Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 247

Applications: Mathematical Proofs and Automated Reasoning 223 Section 5.1 I then present and briefly discuss some of the basic theories mentioned above in Section 5.2 where I give just a minimum background on sets, functions, and relations needed for performing meaningful mathematical reasoning in them The actual proofs will be left as exercises for the reader however, the main purpose of the chapter Section 5.3.1 is supplementary, intended to provide a basic background on Mathematical Induction to the reader who needs it Section 5.3.2 focuses on deductive reasoning in the axiomatic system of Peano Arithmetic, again mainly by means of exercises Logical deductive reasoning is carried out not only manually but also – with increasing popularity and success – using computers That use has lead to the active development of automated reasoning, including automated and interactive theorem proving On the other hand, logic has made a strong methodological contribution to the theory and practice of programming and computing by suggesting the paradigm of logic programming, realized in several programming languages (the most popular of these being Prolog) I discuss these topics very briefly in Section 5.4 5.1 Logical reasoning and mathematical proofs In mathematics the truth of a statement is established by proving it A proof may consist of a simple argument or of much complicated calculations, but essentially no logical reasoning Alternatively, it may consist of a long and intricate argument involving a number of other already-proven statements, conjectures, or logical inferences This section is about the logical aspects and structure of mathematical reasoning and proofs I discuss strategies and tactics for proofs, essentially based on the rules of Natural Deduction, and illustrate them with a few examples I end this section with some brief remarks on Resolution-based automated reasoning 5.1.1 Proof strategies: direct and indirect proofs A typical mathematical statement is of the form: if P1 , , Pn then C where P1 , , Pn (if any) are premises or assumptionsand C is a conclusion While mathematical arguments are very specific to the subject area and the concrete statement, the structure of the logical arguments in a proof only depend on the adopted logical strategy of proof and the logical forms of the premises and the conclusion Here I provide some proof tactics, describing how to go about specific, local steps of the proof, but I first discuss possible proof strategies, describing how proofs are organized globally