as propositional calculus iii - logicamodernapproach.com · propositional calculus iii: reductio ad...
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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
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Logistics …
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Logistics …
Any questions?
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The Starting Code to Purchase in Bookstore
![Page 5: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/5.jpg)
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!
![Page 6: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/6.jpg)
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!
![Page 7: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/7.jpg)
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!Your OS and browser must be fully up-to-date; Chrome is the best choice, browser-wise (though I use Safari).
![Page 8: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/8.jpg)
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!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.
![Page 9: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/9.jpg)
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.
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E-Housekeeping Pts, Redux
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E-Housekeeping Pts, Redux
• Must input your RIN.
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E-Housekeeping Pts, Redux
• Must input your RIN.
• Make sure OS fully up-to-date.
![Page 13: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/13.jpg)
E-Housekeeping Pts, Redux
• Must input your RIN.
• Make sure OS fully up-to-date.
• Make sure browser fully up-to-date.
![Page 14: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/14.jpg)
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).
![Page 15: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/15.jpg)
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).
• Always work in the same browser window with multiple tabs; must do this with email and HyperGraderTM & HyperSlateTM.
![Page 16: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/16.jpg)
Some HyperSlate Pts
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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 …
![Page 18: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/18.jpg)
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.
![Page 19: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/19.jpg)
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.
![Page 20: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/20.jpg)
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?”
![Page 21: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/21.jpg)
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.
![Page 22: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/22.jpg)
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.
• conditional intro, briefly …
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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
![Page 24: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/24.jpg)
Reductio …
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–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.”
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A Greek-shocking Example …
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p
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
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What are rational numbers?p
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
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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
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
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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
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
p2
<latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit>
![Page 31: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/31.jpg)
![Page 32: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/32.jpg)
Prove that:
![Page 33: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/33.jpg)
Prove that:p2
<latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit>
![Page 34: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/34.jpg)
Prove that:p2
<latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit>
is irrational
![Page 35: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/35.jpg)
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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>
?
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And now, what are prime numbers?
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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……
![Page 40: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/40.jpg)
And recall: Euclidean “Magic”
![Page 41: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/41.jpg)
And recall: Euclidean “Magic”
![Page 42: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/42.jpg)
And recall: Euclidean “Magic”
![Page 43: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/43.jpg)
And recall: Euclidean “Magic”
![Page 44: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/44.jpg)
And recall: Euclidean “Magic”
![Page 45: AS propositional calculus III - logicamodernapproach.com · Propositional Calculus III: Reductio ad Absurdum Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department](https://reader035.vdocuments.us/reader035/viewer/2022062507/5fd60c2e5b55b05dca6e11c3/html5/thumbnails/45.jpg)
And recall: Euclidean “Magic”
Notice that this proof uses two applications of indirect proof, and one application of proof by cases.
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And recall: Euclidean “Magic”
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!
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From Algebra 2 in High School …(Pearson Common-Core Compliant Textbook)
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@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}
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Explosion
PC ⊢ ✓
GIVEN. φ ∧ ¬φ{GIVEN} Assume ✓
GOAL. ψ{GIVEN}
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Explosion: Partial Proof Plan
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GreenCheeseMoon1
PC ⊢ ✓
GOAL. G{GIVEN}
GIVEN. ¬(P → (Q → P)){GIVEN} Assume ✓
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GreenCheeseMoon1: Partial Proof Plan
Sub-Proof Here
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Pop Problem