AS propositional calculus III - Rensselaer Polytechnic...

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 Logic2/8/2018; & again 2/12/18 by Rini Palamittam

Selmer Bringsjord

Logistics …

Logistics …Any takers?

Logistics …Any takers? NO

Logistics …Note: Should now have laptop with you and ready to go with Slate installed.

Any takers? NO


And … HyperGrader will debut in class on Feb 12, led by Rini.

Any takers? NO



http://www.logicamodernapproach.com

Any takers? NO





Plenty of problems there to already learn/practice on!

Any takers? NO


Your code for Slate & HyperGrader for the semester:

Your code for Slate & HyperGrader for the semester:

To submit purportedly correct proofs to HyperGrader, you will authenticate with your email address & your code.

Save sleeve & CD, snapshot sleeve & archive!!

What’s on the CD?

What’s on the CD?

LAMA-BDLA

Slate

HyperGrader

What’s on the CD?

Mac OS

LAMA-BDLA

Slate

HyperGrader

What’s on the CD?

Mac OSWindows

LAMA-BDLA

Slate

HyperGrader

What’s on the CD?

Mac OSWindows

textbook

LAMA-BDLA

Slate

HyperGrader

What’s on the CD?

Mac OSWindows

textbook

Complete, sign, email pdf to [email protected]

LAMA-BDLA

Slate

HyperGrader

mailto:[email protected]?subject=

Again: Initial Steps

Again: Initial Steps• Snapshot sleeve with code, and archive.


• Copy the folder from CD to your laptop.



• Eject CD and “bank-vault”-save both!




• Depending upon whether you’re Windows or MacOS, expand the relevant zipped file to obtain Slate.





• Open Slate





• Open Slate

• Today I’ll continue to explain and show inference rules in Slate.

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 Logic2/8/2018; & again on 2/12/18 by Rini Palamittam

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-zerodenominator.

• Rational numbers are a subset of real numbers!

• Examples of irrational numbers?

p






p


p2

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p


p2


e<latexit sha1_base64="8g0P2hkn1tmY+V1wE8v60UKiD0I=">AAAB53icbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWMLxhbaUDbbSbt2swm7G6GE/gIvHlS8+pe8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE3lvxikGMR1IHnFGjZWa2KtU3Zo7A1kmXkGqUKDRq3x1+wnLYpSGCap1x3NTE+RUGc4ETsrdTGNK2YgOsGOppDHqIJ8dOiGnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrRm4r/eZ3MRFdBzmWaGZRsvijKBDEJmX5N+lwhM2JsCWWK21sJG1JFmbHZlG0I3uLLy8Q/r13X3OZFtX5TpFGCYziBM/DgEupwBw3wgQHCM7zCm/PovDjvzse8dcUpZo7gD5zPHzeUjLw=</latexit><latexit sha1_base64="8g0P2hkn1tmY+V1wE8v60UKiD0I=">AAAB53icbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWMLxhbaUDbbSbt2swm7G6GE/gIvHlS8+pe8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE3lvxikGMR1IHnFGjZWa2KtU3Zo7A1kmXkGqUKDRq3x1+wnLYpSGCap1x3NTE+RUGc4ETsrdTGNK2YgOsGOppDHqIJ8dOiGnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrRm4r/eZ3MRFdBzmWaGZRsvijKBDEJmX5N+lwhM2JsCWWK21sJG1JFmbHZlG0I3uLLy8Q/r13X3OZFtX5TpFGCYziBM/DgEupwBw3wgQHCM7zCm/PovDjvzse8dcUpZo7gD5zPHzeUjLw=</latexit><latexit sha1_base64="8g0P2hkn1tmY+V1wE8v60UKiD0I=">AAAB53icbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWMLxhbaUDbbSbt2swm7G6GE/gIvHlS8+pe8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE3lvxikGMR1IHnFGjZWa2KtU3Zo7A1kmXkGqUKDRq3x1+wnLYpSGCap1x3NTE+RUGc4ETsrdTGNK2YgOsGOppDHqIJ8dOiGnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrRm4r/eZ3MRFdBzmWaGZRsvijKBDEJmX5N+lwhM2JsCWWK21sJG1JFmbHZlG0I3uLLy8Q/r13X3OZFtX5TpFGCYziBM/DgEupwBw3wgQHCM7zCm/PovDjvzse8dcUpZo7gD5zPHzeUjLw=</latexit>

Prove that:

Prove that:p2


Prove that:p2


is irrational

What are prime numbers?

What are prime numbers?

• A number that can be divided by only twonumbers, one and itself.

• Must be a whole number.

• Example: 2,3,5,7……

And recall: Euclidean “Magic”

And recall: Euclidean “Magic”

Study it word by work 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

AS propositional calculus III - Rensselaer Polytechnic...

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