Abstract
Taken for granted that knowledge satisfies the “truth-requirement”, K(a,p)→p, it immediately follows that the elementary epistemic attitudes K(a,p), K(a,¬p), together with their respective negations, ¬K(a,p), ¬K(a,¬p), fit into a standard modal “square of knowledge”. In contrast to K(a,p), the operator of belief, B(a,p), does not satisfy the truth-requirement. Even if ‘belief’ is interpreted in the strong sense of certainty (i.e. maximal subjective probability), it still doesn’t follow that whenever subject a is firmly convinced that p, p must therefore be true—as we all know, humans are fallible. Nevertheless, if it is only assumed that the concept of belief satisfies the “consistency principle” B(a,p)→¬B(a,¬p), one obtains a standard “square of belief”. In view of the so-called entailment-thesis, K(a,p)→B(a,p), both squares can be combined into a corresponding “double square” where, interestingly, given certain assumptions of epistemic/doxastic logic, the doxastic “inner square” may be shown to be identical to the following square of complex epistemic modalities:
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Assume that condition 1 is satisfied and that A is true; then B must be false and hence, in view of condition 3, C must be true, i.e. A logically entails C. Similarly, B logically entails D since, if B is true, A must be false (by 1) and hence ¬A, i.e. (by 3) D, must be true. Furthermore, if A and B are contrary to each other, then C and D cannot both be false, because otherwise ¬C and ¬D would both be true, i.e., in view of condition 3, A and B would both be true together.
- 2.
Of course, the possibility operator, ◊p, may be defined as ¬□¬p.
- 3.
Engel [1], 5.
- 4.
In the same way, the C-proposition B(a,p) entails, but is not itself entailed by, the negation of the B-proposition, ¬K(a,¬p).
- 5.
As the reader may verify, also the remaining two “inhomogeneous” variants
$$\begin{array}{c@{\quad}c} \mathrm{K}(a,p) &{\neg}\mathrm{K}(a,p)\\ \mathrm{B}(a,p)&\mathrm{B}(a,{\neg} p) \end{array} \qquad \mbox{and} \qquad \begin{array}{c@{\quad}c} \mathrm{K}(a,p)&\mathrm{K}(a,{\neg} p)\\ \mathrm{B}(a,p)& {\neg}\mathrm{B}(a,p) \end{array}$$fail to satisfy several conditions for a square of opposition.
- 6.
Such a condition at best holds for a supernatural omniscient being like God.
- 7.
- 8.
Cf. Hendricks [2] who refers in particular to Lamarre and Shoham [3]; however, Moore [8] at best put forward the ambiguous principle that believing that p entails believing that one knows that p; but Moore was not aware of the distinction between “weak” belief and conviction; nor did he therefore recognize that the crucial principle does not hold for the operator B(a,p), but only for C(a,p).
References
Engel, P.: Can there be an Epistemic Square of Opposition? In: Béziau, J.-Y., Payette, G. (eds.) Handbook of the First World Congress on the Square of Opposition, Montreux, 2007
Hendricks, V.F.: Hintikka on epistemological axiomatization. In: Kolak, D., Symons, J. (eds.) Questions, Quantifiers and Quantum Physics—Essays on the Philosophy of Jaakko Hintikka, pp. 3–32. Springer, Vienna (2004)
Lamarre, P., Shoham, Y.: Knowledge, belief, and conditionalization. In: Doyle, J., Sandewall, E., Torraso, P. (eds.) Principles of Knowledge Representation and Reasoning, pp. 415–424. Morgan Kaufmann, San Mateo (1994)
Lenzen, W.: Recent Work in Epistemic Logic. Acta Philosophica Fennica. North-Holland, Amsterdam (1978)
Lenzen, W.: Glauben, Wissen und Wahrscheinlichkeit. Springer, Vienna (1980)
Lenzen, W.: Knowledge, belief, and subjective probability: outlines of a unified system of epistemic/doxastic logic. In: Hendricks, V.F., Jørgensen, K.F., Pedersen, S.A. (eds.) Knowledge Contributors, pp. 17–31. Kluwer, Dordrecht (2003)
Lenzen, W.: Epistemic logic. In: Niiniluoto, I., Sintonen, M., Woleński, J. (eds.) Handbook of Epistemology, pp. 963–983. Kluwer, Dordrecht (2004)
Moore, G.E.: Certainty. In: Moore: Philosophical Papers, pp. 226–251. George, Allen & Unwin, London (1959)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this chapter
Cite this chapter
Lenzen, W. (2012). How to Square Knowledge and Belief. In: Béziau, JY., Jacquette, D. (eds) Around and Beyond the Square of Opposition. Studies in Universal Logic. Springer, Basel. https://doi.org/10.1007/978-3-0348-0379-3_21
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0379-3_21
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0378-6
Online ISBN: 978-3-0348-0379-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)