1. Analyze and translate the following passages

1. There would be no need to state logical principles if either there were no disagreements or only a small number of them (since in the latter case each could be considered individually). Conversely, the ability to elucidate logical principles presupposes agreement on some very simple arguments. Without this it is un­likely that communication would be possible at all.

2. Reductio ad absurdum argument against the existence of motion (Zeno's paradox) is most likely to have been inspired by the reductio ad impossible argu­ments of Pythagorean maths. At any rate, we know of no earlier use of this form of argument in philosophy.

3. Its importance is that once the reductio form is learned, it tends to breed discussion and dispute rather than disciples who faithfully accept and promul­gate the master's teachings. The creation of a heritage of discussion proved to be the most important contribution of the pre-Socratic philosophers to our civili­zation; it is sure to be one of the greatest cultural achievements of all time.

4. The idea of an argument having a life of its own, the conclusions of which may be unexpected or unwanted by the formulator of the argument is unlikely to be new with Socrates. But it leads to the open quality of the Socratic dialogues, which in turn heightens interest because the outcome is unknown.

5. Set theory appeared to have been important for math logic in three ways. First, like non-Euclidean geometry, it provided a blow to the notion that our clear and distinct intuitions are criteria for truth. Second, set theory also provided the prototype of a diagonal argument which was to become an important method of math logic. Third, the discovery of logical contradictions within set theory led to new analyses of valid and invalid arguments, as well as new inquiry into the possibility of establishing safeguards to prevent their occurrence in other areas of maths.

6. We require a valid formula to be valid in every domain. We wish our logic to apply to a domain of discourse regardless of what the "facts" about that domain are.

7. The crucial idea which Godel was first to introduce (following a hint of Hilbert's) is that syntactical metamathematical statements can be interpreted arithmetically. This is done by associating some natural number with each sym­bol. This numbering of each symbol (now called Godel numbering) is analogous to Descartes' numbering of the x- and y-axis in the Cartesian plane; it is the ba­sis for showing similarity of structure between two apparently different realms.

8. Math reality is always partly elusive and, therefore, continually fascinat­ing. Had Hilbert's programme actually been carried out, maths would have be­come theoretically boring. Maths demands an intrinsic creativity in order to pro­gressively define the reality to which it refers.

9. Since the same math axioms give rise to different sets of theorems making use of different sets of logical rules, it is necessary to be explicit. It is also historical­ly unfortunate that Euclid did not make explicit his logic. For if the logic of, say, magnitudes being in the same ratio had been investigated, it is very probable that the inadequacy of both Aristotelian and Stoic logic would have been recognized.

10. Any effective finite method by which it can be determined whether or not an arbitrary formula of a formal system is a theorem is called a decision procedure. By effective finite method is meant the same thing that used to be called an algo­rithm. Any system for which there is a decision procedure is called decidable.