Aspects of Modern Logic

ISBN-13: 9789401033343 / Angielski / Miękka / 2012 / 187 str.

E. W. Beth; D. H. J. De Jongh; Susan De Jongh-Kearl

Opis

It is common to consider an area of science as a system of real or sup posed truths which not only continuously extends itself, but also needs periodical revision and therefore tests the inventive capacity of each generation of scholars anew. It sounds highly implausible that a science at one time would be completed, that at that point within its scope there would be no problems left to solve. Indeed, the solution of a scientific problem inevitably raises new questions, so that our eagerness for knowledge will never find lasting satisfaction. Nevertheless there is one science which seems to form an exception to this rule, formal logic, the theory of rigorous argumentation. It seems to have reached the ideal endpoint of every scientific aspiration already very shortly after its inception; using the work of some predecessors, Aristotle, or so it is at least assumed by many, has brought this branch of science once and for all to a conclusion. Of course this doesn't sound that implausible. We apparently know what rigorous argumentation is; otherwise various sciences, in particular pure mathematics, would be completely impossible. And if we know what rigorous argumentation is, then it cannot be difficult to trace once and for all the rules which govern it. The unique subject of formal logic would therefore entail that this science, in variance with the rule which holds for all other sciences, has been able to reach completion at a certain point in history."

Spis treści

I.- I. The Fundamental Criterion for the Soundness of Arguments.- 1. Introduction.- 2. Fundamental criterion for the soundness of arguments.- 3. The formal character of logic.- 4. The transition to mathematical logic.- 5. Construction of a fragment of modern logic.- 6. Natural deduction.- 7. Supplementary considerations.- II. Inferential and Classical Logic.- 8. Semantic and operational aspects of meaning.- 9. Purely implicational logic.- 10. Deduction problems and deductive tableaux.- 11. Truth-value problems and semantic tableaux.- 12. Inferential and classical logic — Peirce’s Law.- 13. Other aspects of meaning.- 14. Informal logic : G. Mannoury, A. Naess, Ch. Perelman.- III. Proof by Contradiction.- 15. Introductory remarks.- 16. Conversion of closed tableaux into natural deductions.- 17. The negation.- 18. Semantic and deductive tableaux.- 19. Final considerations — completeness of classical purely implicational logic.- IV. The Problem of Locke-Berkeley.- 20. An example.- 21. Statement of the problem and attempts at its solution.- 22. The exposition method of Aristotle.- 23. Appeal to the method of semantic tableaux.- 24. The completeness problem - informal and heuristic deduction methods.- V. On the So-Called ‘Thought Machine’.- 25. Introduction.- 26. Binary arithmetic and logic.- 27. Specific operations of the human intellect.- 28. Prehistory.- 29. Difficulties.- 30. Development of modern formal logic.- 31. Automatization of reasoning.- 32. Analysis of the difficulties.- 33. An example.- 34. Heuristics or methodology.- 35. Methodology.- 36. Concluding remarks.- II.- VI. The Paradoxes.- VII. Reason and Intuition.- VIII. Formalized Language and Common Usage.- IX. Considerations about Logical Thought.- X. Constants of Mathematical Thought.- Sources.- Index of Names.- Index of Subjects.