AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad...

Propositional Calculus III:Reductio ad Absurdum

Rensselaer AI & Reasoning (RAIR) LabDepartment of Cognitive ScienceDepartment of Computer Science

Lally School of Management & TechnologyRensselaer Polytechnic Institute (RPI)

Troy, New York 12180 USA

Intro to Logic2/6/2020

Selmer Bringsjord

Logistics …

Logistics …

Any questions?

The Starting Code to Purchase in Bookstore


Once seal broken on envelope, no return. Remember from first class, any reservations, opt for “Stanford” paradigm, with its software instead of LAMATM paradigm!



The email address you enter is case-sensitive!



The email address you enter is case-sensitive!Your OS and browser must be fully up-to-date; Chrome is the best choice, browser-wise (though I use Safari).



The email address you enter is case-sensitive!Your OS and browser must be fully up-to-date; Chrome is the best choice, browser-wise (though I use Safari).Watch that the link doesn’t end up being classified as spam.

The Starting Code Purchased in Bookstore Should By Now’ve Been Used to Register & Subsequently Sign In

Batch of prop. calc. (Homework) Problems in;easiest starting place: switching_conjuncts_fine;

but should be done with some others by now.

E-Housekeeping Pts, Redux


• Must input your RIN.



• Make sure OS fully up-to-date.




• Make sure browser fully up-to-date.





• Chrome best (but I use Safari).





• Chrome best (but I use Safari).

• Always work in the same browser window with multiple tabs; must do this with email and HyperGraderTM & HyperSlateTM.

Some HyperSlate Pts

Some HyperSlate Pts• Don’t fall behind on (Homework) Problems!!

Many more are coming for this self-paced 40%-of-grade part of your learning …



• Some new ones this am, in fact; e.g. Disj_Syll.




• Consider using right-clicking on nodes.





• “Where are my files?”






• We are not covering all inference rules/schemata in class.






• We are not covering all inference rules/schemata in class.

• conditional intro, briefly …

Propositional Calculus III:Reductio ad Absurdum

Rensselaer AI & Reasoning (RAIR) LabDepartment of Cognitive ScienceDepartment of Computer Science

Lally School of Management & TechnologyRensselaer Polytechnic Institute (RPI)

Troy, New York 12180 USA

Intro to Logic2/6/2020

Selmer Bringsjord

Reductio …

–G. H. Hardy

“Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”

A Greek-shocking Example …

p

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What are rational numbers?p


What are rational numbers?• Any number that can be expressed in the form of

such that we have a numerator p and a non-zero denominator.

• Rational numbers are a subset of real numbers!

• Examples of irrational numbers?

p


What are rational numbers?• Any number that can be expressed in the form of

such that we have a numerator p and a non-zero denominator.

• Rational numbers are a subset of real numbers!

• Examples of irrational numbers?

p


p2

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Prove that:

Prove that:p2


Prove that:p2


is irrational

Sizing Some Sets

N<latexit sha1_base64="nr3j8pJXjGFdqDcWNvwqQc1Y3/Y=">AAAB8XicbVDLSsNAFL2pr1pfVZduBovgqiRaqu6KblxJBfvANpTJdNIOnUzCzEQooX/hxoUibv0bd/6NkzSIWg8MHM65lzn3eBFnStv2p1VYWl5ZXSuulzY2t7Z3yrt7bRXGktAWCXkoux5WlDNBW5ppTruRpDjwOO14k6vU7zxQqVgo7vQ0om6AR4L5jGBtpPt+gPXY85Kb2aBcsat2BrRInJxUIEdzUP7oD0MSB1RowrFSPceOtJtgqRnhdFbqx4pGmEzwiPYMFTigyk2yxDN0ZJQh8kNpntAoU39uJDhQahp4ZjJNqP56qfif14u1f+4mTESxpoLMP/JjjnSI0vPRkElKNJ8agolkJisiYywx0aakUlbCRYr698mLpH1SdU6rtdtapXGZ11GEAziEY3DgDBpwDU1oAQEBj/AML5aynqxX620+WrDynX34Bev9C9BlkSY=</latexit>

Z+<latexit sha1_base64="z8AUTvaZctkMTOs7ewyjVitlVWE=">AAAB83icbVDLSsNAFL2pr1pfVZduBosgCCXR4mNXdOOygn1gE8tkOmmHTiZhZiKU0N9w40IRt/6MO//GSVpErQcGDufcyz1z/JgzpW370yosLC4trxRXS2vrG5tb5e2dlooSSWiTRDySHR8rypmgTc00p51YUhz6nLb90VXmtx+oVCwSt3ocUy/EA8ECRrA2kuuGWA99P72b3B/1yhW7audA88SZkQrM0OiVP9x+RJKQCk04Vqrr2LH2Uiw1I5xOSm6iaIzJCA9o11CBQ6q8NM88QQdG6aMgkuYJjXL150aKQ6XGoW8ms4zqr5eJ/3ndRAfnXspEnGgqyPRQkHCkI5QVgPpMUqL52BBMJDNZERliiYk2NZXyEi4ynH5/eZ60jqvOSbV2U6vUL2d1FGEP9uEQHDiDOlxDA5pAIIZHeIYXK7GerFfrbTpasGY7u/AL1vsXBOeRzw==</latexit>

Q<latexit sha1_base64="tWm1V9udxVcW9kU0TpTwB8CTsGU=">AAAB8XicbVDLSsNAFL2pr1pfVZduBovgqiRaqu6Kbly2YB/YhjKZTtqhk0mYmQgl9C/cuFDErX/jzr9xkgZR64GBwzn3MuceL+JMadv+tAorq2vrG8XN0tb2zu5eef+go8JYEtomIQ9lz8OKciZoWzPNaS+SFAcep11vepP63QcqFQvFnZ5F1A3wWDCfEayNdD8IsJ54XtKaD8sVu2pnQMvEyUkFcjSH5Y/BKCRxQIUmHCvVd+xIuwmWmhFO56VBrGiEyRSPad9QgQOq3CRLPEcnRhkhP5TmCY0y9edGggOlZoFnJtOE6q+Xiv95/Vj7l27CRBRrKsjiIz/mSIcoPR+NmKRE85khmEhmsiIywRITbUoqZSVcpah/n7xMOmdV57xaa9Uqjeu8jiIcwTGcggMX0IBbaEIbCAh4hGd4sZT1ZL1ab4vRgpXvHMIvWO9f1PSRKQ==</latexit>

R<latexit sha1_base64="Kq6f/6bdrf7+VEhAWSEpjL7U3o0=">AAAB8XicbVDLSsNAFL2pr1pfVZduBovgqiRaqu6KblxWsQ9sQ5lMJ+3QySTMTIQS+hduXCji1r9x5984SYOo9cDA4Zx7mXOPF3GmtG1/WoWl5ZXVteJ6aWNza3unvLvXVmEsCW2RkIey62FFORO0pZnmtBtJigOP0443uUr9zgOVioXiTk8j6gZ4JJjPCNZGuu8HWI89L7mdDcoVu2pnQIvEyUkFcjQH5Y/+MCRxQIUmHCvVc+xIuwmWmhFOZ6V+rGiEyQSPaM9QgQOq3CRLPENHRhkiP5TmCY0y9edGggOlpoFnJtOE6q+Xiv95vVj7527CRBRrKsj8Iz/mSIcoPR8NmaRE86khmEhmsiIyxhITbUoqZSVcpKh/n7xI2idV57Rau6lVGpd5HUU4gEM4BgfOoAHX0IQWEBDwCM/wYinryXq13uajBSvf2YdfsN6/ANZ5kSo=</latexit>

I<latexit sha1_base64="79DCOKDwvARgvt4ZCvx9FkXGUAQ=">AAAB8XicbVDLSsNAFL2pr1pfVZduBovgqiRaqu6KbnRXwT6wDWUynbRDJ5MwMxFK6F+4caGIW//GnX/jJA2i1gMDh3PuZc49XsSZ0rb9aRWWlldW14rrpY3Nre2d8u5eW4WxJLRFQh7KrocV5UzQlmaa024kKQ48Tjve5Cr1Ow9UKhaKOz2NqBvgkWA+I1gb6b4fYD32vORmNihX7KqdAS0SJycVyNEclD/6w5DEARWacKxUz7Ej7SZYakY4nZX6saIRJhM8oj1DBQ6ocpMs8QwdGWWI/FCaJzTK1J8bCQ6UmgaemUwTqr9eKv7n9WLtn7sJE1GsqSDzj/yYIx2i9Hw0ZJISzaeGYCKZyYrIGEtMtCmplJVwkaL+ffIiaZ9UndNq7bZWaVzmdRThAA7hGBw4gwZcQxNaQEDAIzzDi6WsJ+vVepuPFqx8Zx9+wXr/AsjMkSE=</latexit>

?

And now, what are prime numbers?

And now, what are prime numbers?• A number that can be divided by

only two numbers, one and itself.

• Must be a whole number.

• Example: 2,3,5,7……

And recall: Euclidean “Magic”


Notice that this proof uses two applications of indirect proof, and one application of proof by cases.


Notice that this proof uses two applications of indirect proof, and one application of proof by cases.

Study it word by word until you endorse it with your very soul!

From Algebra 2 in High School …(Pearson Common-Core Compliant Textbook)

@article{moore.proof, Author = {R. C. Moore}, Journal = {Educational Studies in Mathematics}, Pages = {249-266}, Title = {Making the Transition to Formal Proof}, Volume = {27.3}, Year = 1994}

Explosion

PC ⊢ ✓

GIVEN. φ ∧ ¬φ{GIVEN} Assume ✓

GOAL. ψ{GIVEN}

Explosion: Partial Proof Plan

GreenCheeseMoon1

PC ⊢ ✓

GOAL. G{GIVEN}

GIVEN. ¬(P → (Q → P)){GIVEN} Assume ✓

GreenCheeseMoon1: Partial Proof Plan

Sub-Proof Here

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