On-line Logic.Fall - Lecture notes 14 PDF

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21:730:201:90 INTRODUCTION TO LOGIC FALL, 2019 RUTGERS UNIVERSITY INDEX 06313 Office: Conklin Hall 432 INSTRUCTOR: DR. JEFF BUECHNER [email protected]

   Course Description: We will learn various techniques and metatheorems in the proof and model theory of classical sentential and predicate logic. Why is this material important? It provides a reliable tool for objectively evaluating the goodness and cogency of arguments (indeed, of any argument) represented in a formal language. This is important, for showing that an argument—in a natural language, such as English—is valid can be hard to do. The techniques and the representational formal languages are not difficult to understand, but they require a lot of practice to master. Logic provides you with the ability to formulate, evaluate, and communicate conclusions and inferences from quantitative information, and to apply effective and efficient formal processes to reason and to solve problems. Logic employs a formal language whose mastery develops your ability to think abstractly. You will learn that abstract, formal reasoning is within your grasp, and that, as a consequence, so is mathematical reasoning. Formal reasoning abstracts away from words and meanings—leaving only the formal aspects of reasoning.

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Often a natural language gets in the way of understanding because words provoke your emotions, clouding judgment. Logic is the highest form of abstraction known to the human mind. Consider the following English sentences: If Jack studies, then he receives good grades. If Jack does not study, then he enjoys college. If he does not receive good grades, then he does not enjoy college. So it follows that he receives good grades. If you are a normal human being, you will probably not be able to say, with certainty, whether the conclusion does or does not validly follow from those 3 facts about Jack. For one, the words get in the way of seeing the actual structure of the reasoning. Once you learn the tools of logic, with complete certainty, you will be able to easily see that the conclusion validly follows from the premises. Here is a (very) brief outline of the topics that we will cover in this course: The nature and purpose of logic The language of sentence logic Wffs in sentence logic Translating natural language sentences into sentence logic Combinatorial (syntactic) structure, truth, and validity Truth-functions in sentence logic and operators in English Justifying the definitions of truth-functions in terms of truth-tables Truth-tables, Boolean trees, and exponential explosive growth functions Logical Form, the scope of truth-functions, parameterizing scope, and substitution instances Proofs in sentence logic—patterns of valid arguments 1-way inference rules and 2-way inference rules Distribution, quantum logic, and multiplication in arithmetic by way of the Cartesian product Conditional proofs Proof by contradiction Why sentence logic is expressively deficient and what this means The nature of predicate logic and the language of predicate logic Variables, binding, and scope Expansions and the nature of the quantifiers Rules of proof in predicate logic How the world would be different if the rules of proof in predicate logic were different

3 Translating natural language sentences into predicate logic Differential behavior of the quantifiers with respect to the logical connectives ‘and’ and ‘or’ Proving arguments invalid by the method of interpretation An algorithm for finding interpretations in predicate logic Upward and downward isomorphisms Overlapping quantifiers and the logic of n-ary relations Capture conditions and their relation to upwards and downwards isomorphisms

Stress and optimal learning Before talking about course requirements, assessment, and outcomes, I need to say something about a topic of great importance to you—stress. The figures are pretty startling—at least 20% of all undergraduates throughout the USA suffer from high levels of stress. The problem that stress creates for you is that it hinders optimal learning. When you are stressed, you do not learn as well as when you are relatively stress-free. One thing that I can do to help you with stress is to eliminate things that can promote stress. That is why I do not have required due dates for assignments. Each due date is a stress on you. Assignments can be submitted anytime during the semester. The due dates on the syllabus and the course homepage are only suggestive. It is prudent to get your work done on time to avoid a pile-up at the end of the semester. Grades are another source of stress. That is why I allow you to revise your work (without penalty) to improve your grade (the only exception is the final exam—which can only be submitted once). In this way, you can learn from your mistakes without being penalized for them and without receiving a huge dose of stress. My advice to each of you is to eat good meals, get enough sleep, and try to get things done on time. This will help reduce stress, and will help you learn subject matters better. And when you learn better, you feel more confident, and this can also help keep you relatively stress-free.

Course Requirements: There are eight exams (six quizzes, a midterm and a final examination) and two homework assignments. If you do the problems for the homework assignment, you will get 50 (whether the answers are correct or incorrect).

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There are ten assignments (exams and homework), for a total of 1000 points. Each homework is 50 points, the final exam is 200 points, and the midterm and quizzes are each 100 points. You can revise any assignment (except the final exam) as often as you like for a better grade. The due dates for each assignment (except the final exam) are suggestive. You can turn in each assignment (except the final exam) whenever you like during the semester. A B+ B C+ C D F

950-1000 850-949 800-849 750-799 700-749 600-699 0-599

Course Outcomes: What can you expect to come away with once you have finished this course? Certainly 3 interdisciplinary credits! You will also possess skills—the ability to do proofs in sentence logic and in predicate logic. This skill will enable you to decide whether complex arguments that, in English, would be infeasible to consider with respect to the question of their validity, are quite feasible to decide that question. Moreover, you will acquire an ability to make your descriptions of things, events, processes, and ideas more precise. The precision is won easily—as most of the structures, concepts and ideas in logic require extraordinary precision in setting them out. You will also acquire the ability to think abstractly—that is, to abstract away from inessential trappings and think about issues in terms of their essential components. The ability to think precisely will be helpful in all walks of life, most especially in professional areas.

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Homework #1 (Due by the end of the second week of class, Friday, September 13th) Make sure you do this assignment the before the end of the second week of class. Submit it to BLACKBOARD. (You’ll find the place to submit it when you access this course at Blackboard.) Do not e-mail it to my Rutgers email account. When we argue, we make claims about the world (these claims are called the conclusion of the argument) and provide reasons or evidence for those claims (these reasons are called the premises of the argument). Whenever one confronts an argument, the basic question is: are the reasons good? If they are good reasons, it is rational to believe the claim is true. If the reasons are bad, it is rational to withhold belief about the truth of the claim. Homework problem: In 2 pages, tell me how you determine whether the reasons for a claim are good or bad. Be specific. Provide an example of an argument and use your technique to assess how good the reasons are. If you don’t have an argument, here is a simple one to use: Premise1: Jack went to the liquor store intending to buy a bottle of wine. Premise2: Upon entering the store, Jack realized he had left his wallet at home. Conclusion: The guests at Jack’s dinner party tonight will be awfully upset.

Plagiarism Although issues about plagiarism will not arise in this course, you should be aware that plagiarism is a serious offense that can jeopardize your academic career at

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Rutgers and your future career as well. For more information, either consult the University Code of Student Conduct or visit the Student Judicial Affairs website http://judicialaffairs.rutgers.edu

ALL REQUIRED WORK IS DUE Wednesday, December 18th Noon Any work submitted after that time will NOT be accepted. I will submit the final grades to the registrar on Friday, December 20th.  Abs enc e s:Pert heUni ve r s i t y’ sCour s eAt t endanc epol i c y( 10. 2. 7) ,s t ude nt sar er e s pons i bl e f orc ommuni c a t i ngwi t ht hei ri ns t r uc t or sr e gar di ngabs e nc e s .TheDi vi s i onofS t ude ntAffai r si s a vai l abl et ove r i f ye xt e nde dabs e nc e s :( 973 )3535063or [email protected]. ademi cI nt e gr i t y( Thef ol l owi ngs t a t e menti sr ec ommendedf ori nc l us i onona l l  Ac s yl l abi . ) : Asa na c a de mi cc o mmuni t yde d i c a t e dt ot hec r e a t i o n,d i s s e mi na t i o n,a nda ppl i c a t i o no fk no wl e d g e , Rut g e r sUni v e r s i t yi sc o mmi t t e dt of o s t e r i nga ni nt e l l e c t ua la nde t h i c a le nvi r o nme ntb a s e do nt he p r i nc i p l e so fa c a de mi ci nt e g r i t y .Ac a de mi ci nt e gr i t yi se s s e nt i a lt ot h es uc c e s so ft h eUni v e r s i t y’ s e duc a t i o na la ndr e s e a r c hmi s s i o ns ,a ndvi o l a t i o nso fa c a de mi ci nt e gr i t yc o ns t i t ut es e r i o uso ffe ns e sa ga i ns t t hee nt i r ea c a de mi cc o mmuni t y .Thee nt i r eAc a de mi cI nt e gr i t yPo l i c yc a nb ef o undh e r e : ht t p: / / ac ade mi c i nt e gr i t y. r ut g e r s . e du/ ac ade mi c i nt e gr i t ypol i c y/ c ommodat i onandSuppor tSt a t e ment( Thef ol l owi ngs t a t ementandi nf or ma t i oni s  Ac r ec ommendedf ori nc l us i onona l ls yl l abi . ) :

7 Rut ger sUni ve r s i t yNe wa r k( RUN)i sc ommi t t e dt ot hec r e a t i onofa ni nc l us i veands af el ear ni ng e nvi r onme ntf oral ls t udent s .RUN hasi de nt i fie dt hef ol l owi ngr e s our c e st of ur t he rt hemi s s i on ofa c c e s sa nds uppor t : 

St udent swi t hDi s abi l i t i es :Rut ger sUni ve r s i t ywe l c ome ss t ude nt swi t hdi s abi l i t i e si nt o al loft heUni ver s i t y' seduc a t i onalpr ogr ams .TheOffic eofDi s abi l i t ySe r vi c e s( ODS)i s r es pons i bl ef ort hede t er mi na t i onofappr opr i a t eac c ommoda t i onsf ors t ude nt swho e nc ount erbar r i e r sduet odi s abi l i t y .I nor dert or e c e i vec ons i de r a t i onf orr e as onabl e ac c ommoda t i ons , as t ude ntwi t hadi s abi l i t ymus tc ont a c tODS,r e gi s t e r , ha vea ni ni t i al appoi nt ment ,andpr ovi dedoc ume nt a t i on.Onc eas t ude ntha sc ompl e t e dt heODS pr oc es s( r e gi s t r a t i on,i ni t i alappoi nt ment ,anddoc ume nt a t i ons ubmi t t e d)andr e as ona bl e ac c ommoda t i onsar ede t er mi ne dt obenec e s s ar yandappr opr i a t e ,aLe t t erof Ac c ommoda t i on( LOA)wi l lbepr ovi de dt ot hes t ude nt .Thes t udentmus tgi vet heLOA t oe ac hc our s ei ns t r uc t or ,f ol l owe dbyadi s c us s i onwi t ht hei ns t r uc t or .Thi ss houl dbe c ompl e t e dasear l yi nt hes e me s t e raspos s i bl easa c c ommoda t i onsar enotr e t r oac t i ve . ac tODS:( 9 73)35 35375or Mor ei nf or ma t i onc anbef ounda tods.rutgers.edu.Cont [email protected].

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Rel i gi ousHol i da yPol i c yandAc c ommodat i ons :St ude nt sar eadvi s edt opr ovi de t i me l ynot i fic a t i ont oi ns t r uc t or saboutnec e s s ar yabs enc esf orr e l i gi ousobs er vanc e sand ar er e s pons i bl ef orma ki ngupt hewor kore xa msa c c or di ngt oa na gr ee dupons c he dul e . TheDi vi s i onofS t ude ntAffai r si sa vai l abl et ove r i f ya bs e nc e sf orr e l i gi ousobs er va nc e ,as ne ede d:( 973)3 535 [email protected].

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Couns el i ngSer vi c es :Couns e l i ngCent e rRoom 101,Bl ume nt halHal l ,( 9 73)3535805or http://counseling.newark.rutgers.edu/.

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St udent swi t hTempor ar yCondi t i ons / I nj ur i es :St udent se xper i e nc i ngat e mpor ar y c ondi t i onori nj ur yt ha ti sadve r s e l ya ffe c t i ngt he i rabi l i t yt of ul l ypar t i c i pa t ei nt hei r c our s e ss houl ds ubmi tar e que s tf ora s s i s t anc ea t :https://temporaryconditions.rutgers.edu.

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St udent sWhoa r ePr egnant :TheOffic eofTi t l eI XandADACompl i anc ei sa vai l abl et o as s i s ts t ude nt swi t hanyc onc e r nsorpot e nt i alac c ommoda t i onsr e l a t e dt opr e gnanc y: ( 9 73)[email protected].

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GenderorSex Ba s edDi s c r i mi nat i onorHar a s s ment :St udent se xper i enc i nganyf or m ofgende rors e x bas e ddi s c r i mi na t i onorhar as s me nt ,i nc l udi ngs e xualas s aul t ,s e xual har as s me nt ,r e l a t i ons hi pvi ol enc e ,ors t al ki ng,s houl dknow t ha the l pands uppor ta r e a vai l abl e .Tor epor ta ni nc i dent , c ont a c tt heOffic eofTi t l eI XandADACompl i anc e : ubmi tani nc i de ntr e por t : ( 9 73)[email protected] tinyurl.com/RUNReportingForm.Tos pe akwi t has t a ffme mberwhoi sc onfident i aland doesNOTha vear e por t i ngr e s pons i bi l i t y ,c ont ac tt heOffic ef orVi ol e nc ePr e ve nt i onand Vi c t i m As s i s t a nc e :( 973)353 [email protected].

 Lear ni ngRe s our c e s : 

Rut ger sLe ar ni ngCent er( t ut or i ngs er vi c e s ) Room 140,Br adl e yHal l

8 ( 9 73)3535608 https://sasn.rutgers.edu/student-support/tutoring-academic-support/learning-center Wr i t i ngCent er( t ut or i ngandwr i t i ngwor ks hops ) Room 126,Conkl i nHal l ( 9 73)3535847 nwc @r ut ge r s . e du https://sasn.rutgers.edu/student-support/tutoring-academic-support/writing-center 

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