Hostname: page-component-5b777bbd6c-mqssf Total loading time: 0 Render date: 2025-06-21T14:19:07.772Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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