Abstract
In spite of its obvious importance, the notion of information has been strangely neglected by most contemporary epistemologists and philosophers of language. It seems to me more fruitful, however, to try to correct the situation than to lament it. In this paper, I shall point out some largely unexplored possibilities of using the concept of information in logic and especially in the philosophy of logic.
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References
The basic references are R. Carnap and Y. Bar-Hillel, An Outline of a Theory of Semantic Information, Technical Report No. 247, M.I.T., Research Laboratory of Electronics, 1950; reprinted in Y. Bar-Hillel, Language and Information, Addison-Wesley, Reading, Mass., 1964, pp. 221–274; and Y. Bar-Hillel and R. Carnap, ‘Semantic Information’, British Journal for the Philosophy of Science 4(1953) 147–157. (Slightly different version also in Communication Theory: Papers Read at a Symposium on Applications of Communication Theory [ed. by W. Jackson], London 1953, pp. 503–511.)
Cf. my paper ‘On Semantic Information’ in the present volume (forthcoming in Physics, Logic, and History [ed. by W. Yourgrau], The Plenum Press, New York 1970), last few pages.
Karl R. Popper, Logik der Forschung, Springer-Verlag, Vienna, 1934. (English translation, with additions, as Logic of Scientific Discovery, Hutchinson’s, London, 1959.)
See my paper. ‘The Varieties of Information and Scientific Explanation’, in Logic, Methodology and Philosophy of Science III, Proceedings of the 1967 International Congress (ed. by B. van Rootselaar and J. F. Staal ), North-Holland Publishing Company, Amsterdam, 1968, pp. 151–171.
Some aspects of it are dealt with in my paper ‘On Semantic Information’ (present volume) and in the earlier papers of mine on which it is based.
See R. Carnap, Logical Foundations of Probability, University of Chicago Press, Chicago 1950 (2nd ed. 1962 ).
Cf. in the connection my paper ‘Are Logical Truths Tautological?’, in Deskription, Analytizität und Existenz (ed. by P. Weingartner), A. Pustet, Salzburg and Munich, 1966, pp. 215–233, especially pp. 221–223.
With the following discussion, cf. my paper ‘Distributive Normal Forms in First- Order Logic’ in Formal Systems and Recursive Functions. Proceedings of the Eighth Logic Colloquium. Oxford, July 1963 (ed. by J. N. Crossley and M. A. E. Dummett ), North-Holland Publishing Company, Amsterdam 1965, pp. 47–90.
Cf. my papers ‘Toward a Theory of Inductive Generalization’, in Logic, Methodology. and Philosophy of Science, Proceedings of the 1964 International Congress (ed. by Y. Bar-Hillel), North-Holland Publishing Company, Amsterdam 1965, pp. 274–288; and ‘A Two-Dimensional Continuum of Inductive Methods’, in Aspects of Inductive Logic (ed. by Jaakko Hintikka and Patrick Suppes ), North-Holland Publishing Company, Amsterdam 1966, pp. 113–132.
This definition may be sharpened somewhat. Let us consider two quantifiers which occur in the same statement s, let us assume that they contain the bound variables x and y, respectively, and let us assume that the latter quantifier occurs within the scope of the former. Then we shall say that the two quantifiers are connected iff there are within the scope of the former quantifiers (say containing the variables z 1, z 2, z k respectively, such that x and z 1, z i and z i+1 (i = 1, 2, …, k- 1), and z k and y occur in the same atomic expression in s . The depth of s can now be defined as the length of the longest chain of nested and connected quantifiers in s. This sharpened definition does not imply any changes in the subsequent discussion.
For this intuitive meaning of depth, see also my papers ‘Are Logical Truths Analytic?’. Philosophical Review 74 (1965) 178–203; and ‘An Analysis of Analyticity’, in Deskription, Analytizitdt undExistenz (ed. by P. Weingartner ), A. Pustet, Salzburg and Munich 1966, pp. 193–2214.
See once again my paper ‘Distributive Normal Forms in First-Order Logic’ (reference 8 above).
No constituent was inconsistent in the monadic case. If expressions of form (12*) are considered in the monadic case, however, some of them turn out to be trivially inconsistent. But even in this case the only clause that is needed to locate all inconsistencies is the relatively uninteresting third clause (c), which can in fact be dispensed with by treating somewhat differently the relation of quantifiers to identity.
William Hanf, ‘Degrees of Finitely Axiomatizable Theories’, Notices of the American Mathematical Society 9 (1962) 127–128 (abstract).
In ‘Distributive Normal Forms’ (reference 8 above).
In ‘Distributive Normal Forms and Deductive Interpolation’, Zeitschrift fur mathematische Logik und Grund/agen der Mathematik 10 (1964) 185–191. Furthermore, all relations of definability among the predicates of one’s finitely axiomatizable theory can be found in this way.
The nature of constituents in general and of inconsistent constituents in particular is briefly discussed in my paper ‘Are Logical Truths Analytic?’ (reference 11 above).
All this is of course conditional on the assumption that even distribution or some similar weighing principle is used. It seems to me, however, that most natural principles of weighing will yield similar results.
A technical argument to the effect that they cannot be has been given by Hilary Putnam in his contribution to the volume on Carnap in the Library of Living Philosophers, entitled ‘Degree of Confirmation and Inductive Logic’, in The Philosophy of Rudolf Carnap (ed. by P. A. Schilpp), Open Court, La Salle, 111., 1963. Cf. also Hilary Putnam, ‘Probability and Confirmation’, in Philosophy of Science Today (ed. by S. Morgenbesser ), Basic Books, New York 1967.
The basic ideas of this justification go back to Frank Ramsey and Bruno de Finetti, whose fundamental papers arc conveniently reprinted in H. E. Kyburg and H. E. Smokier (eds.), Studies in Subjective Probability, John Wiley, New York 1964. More recent treatments are contained in the following papers: A. Shimony, ‘Coherence and the Axioms of Confirmation’, Journal of Symbolic Logic 20 (1955) 1–28; R. S. Lehman, ‘On Confirmation and Rational Betting’, ibid., 251–262; J. G. Kemeny, ‘Fair Bets and Inductive Probabilities’, ibid., 263–273.
Leonard J. Savage, ‘Difficulties in the Theory of Personal Probability’, Philosophy of Science 34 (1967) 305–310; cf. Ian Hacking, ‘Slightly More Realistic Personal Probability’, ibid., 311–325.
Further arguments to the same effect can be given. Consider, for instance, the disjunction of all consistent constituents of a given depth. At this depth, it appears to convey some information, for it excludes prima facie acceptable alternatives (viz. those specified by inconsistent but not trivially inconsistent constituents of this depth). When the depth of our constituents is increased, it gradually becomes clearer and clearer that our disjunction does not exclude any consistent alternatives and contains therefore no (depth) information.
See my paper ‘The Varieties of Information’ (reference 4 above).
With this section, cf. my papers in Deskription. Analytizitat und Existenz (cd. by P. Weingartner), A. Pustet, Salzburg and Munich, 1966, especially ‘Are Logical Truths Tautologies?’ (pp. 213–233) and ‘Kant Vindicated’ (pp. 234–253).
Cf. e. g. Carl G. Hempel, ‘Geometry and Empirical Science’, American Mathematical Monthly 52 (1946), reprinted in Readings in Philosophical Analysis, New York 1949, pp. 238–249 (especially p. 241); A. J. A. er, Language. Truth and Logic, London 1936, p. 80.
Ernst Mach, Erkenntnis und Irrtum: Skizzen zur Psychologie der Forschung,Leipzig 1905, pp. 300 and 302.
See Gottlob Frege, Die Grundlagen der Arithmetik, Wilhelm Koebner, Breslau 1884 (§88, pp. 100–101).
In addition to the papers mentioned earlier, see ‘Kant and the Tradition of Analysis’ in Deskription. Analytizitat und Existenz (references 11 and 24 above), pp. 254–272.
In my contribution to the symposium ‘Are Mathematical Truths Synthetic A Priori?’, Journal of Philosophy 65 (1968) 640–651.
Here the familiar distinction between information and the amount of information seems to be needed. It does not make any essential difference for my purposes, however.
If this is not the case, the very concept of meaning will lose much of its usefulness. Its connection with what someone means or can mean is broken, and so is all connection with the concept of intention, for intention and intended meaning arc something which one actually has or at least can always in principle recall to one’s mind. An entity that cannot be effectively found cannot play this role.
I have discussed these difficulties in Knowledge and Belief, Cornell University Press, Ithaca, New York, 1962, chapter ii. The solution I proposed there does not involve the unrealistic idealizing assumption that people always believe (hope, know, remember, etc.) ail the logical consequences of what they believe (hope, know, remember, etc.). However, it makes the limits of applicability of one’s epistemic logic to what people actually believe unclear, and hence leaves a great deal to be desired.
It might even be suggested that the failure of an operator to be invariant with respect to logical equivalence is an indication of the intentional (or, as some philosophers prefer to say, ‘psychological’) character expressed by this operator. This suggestion would certainly deserve further attention.
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© 1970 D. Reidel Publishing Company, Dordrecht-Holland
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Hintikka, J. (1970). Surface Information and Depth Information. In: Hintikka, J., Suppes, P. (eds) Information and Inference. Synthese Library, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-3296-4_8
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