SB propositional calculus II · Propositional Calculus II: Two more Rules of Inference/Inference...

Propositional Calculus II:Two more Rules of Inference/Inference Schemata

(conditional elim = modus ponens, proof by cases), Application to Additional Motivating Problems

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 Logic2/3/2020

Selmer Bringsjord

No, it’s not logical to throw a bomb from the Kansas City 49-yard line @ 3rd and 10, with 1:49 to go.

Re-re-re…orientation w.r.t. web pages …

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

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

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.

Propositional Calculus II:More Rules of Inference

(conditional elim = modus ponens, proof by cases), Application to Additional Motivating Problems

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 Logic2/3/2020

Selmer Bringsjord

Last time we introduced and and lauded the power of oracles, and questions … and now … picking up where we left off …

“NYS 3” RevisitedGiven the statements¬¬cc → a¬a ∨ bb → d¬(d ∨ e)

which one of the following statements must also be true?

¬ceh¬aall of the above

After last class, should have done …Exercise: Show in HyperSlateTM that each of the first four options can be proved using the PC entailment oracle.

(and (not C) E H (not A))

Proof Plan …

Sub-Proof

Proof Plan …

Sub-ProofInstead of using the PC oracle, we can use a new inference rule:

Proof Plan …

conditional elima.k.a., modus ponens

Proof Plan …

conditional elima.k.a., modus ponens

The Universe of Logics

Infinitary Finitary

Intensional ExtensionalDeductive InductiveDeductive Inductive

Symbolic Diagram Symbolic Diagram Symbolic Diagram Symbolic Diagram

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L1 = first-order logic<latexit sha1_base64="45ddNPsZu/fZxPjqZFRqcoCgy/Q=">AAACJ3icbVDLSsNAFJ34rPUVdelmsAjdWBIRdCMU3bhwUcE+oAlhMp20QyeZMDMRS8gv+B/u3eovuBNduvMznKRZ2NYDA4dz7muOHzMqlWV9GUvLK6tr65WN6ubW9s6uubffkTwRmLQxZ1z0fCQJoxFpK6oY6cWCoNBnpOuPr3O/+0CEpDy6V5OYuCEaRjSgGCkteWbdCZEaSSzS28yz4SV0Qp8/pgEVUp1wMSACMj6kOPPMmtWwCsBFYpekBkq0PPPHGXCchCRSmCEp+7YVKzdFQlHMSFZ1EklihMdoSPqaRigk0k2LH2XwWCsDGHChX6Rgof7tSFEo5ST0dWVx/7yXi/95/UQFF25KozhRJMLTRUHCoOIwjwcOqCBYsYkmCAuqb4V4hATCSoc4syWfLWQgs6qOxp4PYpF0Thu21bDvzmrNqzKkCjgER6AObHAOmuAGtEAbYPAEXsAreDOejXfjw/icli4ZZc8BmIHx/QuIhqbm</latexit><latexit sha1_base64="45ddNPsZu/fZxPjqZFRqcoCgy/Q=">AAACJ3icbVDLSsNAFJ34rPUVdelmsAjdWBIRdCMU3bhwUcE+oAlhMp20QyeZMDMRS8gv+B/u3eovuBNduvMznKRZ2NYDA4dz7muOHzMqlWV9GUvLK6tr65WN6ubW9s6uubffkTwRmLQxZ1z0fCQJoxFpK6oY6cWCoNBnpOuPr3O/+0CEpDy6V5OYuCEaRjSgGCkteWbdCZEaSSzS28yz4SV0Qp8/pgEVUp1wMSACMj6kOPPMmtWwCsBFYpekBkq0PPPHGXCchCRSmCEp+7YVKzdFQlHMSFZ1EklihMdoSPqaRigk0k2LH2XwWCsDGHChX6Rgof7tSFEo5ST0dWVx/7yXi/95/UQFF25KozhRJMLTRUHCoOIwjwcOqCBYsYkmCAuqb4V4hATCSoc4syWfLWQgs6qOxp4PYpF0Thu21bDvzmrNqzKkCjgER6AObHAOmuAGtEAbYPAEXsAreDOejXfjw/icli4ZZc8BmIHx/QuIhqbm</latexit><latexit sha1_base64="45ddNPsZu/fZxPjqZFRqcoCgy/Q=">AAACJ3icbVDLSsNAFJ34rPUVdelmsAjdWBIRdCMU3bhwUcE+oAlhMp20QyeZMDMRS8gv+B/u3eovuBNduvMznKRZ2NYDA4dz7muOHzMqlWV9GUvLK6tr65WN6ubW9s6uubffkTwRmLQxZ1z0fCQJoxFpK6oY6cWCoNBnpOuPr3O/+0CEpDy6V5OYuCEaRjSgGCkteWbdCZEaSSzS28yz4SV0Qp8/pgEVUp1wMSACMj6kOPPMmtWwCsBFYpekBkq0PPPHGXCchCRSmCEp+7YVKzdFQlHMSFZ1EklihMdoSPqaRigk0k2LH2XwWCsDGHChX6Rgof7tSFEo5ST0dWVx/7yXi/95/UQFF25KozhRJMLTRUHCoOIwjwcOqCBYsYkmCAuqb4V4hATCSoc4syWfLWQgs6qOxp4PYpF0Thu21bDvzmrNqzKkCjgER6AObHAOmuAGtEAbYPAEXsAreDOejXfjw/icli4ZZc8BmIHx/QuIhqbm</latexit><latexit sha1_base64="45ddNPsZu/fZxPjqZFRqcoCgy/Q=">AAACJ3icbVDLSsNAFJ34rPUVdelmsAjdWBIRdCMU3bhwUcE+oAlhMp20QyeZMDMRS8gv+B/u3eovuBNduvMznKRZ2NYDA4dz7muOHzMqlWV9GUvLK6tr65WN6ubW9s6uubffkTwRmLQxZ1z0fCQJoxFpK6oY6cWCoNBnpOuPr3O/+0CEpDy6V5OYuCEaRjSgGCkteWbdCZEaSSzS28yz4SV0Qp8/pgEVUp1wMSACMj6kOPPMmtWwCsBFYpekBkq0PPPHGXCchCRSmCEp+7YVKzdFQlHMSFZ1EklihMdoSPqaRigk0k2LH2XwWCsDGHChX6Rgof7tSFEo5ST0dWVx/7yXi/95/UQFF25KozhRJMLTRUHCoOIwjwcOqCBYsYkmCAuqb4V4hATCSoc4syWfLWQgs6qOxp4PYpF0Thu21bDvzmrNqzKkCjgER6AObHAOmuAGtEAbYPAEXsAreDOejXfjw/icli4ZZc8BmIHx/QuIhqbm</latexit>

L0 = zeroth-order logic<latexit sha1_base64="rae+VGvksk3nCsYockaB25wnhnQ=">AAACKHicbVDLSsNAFJ3UV62vqks3g0XQhSURQTdC0Y0LFxXsA9pSJtObdugkE2YmYg35Bv/DvVv9BXfSrSs/w0mbhW09MHA45z7mHjfkTGnbHlu5peWV1bX8emFjc2t7p7i7V1cikhRqVHAhmy5RwFkANc00h2Yogfguh4Y7vEn9xiNIxUTwoEchdHzSD5jHKNFG6hZP2j7RA0VlfJd0bXyF274rnuJnkEIPToXsgcRc9BlNusWSXbYnwIvEyUgJZah2iz/tnqCRD4GmnCjVcuxQd2IiNaMckkI7UhASOiR9aBkaEB9UJ56clOAjo/SwJ6R5gcYT9W9HTHylRr5rKicHzHup+J/XirR32YlZEEYaAjpd5EUca4HTfHCPSaCajwwhVDLzV0wHRBKqTYozW9LZUnkqKZhonPkgFkn9rOzYZef+vFS5zkLKowN0iI6Rgy5QBd2iKqohil7QG3pHH9ar9Wl9WeNpac7KevbRDKzvX3aSp2M=</latexit><latexit sha1_base64="rae+VGvksk3nCsYockaB25wnhnQ=">AAACKHicbVDLSsNAFJ3UV62vqks3g0XQhSURQTdC0Y0LFxXsA9pSJtObdugkE2YmYg35Bv/DvVv9BXfSrSs/w0mbhW09MHA45z7mHjfkTGnbHlu5peWV1bX8emFjc2t7p7i7V1cikhRqVHAhmy5RwFkANc00h2Yogfguh4Y7vEn9xiNIxUTwoEchdHzSD5jHKNFG6hZP2j7RA0VlfJd0bXyF274rnuJnkEIPToXsgcRc9BlNusWSXbYnwIvEyUgJZah2iz/tnqCRD4GmnCjVcuxQd2IiNaMckkI7UhASOiR9aBkaEB9UJ56clOAjo/SwJ6R5gcYT9W9HTHylRr5rKicHzHup+J/XirR32YlZEEYaAjpd5EUca4HTfHCPSaCajwwhVDLzV0wHRBKqTYozW9LZUnkqKZhonPkgFkn9rOzYZef+vFS5zkLKowN0iI6Rgy5QBd2iKqohil7QG3pHH9ar9Wl9WeNpac7KevbRDKzvX3aSp2M=</latexit><latexit sha1_base64="rae+VGvksk3nCsYockaB25wnhnQ=">AAACKHicbVDLSsNAFJ3UV62vqks3g0XQhSURQTdC0Y0LFxXsA9pSJtObdugkE2YmYg35Bv/DvVv9BXfSrSs/w0mbhW09MHA45z7mHjfkTGnbHlu5peWV1bX8emFjc2t7p7i7V1cikhRqVHAhmy5RwFkANc00h2Yogfguh4Y7vEn9xiNIxUTwoEchdHzSD5jHKNFG6hZP2j7RA0VlfJd0bXyF274rnuJnkEIPToXsgcRc9BlNusWSXbYnwIvEyUgJZah2iz/tnqCRD4GmnCjVcuxQd2IiNaMckkI7UhASOiR9aBkaEB9UJ56clOAjo/SwJ6R5gcYT9W9HTHylRr5rKicHzHup+J/XirR32YlZEEYaAjpd5EUca4HTfHCPSaCajwwhVDLzV0wHRBKqTYozW9LZUnkqKZhonPkgFkn9rOzYZef+vFS5zkLKowN0iI6Rgy5QBd2iKqohil7QG3pHH9ar9Wl9WeNpac7KevbRDKzvX3aSp2M=</latexit><latexit sha1_base64="rae+VGvksk3nCsYockaB25wnhnQ=">AAACKHicbVDLSsNAFJ3UV62vqks3g0XQhSURQTdC0Y0LFxXsA9pSJtObdugkE2YmYg35Bv/DvVv9BXfSrSs/w0mbhW09MHA45z7mHjfkTGnbHlu5peWV1bX8emFjc2t7p7i7V1cikhRqVHAhmy5RwFkANc00h2Yogfguh4Y7vEn9xiNIxUTwoEchdHzSD5jHKNFG6hZP2j7RA0VlfJd0bXyF274rnuJnkEIPToXsgcRc9BlNusWSXbYnwIvEyUgJZah2iz/tnqCRD4GmnCjVcuxQd2IiNaMckkI7UhASOiR9aBkaEB9UJ56clOAjo/SwJ6R5gcYT9W9HTHylRr5rKicHzHup+J/XirR32YlZEEYaAjpd5EUca4HTfHCPSaCajwwhVDLzV0wHRBKqTYozW9LZUnkqKZhonPkgFkn9rOzYZef+vFS5zkLKowN0iI6Rgy5QBd2iKqohil7QG3pHH9ar9Wl9WeNpac7KevbRDKzvX3aSp2M=</latexit>

LPropCalc = propositional calculus<latexit sha1_base64="DE1nAlInoZB4QbARKY9i/o//6a4=">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</latexit><latexit sha1_base64="DE1nAlInoZB4QbARKY9i/o//6a4=">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</latexit><latexit sha1_base64="DE1nAlInoZB4QbARKY9i/o//6a4=">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</latexit><latexit sha1_base64="DE1nAlInoZB4QbARKY9i/o//6a4=">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</latexit>

DCEC⇤<latexit sha1_base64="jH7bxOPxNLvK9MqLWOo6fnj1hGk=">AAACEnicbVDLSsNAFJ3UV62vaJduBovgqiQi6LJYBZcV7AOaWG6mk3bo5MHMRCghf+Herf6CO3HrD/gHfoaTNgvbemDgcM69cw/HizmTyrK+jdLa+sbmVnm7srO7t39gHh51ZJQIQtsk4pHoeSApZyFtK6Y47cWCQuBx2vUmzdzvPlEhWRQ+qGlM3QBGIfMZAaWlgVl1AlBjAjy9ad42s0cHpBqYNatuzYBXiV2QGirQGpg/zjAiSUBDRThI2betWLkpCMUIp1nFSSSNgUxgRPuahhBQ6aaz8Bk+1coQ+5HQL1R4pv7dSCGQchp4ejKPKpe9XPzP6yfKv3JTFsaJoiGZH/ITjlWE8ybwkAlKFJ9qAkQwnRWTMQggSve1cCX/W0hfZhVdjb1cxCrpnNdtq27fX9Qa10VJZXSMTtAZstElaqA71EJtRNAUvaBX9GY8G+/Gh/E5Hy0ZxU4VLcD4+gWz5p4L</latexit><latexit sha1_base64="jH7bxOPxNLvK9MqLWOo6fnj1hGk=">AAACEnicbVDLSsNAFJ3UV62vaJduBovgqiQi6LJYBZcV7AOaWG6mk3bo5MHMRCghf+Herf6CO3HrD/gHfoaTNgvbemDgcM69cw/HizmTyrK+jdLa+sbmVnm7srO7t39gHh51ZJQIQtsk4pHoeSApZyFtK6Y47cWCQuBx2vUmzdzvPlEhWRQ+qGlM3QBGIfMZAaWlgVl1AlBjAjy9ad42s0cHpBqYNatuzYBXiV2QGirQGpg/zjAiSUBDRThI2betWLkpCMUIp1nFSSSNgUxgRPuahhBQ6aaz8Bk+1coQ+5HQL1R4pv7dSCGQchp4ejKPKpe9XPzP6yfKv3JTFsaJoiGZH/ITjlWE8ybwkAlKFJ9qAkQwnRWTMQggSve1cCX/W0hfZhVdjb1cxCrpnNdtq27fX9Qa10VJZXSMTtAZstElaqA71EJtRNAUvaBX9GY8G+/Gh/E5Hy0ZxU4VLcD4+gWz5p4L</latexit><latexit sha1_base64="jH7bxOPxNLvK9MqLWOo6fnj1hGk=">AAACEnicbVDLSsNAFJ3UV62vaJduBovgqiQi6LJYBZcV7AOaWG6mk3bo5MHMRCghf+Herf6CO3HrD/gHfoaTNgvbemDgcM69cw/HizmTyrK+jdLa+sbmVnm7srO7t39gHh51ZJQIQtsk4pHoeSApZyFtK6Y47cWCQuBx2vUmzdzvPlEhWRQ+qGlM3QBGIfMZAaWlgVl1AlBjAjy9ad42s0cHpBqYNatuzYBXiV2QGirQGpg/zjAiSUBDRThI2betWLkpCMUIp1nFSSSNgUxgRPuahhBQ6aaz8Bk+1coQ+5HQL1R4pv7dSCGQchp4ejKPKpe9XPzP6yfKv3JTFsaJoiGZH/ITjlWE8ybwkAlKFJ9qAkQwnRWTMQggSve1cCX/W0hfZhVdjb1cxCrpnNdtq27fX9Qa10VJZXSMTtAZstElaqA71EJtRNAUvaBX9GY8G+/Gh/E5Hy0ZxU4VLcD4+gWz5p4L</latexit><latexit sha1_base64="jH7bxOPxNLvK9MqLWOo6fnj1hGk=">AAACEnicbVDLSsNAFJ3UV62vaJduBovgqiQi6LJYBZcV7AOaWG6mk3bo5MHMRCghf+Herf6CO3HrD/gHfoaTNgvbemDgcM69cw/HizmTyrK+jdLa+sbmVnm7srO7t39gHh51ZJQIQtsk4pHoeSApZyFtK6Y47cWCQuBx2vUmzdzvPlEhWRQ+qGlM3QBGIfMZAaWlgVl1AlBjAjy9ad42s0cHpBqYNatuzYBXiV2QGirQGpg/zjAiSUBDRThI2betWLkpCMUIp1nFSSSNgUxgRPuahhBQ6aaz8Bk+1coQ+5HQL1R4pv7dSCGQchp4ejKPKpe9XPzP6yfKv3JTFsaJoiGZH/ITjlWE8ybwkAlKFJ9qAkQwnRWTMQggSve1cCX/W0hfZhVdjb1cxCrpnNdtq27fX9Qa10VJZXSMTtAZstElaqA71EJtRNAUvaBX9GY8G+/Gh/E5Hy0ZxU4VLcD4+gWz5p4L</latexit>

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The Physical Universe

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Non-Physical

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Non-Physical

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Non-Physical

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�<latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AAAB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8mm2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzzqmiTTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX355lXQvmr7X9O8vG62bMo4qnMApnIMPV9CCO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIjj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AAAB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8mm2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzzqmiTTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX355lXQvmr7X9O8vG62bMo4qnMApnIMPV9CCO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIjj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AAAB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8mm2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzzqmiTTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX355lXQvmr7X9O8vG62bMo4qnMApnIMPV9CCO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIjj8=</latexit><latexit sha1_base64="vMCmCBvrQqvMkcZe5c78yK19qu0=">AAAB63icbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeiF48VbC20S8mm2W5okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemApurOd9o8ra+sbmVnW7trO7t39QPzzqmiTTlHVoIhLdC4lhgivWsdwK1ks1IzIU7DGc3Bb+4xPThifqwU5TFkgyVjzilNhCGqQxH9YbXtObA68SvyQNKNEe1r8Go4RmkilLBTGm73upDXKiLaeCzWqDzLCU0AkZs76jikhmgnx+6wyfOWWEo0S7UhbP1d8TOZHGTGXoOiWxsVn2CvE/r5/Z6DrIuUozyxRdLIoygW2Ci8fxiGtGrZg6Qqjm7lZMY6IJtS6emgvBX355lXQvmr7X9O8vG62bMo4qnMApnIMPV9CCO2hDByjE8Ayv8IYkekHv6GPRWkHlzDH8Afr8AROIjj8=</latexit>

modus ponens

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Next problem(King-Ace) …

The Original King-AceSuppose that the following premise is true:

If there is a king in the hand, then there is an acein the hand, or else if there isn’t a king in the hand,then there is an ace.

What can you infer from this premise?

There is an ace in the hand.

There is an ace in the hand.NO!

There is an ace in the hand.NO! NO!

In fact, what you can infer is that there isn’t an ace in the hand!

King-Ace Solved

Proposition: There is not an ace in the hand.

Proof: We know that at least one of the if-thens (i.e., at least one of the conditionals) is false. So we have two cases to consider, viz., that K => A is false, and that ¬K => A is false. Take first the first case; accordingly, suppose that K => A is false. Then it follows that K is true (since when a conditional is false, its antecedent holds but its consequent doesn’t), and A is false. Now consider the second case, which consists in ¬K => A being false. Here, in a direct parallel, we know ¬K and, once again, ¬A. In both of our two cases, which are exhaustive, there is no ace in the hand. The proposition is established. QED

King-Ace 2Suppose that the following premise is true:

If there is a king in the hand, then there is an ace in the hand; or if there isn’t a king in the hand, then there is an ace; but not both of these if-then statements are true.

There is an ace in the hand.NO!

Study the S-expression

We need another rule of inference to crack this problem … …

disjunction elimination

From ~ p. 54 in LAMA-BDLA

Exercise (on HyperGraderTM): Finish the proof in HyperSlateTM — with no remaining use of an oracle.

Pop Problem

SB propositional calculus II · Propositional Calculus II: Two more Rules of Inference/Inference...

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