A sketch of logic without truth

ABSTRACT

1. Jøgensen's Dilemma - 2. The Proposed Solution - 3. Short History of a Philosophical Prejudice - 4. The Abstract, Syntactical and Semantic Notion of Consequence - 4.1. The Abstract Notion of Consequence - 4.2. The Syntactical Notion of Consequence - 4.3. The Semantic Notion of Consequence - 4.4. The Meaning Given by Rules of Use in a Context - 4.5. Which Operators does Logic Require? - 5. Deontic Logic - 6. The Consequences for Computer Scientists.

REFERENCES

• 1: J. J~rgensen, Imperatives and Logic, "Erkenntnis", 7, 1937/38. A. Ross, Directives and Norms, Routledge & Kegal Paul, London, 1967, chapt. VI, p. 30, the name of "J~rgensen's dilemma" is given in Imperative and Logic,"Theoria",p. 53 ff.
• 2: E. Mally, Grundgesetze des Sollens. Elemente der Logic des Willens, Leuschner & Lubenski, Graz, 1926.
• 3: G. Kalinowski, Introduction a la logique juridique, Bibliotheque de Philosophie du Droit, Paris, 1965.
• 4: G.H. von Wright, Norm and Action, Routledge & Kegan, Paul, London, 1963; A. G. Conte, Studio per una teoria della validitY, 1980, p. 382 ff.
• 5: Cfr. H. Kelsen, Th~orie Pure du Droit, Dalloz, Paris, 1962.
• 6: H. Neri CastaSeda, in Thinking and Doing: The Philosophical Foundations on Institutions (Reidel, Dordrecht, 1975), argues that although norms are true or false, what falsifies norms or makes them true is the legitimacy or illegitimacy of the prescriptions contained in them.
• 7: Hao Wang, From Mathematics to Philosophy, Routledge & Kegan Paul, London, 1974, p. 8.
• 8: Aristotle, De Interpretatione, chapt. 4-17a, included in the works of Aristotle, translated and the edited by W.D. Ross, Vol. I, Oxford University Press, London, 1928- ''Every sentence has meaning, not as being the natural means by which a physical faculty is realized, but, as we have said, by convention. Yet every sentence is not a proposition; only such are propositions as have in them either truth or falsity. Thus a prayer is a sentence, but is neither true or false."
• 9: A. Tarski in Logic, Semantics and Metamathematics, Oxford, 1956, chapt. VIII- The Concept of Truth in Formalized Language, is plainly aware that he is concerned only with declarative sentences. See also, The Semantic Conception of Truth, "Philosophy and Phenomenological Research" 4 1944 p 341 ff '
• 10: L. Wittgenstein, Tractatus logico-philosophicus, with translation by D.F. Pears & B.F. McGuinness, Routledge and Kegan Paul, London, 1961: "2.2 A picture has logico-ictorial form in common with what it depicts. 2.201 A picture depicts reality by representing a possibility of the existence and non-existence of states of affairs. 2.21 A picture agrees with reality or fails to agree; it is correct or incorrect, true or false".
• 11: Ibidem, 4.31
• 12: R. Carnap, Foundations of Logic and Mathematics, University of Chicago Press, Chicago, 1938, chapt. 2, 12, International Encyclopedia of Unified Science, Vol. 1, No. 3: "The result of our discussion is the following- logic, or the rules of deduction (in our terminology, the syntactical rules of transformation), can be chosen arbitrarily and hence are conventional if they are taken as the basis of the construction of the language system and if the interpretation of the system is later superimposed. On the other hand, a system of logic is not a matter of choice, but either right or wrong, if an interpretation of the logical signs is given in advance. But even here, conventions are of fundamental importance- for the basis on which logic is constructed, namely the interpretation of logical symbols (through, for example, a determination of the conditions of truth) can be chosen freely".
• 13: A.N. Prior, The Runabout Inference-Ticket,"Analysis" Vol.21 (Blackwell, 1960), pp. 38-39.
• 14: G. Gentzen, Untersuchungen 'liber das logische Schliessen, "Mathematische Zeitschrift" 1934 Vol 39 p 176 ff
• 15: G.H. von Wright has said on more than one occasion that logic goes beyond truth and falsity; F. Mir6 Quesada maintains that the subject of logic is not immediately concerned with truth and falsity but, rather, the inheritability of any value transmitted through the notion of consequence. Tarski, Jaskowski, Gentzen, Wittgenstein and Belnap, even though they do not express themselves in the same terms as us, created all the conditions and criteria for making this notion explicit.
• 16: It is often said that the the semantic notion of consequence formulated by Tarski could already be found in GSdel, and that this, in turn, was taken from Skolem. For this reason, many say that all of modern semantic logic can be traced back to Skolem.
• 17: In Tarski it is represented as follows: 1. A c Cn (A); 2. Cn (A) = Cn (Cn (A)); 3. Cn {A) - U {Cn (B) ~ B c A & B is finite}. This last conditions enables us to obtain: 3.1 If AcB then Cn(A) c Cn(B) (monotonicity) and 3.2 if x c Cn (A) then there is a BcA such that B is finite and x s Cn(B) (compactness). A. Tarski, Logic, Semantics and Metamathematics, op. cit., chapt. IIi. Foundamental Concepts of Metamathematics, p. 30 and see, Fundamental Concepts of the Methodology of Deductive Sciences, p. 60.
• 18: This is a reconstruction of the same abstract notion of 'consequence', only that in Tarski it was an operation whereas here it is presented as a relation in Gentzen's conception Untersuchunqen ..., op. cit.
• 19: A. Tarski, Logic, Semantics and Metamathematics, op. cit., chapt. XVi, On the Concept of the Logical Consequence, p. 409.
• 20: L. Wittgenstein, Philosophische Untersuchungen , Mac Millan Publishing Co., New York, 1955 with translation by G.E.M. Anscombe, Oxford, 1953, "23. Wieviele Arten der S~tze gibt es aber? Etwa Behauptung, Frage und Befehl? - Es gibt unz~hlige solcher Arten: unz~hlige verschiedene Arten der Verwendung alles dessen, was wir "Zeichen" , "Worte", "S~tze", nennen. Und diese Mannigfaltigkeit ist nichts Festes, ein f~r allemal Gegebenes; sondern neue Typen der Sprache, neue Sprachspiele, wie wir sagen k~nnen, entstehen und andre veralten und werden vergessen. (Ein ungeg~hres Bild davon kSnnen uns die Wandlungen der Mathematik geben).Das Wort "Sprachspiel" soll hier hervorheben, dass das Sprechen der Sprache ein Teil ist einer T~tigkeit, oder einer Lebensform." "But how many kinds of sentences are there? Say assertion, question and command? - There are countless kinds of use of what we call "symbols", "words", "sentences" ... the speaking of language is part of an activity or a form of life ...". And in propsition 22 (at the end): "If I hear someone say "it's raining", but I have not heard the beginning or the end of the phrase, this proposition is not yet, for me, a means of communication".
• 21: See, Note 13.
• 22: It is to be noted, for example, that Wang's algorithm is presented in true and false terms, when it actually works with sequences. For example- M.L. Schagrin, W.J. Rapaport and R.R. Dipert, Logic: A Computer Approach, 1985, McGraw-Hill, chapt. 6.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE

Additional Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE

General Terms:
Design, Legal Aspects, Theory