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Language, Proof and Logic. Introduction . Introduction. 0.a. What do the fields of astronomy, economics, finance, law, mathematics, medicine, physics and sociology have in common? Not much in the way of subject matter, that’s for sure. And not all that much in the way of methodology.
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Language, Proof and Logic Introduction Introduction
0.a What do the fields of astronomy, economics, finance, law, mathematics, medicine, physics and sociology have in common? Not much in the way of subject matter, that’s for sure. And not all that much in the way of methodology. What they do have in common, with each other and with all other areas of intellectual activity, is their dependence on a certain standard of rationality. In other words, they presuppose an underlying acceptance of basic principles of logic. The special role of logic in rational inquiry In this sense logic --- the study of correct ways of reasoning --- is most basic of all sciences. Its laws are even more fundamental and less subject to repeal than the laws of physics. We can imagine a world where water flows up hill. But we can’t imagine a world where both are and are not nine planets. Logic is considered a full-fledged subfield of three disciplines: philosophy, mathematics, and computer science. It is mainly philosophy by subject matter, mainly mathematics by methodology, and mainly computer science by applications. This is why the course is cross-listed under CSC, MAT and PHI.
0.b What is logic? Logic = Science about correct reasoning. As such, it is only interested in the form rather than content. Every human is mortal Socrates is a human ---------------------------- Socrates is mortal Every H is M S is an H ---------------------------- S is M
0.b What is logic? Logic = Science about correct reasoning. As such, it is only interested in the form rather than content. Every hemin is melin Solik is a hemin ---------------------------- Solik is melin Every H is M S is an H ---------------------------- S is M
0.b What is logic? Logic = Science about correct reasoning. As such, it is only interested in the form rather than content. Every human is mortal Socrates is a human ---------------------------- Socrates is mortal Some humans are mean Socrates is a human ---------------------------- Socrates is mean Every H is M S is an H ---------------------------- S is M Some H are M S is an H ---------------------------- S is M
0.c True or false? There is a student in this class who likes all those (and only those) who do not like themselves.
0.d Liar’s paradox True or false? The sentence written in this frame is false.
0.e PRISONER Can I be executed on Saturday? Friday? Thursday? Wednesday? Tuesday? Monday? Today (Sunday)? JUDGE Prisoner’s paradox “You will be executed at noon on one of the days of this week; yet so that in that morning you will not know it is your last day.” No! No! No! No! EXECUTED! No! No! No!
0.f The main aims of this course The main goal of this course is to introduce you to some of the most important concepts and tools of logic. More particularly, there are two main aims: 1. To help you learn a new language, the language of first-order logic (FOL). It serves as a prototypal example of an artificial language, and is used every day by mathematicians, philosophers, computer scientists, linguists, and practitioners of artificial intelligence. Like English, it is a truly universal language: there is nothing on earth you cannot say in it. But, unlike English, it is mathematically precise and hence totally free of any ambiguities, abundant in natural languages. 2. To help you learn about the notion of logical consequence, and about how one goes about establishing some claim does or does not logically follow from other accepted claims. Modern logic teaches us that one claim is a logical consequence of another if there is no way the latter could be true without the former also being true.
0.g Essential instructions about homework exercises Read the same-title subsection of the textbook. You should familiarize yourself with Tarski’s World, Fitch, Boole, Submit, and Grade Grinder. We will be heavily relying on these software tools throughout the course. Heaven forbid you copy somebody’s file, or even just a tiny fragment of such. The Grade Grinder is smart, and will immediately alarm the instructor about suspicious activity. Such a report will be considered a proof of cheating, no matter what actual story stands behind it.
