yorker, 04 21, 1964.Google Scholar