Propositional Calculus II: More Rules of Inference, Application to Additional Motivating Problems
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1 Propositional Calculus II: More Rules of Inference, Application to Additional Motivating Problems Selmer Bringsjord Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department of Computer Science Lally School of Management & Technology Rensselaer Polytechnic Institute (RPI) Troy, New York USA Intro to Logic 2/5/2018
2 Logistics
3 Any takers? Logistics
4 Any takers? NO Logistics
5 Any takers? NO Logistics
6 Any takers? NO Logistics Note: Should now have laptop with you and ready to go with Slate installed.
7 Any takers? NO Logistics Note: Should now have laptop with you and ready to go with Slate installed.
8 Any takers? NO Logistics Note: Should now have laptop with you and ready to go with Slate installed. And HyperGrader will debut in class on Feb 12, led by Rini.
9 Any takers? NO Logistics Note: Should now have laptop with you and ready to go with Slate installed. And HyperGrader will debut in class on Feb 12, led by Rini.
10 Any takers? NO Logistics Note: Should now have laptop with you and ready to go with Slate installed. And HyperGrader will debut in class on Feb 12, led by Rini. Plenty of problems there to learn/practice on!
11 Your code for Slate & HyperGrader for the semester:
12 Your code for Slate & HyperGrader for the semester:
13 Save sleeve & CD, snapshot sleeve & archive!!
14 Save sleeve & CD, snapshot sleeve & archive!!
15 What s on the CD?
16 What s on the CD?
17 What s on the CD? Mac OS
18 What s on the CD? Windows Mac OS
19 What s on the CD? textbook Windows Mac OS
20 What s on the CD? textbook Windows Mac OS
21 What s on the CD? textbook Windows Mac OS Complete, sign, pdf to Selmer.Bringsjord@gmail.com
22 Again: Initial Steps
23 Again: Initial Steps Snapshot sleeve with code, and archive.
24 Again: Initial Steps Snapshot sleeve with code, and archive. Copy the folder from CD to your laptop.
25 Again: Initial Steps Snapshot sleeve with code, and archive. Copy the folder from CD to your laptop. Eject CD and bank-vault -save both!
26 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.
27 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
28 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 Today I ll continue to explain and show inference rules in Slate.
29 Propositional Calculus II: More Rules of Inference, Application to Additional Motivating Problems Selmer Bringsjord Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department of Computer Science Lally School of Management & Technology Rensselaer Polytechnic Institute (RPI) Troy, New York USA Intro to Logic 2/5/2018
30 Where we are in the schedule Go to syllabus
31 Last time we introduced and and lauded the power of oracles, and now picking up where we left off
32 Given the statements c c a a b b d (d e) NYS 3 Revisited which one of the following statements must also be true? c e h a all of the above
33 Given the statements c c a a b b d (d e) NYS 3 Revisited which one of the following statements must also be true? c e h a all of the above
34 NYS 3 Revisited Given the statements c c a After last class, should have done a b Exercise: Show in Slate that each of b d the first four options can be proved (d e) using the PC entailment oracle. which one of the following statements must also be true? c e h a all of the above
35 (and (not C) E H (not A))
36 (and (not C) E H (not A))
37 (and (not C) E H (not A))
38 (and (not C) E H (not A))
39 (and (not C) E H (not A))
40 (and (not C) E H (not A))
41 (and (not C) E H (not A))
42
43 Proof Plan
44 Proof Plan
45 Proof Plan
46 Proof Plan
47 Proof Plan Sub-Proof
48 Proof Plan Sub-Proof New Inference Rule:
49 Proof Plan Sub-Proof New Inference Rule:
50 Proof Plan Sub-Proof New Inference Rule: conditional elim a.k.a., modus ponens
51 Proof Plan Sub-Proof New Inference Rule: conditional elim a.k.a., modus ponens
52 The Original King-Ace Suppose that the following premise is true: If there is a king in the hand, then there is an ace in 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?
53 The Original King-Ace Suppose that the following premise is true: If there is a king in the hand, then there is an ace in 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.
54 The Original King-Ace Suppose that the following premise is true: If there is a king in the hand, then there is an ace in 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.
55 The Original King-Ace Suppose that the following premise is true: If there is a king in the hand, then there is an ace in 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? NO! There is an ace in the hand.
56 The Original King-Ace Suppose that the following premise is true: If there is a king in the hand, then there is an ace in 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? NO! There is an ace in the hand. NO!
57 The Original King-Ace Suppose that the following premise is true: If there is a king in the hand, then there is an ace in 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? NO! There is an ace in the hand. NO! In fact, what you can infer is that there isn t an ace in the hand!
58 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
59 King-Ace 2 Suppose 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. What can you infer from this premise?
60 King-Ace 2 Suppose 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. What can you infer from this premise? There is an ace in the hand.
61 King-Ace 2 Suppose 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. What can you infer from this premise? There is an ace in the hand.
62 King-Ace 2 Suppose 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. What can you infer from this premise? NO! There is an ace in the hand.
63 King-Ace 2 Suppose 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. What can you infer from this premise? NO! There is an ace in the hand. NO!
64 King-Ace 2 Suppose 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. What can you infer from this premise? NO! There is an ace in the hand. NO! In fact, what you can infer is that there isn t an ace in the hand!
65 King-Ace 2 Suppose 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. What can you infer from this premise? NO! There is an ace in the hand. NO! In fact, what you can infer is that there isn t an ace in the hand!
66 Study the S-expression
67 Study the S-expression
68 We need another rule of inference to crack this problem
69 We need another rule of inference to crack this problem disjunction elimination
70 From p. 54
71 From p. 54
72 From p. 54
73 From p. 54?
74 From p. 54?
75 From p. 54?
76 King-Ace 2 Suppose 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. What can you infer from this premise? NO! There is an ace in the hand. NO! In fact, what you can infer is that there isn t an ace in the hand!
77 King-Ace 2 Suppose 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. What can you infer from this premise? NO! There is an ace in the hand. NO! In fact, what you can infer is that there isn t an ace in the hand!
78 King-Ace 2 Suppose 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. What can you infer from this premise? NO! There is an ace in the hand. NO! In fact, what you can infer is that there isn t an ace in the hand!
79 King-Ace 2 Suppose 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 Exercise: an ace; but Finish not both the proof of these in Slate if-then statements with are no true. remaining use of an oracle. What can you infer from this premise? NO! There is an ace in the hand. NO! In fact, what you can infer is that there isn t an ace in the hand!
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