Derivations, part 2. Let s dive in to some derivations that require the use of the last four rules:
|
|
- Dustin Harrison
- 5 years ago
- Views:
Transcription
1 Derivations, part 2 Let s dive in to some derivations that require the use of the last four rules: 1. I Derivations: Let s start with some derivations that use conditional-introduction. (a) Here s an easy one: Imagine that your friend says, If you take that job, then you ll move away. And if you move away, then I ll be sad. So, if you take that job, then I ll be sad. This is known as a hypothetical syllogism, and your textbook introduces it as the following sequent: S89: P Q, Q R Ⱶ P R Somehow, we ll need to get from (1) and (2) to (n) in the following derivation: 2 (2) Q R A 1, 2 (n) P R? We ll want to introduce the conditional on line (n), so we ll need to assume it s antecedent P. It turns out that that s all we need. Here s the derivation: 2 (2) Q R A 3 (3) P Ass. ( I) 1, 3 (4) Q 1, 3, E 1, 2, 3 (5) R 2, 4, E 1, 2 (6) P R 3, 5, I We were able to derive R from our assumption P, so we can just introduce an arrow to get P R, which discharges our assumption P. Let s do a slightly harder one: (b) Imagine that, due to another love triangle, someone says, If Paul shows up to the party, then Quinn will be really upset, but Rebecca will be thrilled. Therefore, if Paul shows up to the party, then Quinn will be really upset, and, if Paul shows up to the party, then Rebecca will be thrilled. That makes sense. Here s the sequent to be proved: S19: P (Q R) Ⱶ (P Q) (P R) Let s begin: 1
2 1 (n) (P Q) (P R)? If we assume P for the sake of later doing a I, then we can get both of the conditionals in the conclusion. Like this: 2 (2) P Ass. ( I) 1, 2 (3) Q R 1, 2, E 1, 2 (4) Q 3, E 1, 2 (5) R 3, E 1 (6) P Q 2, 4, I 1 (7) P R 2, 5, I 1 (8) (P Q) (P R) 6, 7, I Neither (6) nor (7) relies on our assumption P, since they packed P into a conditional. So, the assumption P has been discharged. Then, we simply combine the two conditionals in the last line using I to get the desired conclusion. (c) Here s another one: If Peter and Quentin both bring presents, then Rachel will have a great birthday. Therefore, if Peter brings presents, then, if Quentin brings presents, then Rachel will have a great birthday. Or: S20: (P Q) R Ⱶ P (Q R) Let s begin: 1 (1) (P Q) R A 1 (n) P (Q R)? We ll have to assume TWO sentences, P AND Q, and discharge them one at a time: 1 (1) (P Q) R A 2 (2) P Ass. ( I) #1 * 3 (3) Q Ass. ( I) #2 2, 3 (4) P Q 2, 3, I 1, 2, 3 (5) R 1, 4, E 1, 2 (6) Q R 3, 5, I (#2) ** 1 (7) P (Q R) 2, 6, I (#1) 2 * Note: To help keep track of my assumptions, I like to number them whenever there are several assumptions of the same type. ** (and I also like to note which assumption was discharged as I discharge them)
3 Here, we ve combined the two assumptions into a conjunction in order to get the antecedent of the conditional in (1). Then, once we obtained R, we discharge the assumption Q first by deriving the conditional Q R using I. Finally, we discharge the assumption P by deriving the conditional in the conclusion using another I. 2. I Derivations: Now, let s do one that involves introducing bi-conditionals. S36: P Q, Q R Ⱶ P R For instance, someone might say, Paul will go to the party if and only if Quinn goes to the party. Also, Quinn will go to the party if and only if Rebecca invites him. Therefore, Paul will go to the party if and only if Rebecca invites Quinn. We ll start: 2 (2) Q R A 1, 2 (n) P R? Let s start by breaking up the two bi-conditionals, and the breaking up the conjunctions that result (that is, let s perform E and then E ). 2 (2) Q R A 1 (3) (P Q) (Q P) 1, E 1 (4) P Q 3, E 1 (5) Q P 3, E 2 (6) (Q R) (R Q) 2, E 2 (7) Q R 6, E 2 (8) R Q 6, E 1, 2 (n) P R? Now for the tricky part. To get to the conclusion, P R, we re going to need to derive the conjunction, (P R) (R P). So, we re going to need to derive each of those two conditionals separately. We derive each one by assuming their antecedents. In other words, we ll assume P to obtain P R, and we ll assume R to get R P. Like this: 3
4 2 (2) Q R A 1 (3) (P Q) (Q P) 1, E 1 (4) P Q 3, E 1 (5) Q P 3, E 2 (6) (Q R) (R Q) 2, E 2 (7) Q R 6, E 2 (8) R Q 6, E 9 (9) P Ass. ( I) #1 1, 9 (10) Q 4, 9, E 1, 2, 9 (11) R 7, 11, E 1, 2 (12) P R 9, 11, I (#1) 13 (13) R Ass. ( I) #2 2, 13 (14) Q 8, 13, E 1, 2, 13 (15) P 5, 15, E 1, 2 (16) R P 13, 15, I (#2) 1, 2 (17) (P R) (R P) 12, 16, I 1, 2 (18) P R 17, I Whew! Take a look. We introduced P in order to get P R by using I. THEN we introduced R in order to get R P by using I again. THEN we combined the two conditionals into a conjunction of conditionals so that we could introduce the. 3. E Derivations: On to disjunction-elimination. Either Paul won t fail to show up, or Quinn won t fail to show up. Therefore, either Paul or Quinn will show up. Here s the sequent: S42: P Q Ⱶ P Q So, we have: 1 (1) P Q A 1 (n) P Q? This one will use disjunction-elimination. Remember that, in order to eliminate a disjunction, you ll need to show that, if the first disjunct WERE true, then the conclusion would follow, and also that, if the SECOND disjunct WERE true, then the conclusion would ALSO follow. So, let s assume each of the two disjuncts one at a time: 4
5 1 (1) P Q A 2 (2) P Ass. ( I) 2 (3) P 2, E 2 (4) P Q 3, I - (5) P (P Q) 2, 4, I 1 (n) P Q? Here, we ve assumed the first disjunct of (1), P, and shown by I and I that, if true, P would entail the disjunction P Q. There are no numbers to the left of (5) because we DISCHARGED the assumption that it relies on. Let s do the same for the other disjunct now: 1 (1) P Q A 2 (2) P Ass. ( I) #1 2 (3) P 2, E 2 (4) P Q 3, I - (5) P (P Q) 2, 4, I (#1) 6 (6) Q Ass. ( I) #2 6 (7) Q 6, E 6 (8) P Q 7, I - (9) Q (P Q) 6, 8, I (#2) 1 (10) P Q 1, 5, 9, E Note how we have performed exactly the same operations for the first disjunct and the second disjunct of line (1) in order to perform E. Since we know from line (5) that P entails P Q, and also from line (9) that Q ALSO entails P Q, and we ALSO know that either P or Q is true from our premise in line (1), we can infer on the final line that P Q must be true. Note that this is a little confusing because we didn t really eliminate the at the end. What the disjunction-elimination rule says is that if both disjuncts lead to SOMETHING (Δ), then we can just infer that that something (Δ) must be true. But, Δ might itself be a disjunction as is the case here in which case the symbol remains. Here s another: Peggy is bringing wine, and either Quinn or Rob is bringing beer. Therefore, either Peggy is bringing wine and Quinn is bringing beer, or Peggy is bringing wine and Rob is bringing beer. Here s the sequent to be proved: S43: P (Q R) Ⱶ (P Q) (P R) Or, in derivation form: 5
6 1 (n) (P Q) (P R)? Let s start by breaking down the conjunction in (1). 1 (2) P 1, E 1 (3) Q R 1, E 1 (n) (P Q) (P R)? Now what? Well, we have a disjunction on line (3), and what we need to do is show that, EITHER WAY, the conclusion (n) would be true. That is, whether Q is true, or R is true, it will follow that (P Q) (P R) is true. So, we re going to need to introduce some conditionals with I. Here s how: 1 (2) P 1, E 1 (3) Q R 1, E 4 (4) Q Ass. ( I) 1, 4 (5) P Q 2, 4, I 1 (n) (P Q) (P R)? So far, we ve shown that, if Q were true, then we could derive P Q. But, what we really want to get is the entire conclusion. How do we get the second disjunct ( P R )? That s easy, we can just tack it on using I. Remember, if we have ONE disjunct, we can add WHATEVER WE D LIKE as a second disjunct. (Recall that, I m either going to the bars tonight OR I m staying home to watch paint dry is still true, even if you know for sure that you re going to the bars.) Like this: 1 (2) P 1, E 1 (3) Q R 1, E 4 (4) Q Ass. ( I) 1, 4 (5) P Q 2, 4, I 1, 4 (6) (P Q) (P R) 5, I 1 (7) Q [(P Q) (P R)] 4, 6, I 6
7 1 (n) (P Q) (P R)? So, on line (7) we were able to discharge our assumption Q using I. Let s do the same for the other disjunct of line (3), R : 1 (2) P 1, E 1 (3) Q R 1, E 4 (4) Q Ass. ( I) #1 1, 4 (5) P Q 2, 4, I 1, 4 (6) (P Q) (P R) 5, I 1 (7) Q [(P Q) (P R)] 4, 6, I (#1) 8 (8) R Ass. ( I) #2 1, 8 (9) P R 2, 8, I 1, 8 (10) (P Q) (P R) 9, I 1 (11) R [(P Q) (P R)] 8, 10, I (#2) 1 (12) (P Q) (P R) 3, 7, 11, E Note how we have performed exactly the same operations for the first disjunct and the second disjunct of line (3) in order to perform E. We can infer the conclusion on line (12) because we have shown that, no matter which disjunct of the disjunction in line (3) is true, they both lead to the conclusion on line (12). 4. I Derivations: And finally, negation-introduction. Let s start with an easy one: Peggy will pass. So, it is not the case that Peggy will not pass. The sequent is: S61: P Ⱶ P Here s the derivation: 1 (1) P A 2 (2) P Ass. (Red.) 1, 2 (3) P P 1, 2, I 1 (4) P 2, 3, I In line (2), we introduced P for the sake of performing a reductio. As we can see, lines (1) and (2) combined form a syntactic contradiction in line (3). So, we can conclude that the negation of our assumption is true, using I in line (4). They get harder. Try this one: It is not the case that both Peggy and Quinn will fail. Therefore, either Peggy or Quinn will pass. The sequent is as follows: 7
8 S62: ( P Q) Ⱶ P Q In derivation form: 1 (1) ( P Q) A 1 (n) P Q? How do we begin? It doesn t look like we can do much to (1), so perhaps the best strategy is to assume the NEGATION of the conclusion and try to derive a contradiction. If we can do that, then we ll have proved the conclusion. Let s begin: 1 (1) ( P Q) A 2 (2) (P Q) Ass. (Red.)? (n-2) ( P Q) ( P Q)?? (n-1) (P Q)? 1 (n) P Q? But, how do we derive the contradiction? In other words, how do we get the first conjunct in line (n-2) that contradicts the second conjunct which is the premise from line (1)? Well, as it turns out, we re going to have to make TWO more assumptions for TWO more reductios. Let s assume P AND Q. 1 (1) ( P Q) A 2 (2) (P Q) Ass. (Red.) #1 3 (3) P Ass. (Red.) #2 3 (4) P Q 3, I 2, 3 (5) (P Q) (P Q) 2, 4, I 2 (6) P 3, 5, I 7 (7) Q Ass. (Red.) #3 7 (8) P Q 7, I 2, 7 (9) (P Q) (P Q) 2, 8, I 2 (10) Q 7, 9, I 2 (11) P Q 6, 10, I 1, 2 (12) ( P Q) ( P Q) 1, 11, I 1 (13) (P Q) 2, 12, I 1 (14) P Q 13, E The red lines denote our primary reductio proof, while the green and blue lines denote our secondary reductio proofs (to prove P and Q, respectively). 8
9 Here s the same sequent in the reverse order: S63: P Q Ⱶ ( P Q) 1 (n) ( P Q)? Since we re trying to get from a disjunction to a conclusion, it is natural to suspect that we will try to get from P (the first disjunct of the premise) to the conclusion, and also from Q (the second disjunct of the premise) to the conclusion. But, assuming P and assuming Q clearly won t be enough. We re going to ALSO have to assume the opposite of the conclusion for the purposes of a reductio TWICE. Here it is for P : 2 (2) P Ass. ( I) 3 (3) P Q Ass. (Red) 3 (4) P 3, E 2, 3 (5) P P 2, 4, I 2 (6) ( P Q) 3, 5, I - (7) P [ ( P Q)] 2, 6, I 1 (n) ( P Q)? Here, we ve assumed the first disjunct of (1) in order to show that it entails the conclusion. But, to do that, we had to assume the OPPOSITE of the conclusion in order to derive a contradiction. Follow these same steps for the second disjunct, like this: 2 (2) P Ass. ( I) 3 (3) P Q Ass. (Red) 3 (4) P 3, E 2, 3 (5) P P 2, 4, I 2 (6) ( P Q) 3, 5, I - (7) P [ ( P Q)] 2, 6, I 8 (8) Q Ass. ( I) 9 (9) P Q Ass. (Red) * 9 (10) Q 9, E 8, 9 (11) Q Q 8, 10, I 8 (12) ( P Q) 9, 11, I - (13) Q [ ( P Q)] 8, 12, I 1 (14) ( P Q) 1, 7, 13, E * Note that you do not need to assume this AGAIN. You could in fact use the assumption from line (3). If you do, this derivation will be one line shorter. (I chose to assume it again here for educational purposes, since this is slightly less confusing.) 9
10 Last one: S64: P Q Ⱶ Q P Derivation: 1 (n) Q P? Well, the obvious thing to try first is to break down (1) into two separate conditionals. And keep in mind that what we probably WANT is a conjunction of two conditionals that will give us the conclusion: 1 (2) (P Q) (Q P) 1, E 1 (3) P Q 2, E 1 (4) Q P 2, E 1 (n-1) ( Q P) ( P Q)? 1 (n) Q P? Now, if we can somehow show that Q entails P, then we ll be able to get the first conjunct of line (n-1); and vice versa for the second conjunct. To do that, we ll have to assume Q and hope that we can get it to entail P but to do THAT, we ll have to assume P for the purpose of a reductio. Like this: 1 (2) (P Q) (Q P) 1, E 1 (3) P Q 2, E 1 (4) Q P 2, E 5 (5) Q Ass. ( I) 5 (6) Q 5, E 1, 5 (7) P 4, 6, E 8 (8) P Ass. (Red) 1, 5, 8 (9) P P 7, 8, I 1, 5 (10) P 8, 9, I 1 (11) Q P 5, 10, I 1 (n-1) ( Q P) ( P Q)? 1 (n) Q P? 10
11 Now, just to the same thing for P. Like this: 1 (2) (P Q) (Q P) 1, E 1 (3) P Q 2, E 1 (4) Q P 2, E 5 (5) Q Ass. ( I) 5 (6) Q 5, E 1, 5 (7) P 4, 6, E 8 (8) P Ass. (Red) 1, 5, 8 (9) P P 7, 8, I 1, 5 (10) P 8, 9, I 1 (11) Q P 5, 10, I 12 (12) P Ass. ( I) 12 (13) P 12, E 1, 12 (14) Q 3, 13, E 15 (15) Q Ass. (Red) 1, 12, 15 (16) Q Q 14, 14, I 1, 12 (17) Q 15, 16, I 1 (18) P Q 12, 17, I 1 (19) ( Q P) ( P Q) 11, 18, I 1 (20) Q P 19, I 11
Single-Predicate Derivations
Single-Predicate Derivations Let s do some derivations. Start with an easy one: Practice #1: Fb, Gb Ⱶ (ꓱx)(Fx Gx) Imagine that I have a frog named Bob. The above inference might go like this: Bob is friendly.
More informationTruth Table Definitions of Logical Connectives
Truth Table Definitions of Logical Connectives 1. Truth Functions: Logicians DEFINE logical operators in terms of their relation to the truth or falsehood of the statement(s) that they are operating on.
More informationCHAPTER 6 - THINKING ABOUT AND PRACTICING PROPOSITIONAL LOGIC
1 CHAPTER 6 - THINKING ABOUT AND PRACTICING PROPOSITIONAL LOGIC Here, you ll learn: what it means for a logic system to be finished some strategies for constructing proofs Congratulations! Our system of
More informationTruth Tables for Propositions
Truth Tables for Propositions 1. Truth Tables for 2-Letter Compound Statements: We have learned about truth tables for simple statements. For instance, the truth table for A B is the following: Conditional
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationIntermediate Logic. Natural Deduction for TFL
Intermediate Logic Lecture Two Natural Deduction for TFL Rob Trueman rob.trueman@york.ac.uk University of York The Trouble with Truth Tables Natural Deduction for TFL The Trouble with Truth Tables The
More informationProof Worksheet 2, Math 187 Fall 2017 (with solutions)
Proof Worksheet 2, Math 187 Fall 2017 (with solutions) Dr. Holmes October 17, 2017 The instructions are the same as on the first worksheet, except you can use all the rules in the strategies handout. We
More informationArguments and Proofs. 1. A set of sentences (the premises) 2. A sentence (the conclusion)
Arguments and Proofs For the next section of this course, we will study PROOFS. A proof can be thought of as the formal representation of a process of reasoning. Proofs are comparable to arguments, since
More informationNatural deduction for truth-functional logic
Natural deduction for truth-functional logic Phil 160 - Boston University Why natural deduction? After all, we just found this nice method of truth-tables, which can be used to determine the validity or
More informationCS 3110: Proof Strategy and Examples. 1 Propositional Logic Proof Strategy. 2 A Proof Walkthrough
CS 3110: Proof Strategy and Examples 1 Propositional Logic Proof Strategy The fundamental thing you have to do is figure out where each connective is going to come from. Sometimes the answer is very simple;
More informationMathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras. Lecture - 15 Propositional Calculus (PC)
Mathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras Lecture - 15 Propositional Calculus (PC) So, now if you look back, you can see that there are three
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationFormal Logic. Critical Thinking
ormal Logic Critical hinking Recap: ormal Logic If I win the lottery, then I am poor. I win the lottery. Hence, I am poor. his argument has the following abstract structure or form: If P then Q. P. Hence,
More information4 Derivations in the Propositional Calculus
4 Derivations in the Propositional Calculus 1. Arguments Expressed in the Propositional Calculus We have seen that we can symbolize a wide variety of statement forms using formulas of the propositional
More informationA Quick Lesson on Negation
A Quick Lesson on Negation Several of the argument forms we have looked at (modus tollens and disjunctive syllogism, for valid forms; denying the antecedent for invalid) involve a type of statement which
More informationWarm-Up Problem. Write a Resolution Proof for. Res 1/32
Warm-Up Problem Write a Resolution Proof for Res 1/32 A second Rule Sometimes throughout we need to also make simplifications: You can do this in line without explicitly mentioning it (just pretend you
More information(1) If Bush had not won the last election, then Nader would have won it.
24.221 Metaphysics Counterfactuals When the truth functional material conditional (or ) is introduced, it is normally glossed with the English expression If..., then.... However, if this is the correct
More informationManual of Logical Style (fresh version 2018)
Manual of Logical Style (fresh version 2018) Randall Holmes 9/5/2018 1 Introduction This is a fresh version of a document I have been working on with my classes at various levels for years. The idea that
More informationChapter 2. Mathematical Reasoning. 2.1 Mathematical Models
Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................
More informationSupplementary Logic Notes CSE 321 Winter 2009
1 Propositional Logic Supplementary Logic Notes CSE 321 Winter 2009 1.1 More efficient truth table methods The method of using truth tables to prove facts about propositional formulas can be a very tedious
More informationPHIL012. SYMBOLIC LOGIC PROPOSITIONAL LOGIC DERIVATIONS
HIL012 SYMBOLIC LOGIC ROOSITIONL LOGIC DERIVTIONS When we argue, what we want are (i) clearly specifiable rules, (ii) that apply to any particular subject matter, and (iii) that legitimate transitions
More informationFor a horseshoe statement, having the matching p (left side) gives you the q (right side) by itself. It does NOT work with matching q s.
7.1 The start of Proofs From now on the arguments we are working with are all VALID. There are 18 Rules of Inference (see the last 2 pages in Course Packet, or front of txt book). Each of these rules is
More informationDISCRETE MATHEMATICS BA202
TOPIC 1 BASIC LOGIC This topic deals with propositional logic, logical connectives and truth tables and validity. Predicate logic, universal and existential quantification are discussed 1.1 PROPOSITION
More informationManual of Logical Style
Manual of Logical Style Dr. Holmes January 9, 2015 Contents 1 Introduction 2 2 Conjunction 3 2.1 Proving a conjunction...................... 3 2.2 Using a conjunction........................ 3 3 Implication
More informationSolution to Proof Questions from September 1st
Solution to Proof Questions from September 1st Olena Bormashenko September 4, 2011 What is a proof? A proof is an airtight logical argument that proves a certain statement in general. In a sense, it s
More informationNatural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
More informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.
More informationIntroducing Proof 1. hsn.uk.net. Contents
Contents 1 1 Introduction 1 What is proof? 1 Statements, Definitions and Euler Diagrams 1 Statements 1 Definitions Our first proof Euler diagrams 4 3 Logical Connectives 5 Negation 6 Conjunction 7 Disjunction
More informationAnnouncements For Methods of Proof for Boolean Logic Proof by Contradiction. Outline. The Big Picture Where is Today? William Starr
Announcements For 09.22 Methods of for Boolean Logic William Starr 1 HW1 grades will be on Bb by end of week 2 HW4 is due on Tuesday This one is mostly written Feel free to type it out! 3 If you have problems
More informationcis32-ai lecture # 18 mon-3-apr-2006
cis32-ai lecture # 18 mon-3-apr-2006 today s topics: propositional logic cis32-spring2006-sklar-lec18 1 Introduction Weak (search-based) problem-solving does not scale to real problems. To succeed, problem
More informationPropositional Logic: Part II - Syntax & Proofs 0-0
Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems
More informationDeduction by Daniel Bonevac. Chapter 3 Truth Trees
Deduction by Daniel Bonevac Chapter 3 Truth Trees Truth trees Truth trees provide an alternate decision procedure for assessing validity, logical equivalence, satisfiability and other logical properties
More informationLogic for Computer Science - Week 4 Natural Deduction
Logic for Computer Science - Week 4 Natural Deduction 1 Introduction In the previous lecture we have discussed some important notions about the semantics of propositional logic. 1. the truth value of a
More informationThe Natural Deduction Pack
The Natural Deduction Pack Alastair Carr March 2018 Contents 1 Using this pack 2 2 Summary of rules 3 3 Worked examples 5 31 Implication 5 32 Universal quantifier 6 33 Existential quantifier 8 4 Practice
More informationFirst-Degree Entailment
March 5, 2013 Relevance Logics Relevance logics are non-classical logics that try to avoid the paradoxes of material and strict implication: p (q p) p (p q) (p q) (q r) (p p) q p (q q) p (q q) Counterintuitive?
More informationIn this chapter, we specify a deductive apparatus for PL.
Handout 5 PL Derivations In this chapter, we specify a deductive apparatus for PL Definition deductive apparatus A deductive apparatus for PL is a set of rules of inference (or derivation rules) that determine
More informationTruth Tables for Arguments
ruth ables for Arguments 1. Comparing Statements: We ve looked at SINGLE propositions and assessed the truth values listed under their main operators to determine whether they were tautologous, self-contradictory,
More information1.1 Statements and Compound Statements
Chapter 1 Propositional Logic 1.1 Statements and Compound Statements A statement or proposition is an assertion which is either true or false, though you may not know which. That is, a statement is something
More informationPHIL12A Section answers, 16 February 2011
PHIL12A Section answers, 16 February 2011 Julian Jonker 1 How much do you know? 1. Show that the following sentences are equivalent. (a) (Ex 4.16) A B A and A B A B (A B) A A B T T T T T T T T T T T F
More informationSection 2.6 Solving Linear Inequalities
Section 2.6 Solving Linear Inequalities INTRODUCTION Solving an inequality is much like solving an equation; there are, though, some special circumstances of which you need to be aware. In solving an inequality
More information3 The Semantics of the Propositional Calculus
3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical
More information15414/614 Optional Lecture 1: Propositional Logic
15414/614 Optional Lecture 1: Propositional Logic Qinsi Wang Logic is the study of information encoded in the form of logical sentences. We use the language of Logic to state observations, to define concepts,
More informationPreptests 59 Answers and Explanations (By Ivy Global) Section 1 Analytical Reasoning
Preptests 59 Answers and Explanations (By ) Section 1 Analytical Reasoning Questions 1 5 Since L occupies its own floor, the remaining two must have H in the upper and I in the lower. P and T also need
More informationOverview of Today s Lecture
Branden Fitelson Philosophy 4515 (Advanced Logic) Notes 1 Overview of Today s Lecture Administrative Stuff HW #1 grades and solutions have been posted Please make sure to work through the solutions HW
More informationThe statement calculus and logic
Chapter 2 Contrariwise, continued Tweedledee, if it was so, it might be; and if it were so, it would be; but as it isn t, it ain t. That s logic. Lewis Carroll You will have encountered several languages
More informationDEDUCTIVE REASONING Propositional Logic
7 DEDUCTIVE REASONING Propositional Logic Chapter Objectives Connectives and Truth Values You will be able to understand the purpose and uses of propositional logic. understand the meaning, symbols, and
More informationCHAPTER 5 - DISJUNCTIONS
1 CHAPTER 5 - DISJUNCTIONS Here, you ll learn: How to understand complex sentences by symbolizing disjunctions applying truth conditions for disjunctions How to assess arguments for validity / invalidity
More informationChapter 1: Formal Logic
Chapter 1: Formal Logic Dr. Fang (Daisy) Tang ftang@cpp.edu www.cpp.edu/~ftang/ CS 130 Discrete Structures Logic: The Foundation of Reasoning Definition: the foundation for the organized, careful method
More informationPL: Truth Trees. Handout Truth Trees: The Setup
Handout 4 PL: Truth Trees Truth tables provide a mechanical method for determining whether a proposition, set of propositions, or argument has a particular logical property. For example, we can show that
More informationSection 4.6 Negative Exponents
Section 4.6 Negative Exponents INTRODUCTION In order to understand negative exponents the main topic of this section we need to make sure we understand the meaning of the reciprocal of a number. Reciprocals
More informationLecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
More informationCHAPTER 1 - LOGIC OF COMPOUND STATEMENTS
CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS 1.1 - Logical Form and Logical Equivalence Definition. A statement or proposition is a sentence that is either true or false, but not both. ex. 1 + 2 = 3 IS a statement
More informationa. ~p : if p is T, then ~p is F, and vice versa
Lecture 10: Propositional Logic II Philosophy 130 3 & 8 November 2016 O Rourke & Gibson I. Administrative A. Group papers back to you on November 3. B. Questions? II. The Meaning of the Conditional III.
More informationWarm-Up Problem. Is the following true or false? 1/35
Warm-Up Problem Is the following true or false? 1/35 Propositional Logic: Resolution Carmen Bruni Lecture 6 Based on work by J Buss, A Gao, L Kari, A Lubiw, B Bonakdarpour, D Maftuleac, C Roberts, R Trefler,
More informationPROPOSITIONAL CALCULUS
PROPOSITIONAL CALCULUS A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both. These are not propositions! Connectives and
More information(p == train arrives late) (q == there are taxis) (r == If p and not q, then r. Not r. p. Therefore, q. Propositional Logic
Propositional Logic The aim of logic in computer science is to develop languages to model the situations we encounter as computer science professionals Want to do that in such a way that we can reason
More informationCHAPTER 1 LINEAR EQUATIONS
CHAPTER 1 LINEAR EQUATIONS Sec 1. Solving Linear Equations Kids began solving simple equations when they worked missing addends problems in first and second grades. They were given problems such as 4 +
More informationPartial Fractions. June 27, In this section, we will learn to integrate another class of functions: the rational functions.
Partial Fractions June 7, 04 In this section, we will learn to integrate another class of functions: the rational functions. Definition. A rational function is a fraction of two polynomials. For example,
More informationAlgebra & Trig Review
Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The
More informationAxiomatic systems. Revisiting the rules of inference. Example: A theorem and its proof in an abstract axiomatic system:
Axiomatic systems Revisiting the rules of inference Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 2.1,
More informationOne sided tests. An example of a two sided alternative is what we ve been using for our two sample tests:
One sided tests So far all of our tests have been two sided. While this may be a bit easier to understand, this is often not the best way to do a hypothesis test. One simple thing that we can do to get
More informationNotes on Propositional and First-Order Logic (CPSC 229 Class Notes, January )
Notes on Propositional and First-Order Logic (CPSC 229 Class Notes, January 23 30 2017) John Lasseter Revised February 14, 2017 The following notes are a record of the class sessions we ve devoted to the
More informationFinal Exam Theory Quiz Answer Page
Philosophy 120 Introduction to Logic Final Exam Theory Quiz Answer Page 1. (a) is a wff (and a sentence); its outer parentheses have been omitted, which is permissible. (b) is also a wff; the variable
More informationIdentity. We often use the word identical to simply mean looks the same. For instance:
Identity 1. Introduction: In this lesson, we are going to learn how to express statements about identity; or, namely, we re going to learn how to say that two things are identical. We often use the word
More informationLogical Representations. LING 7800/CSCI 7000 Martha Palmer 9/30/2014
Logical Representations LING 7800/CSCI 7000 Martha Palmer 9/30/2014 1 The Semantic Wall Physical Symbol System World +BLOCKA+ +BLOCKB+ +BLOCKC+ P 1 :(IS_ON +BLOCKA+ +BLOCKB+) P 2 :((IS_RED +BLOCKA+) Truth
More informationWriting proofs. Tim Hsu, San José State University. May 31, Definitions and theorems 3. 2 What is a proof? 3. 3 A word about definitions 4
Writing proofs Tim Hsu, San José State University May 31, 2006 Contents I Fundamentals 3 1 Definitions and theorems 3 2 What is a proof? 3 3 A word about definitions 4 II The structure of proofs 6 4 Assumptions
More information6. Logical Inference
Artificial Intelligence 6. Logical Inference Prof. Bojana Dalbelo Bašić Assoc. Prof. Jan Šnajder University of Zagreb Faculty of Electrical Engineering and Computing Academic Year 2016/2017 Creative Commons
More informationProof strategies, or, a manual of logical style
Proof strategies, or, a manual of logical style Dr Holmes September 27, 2017 This is yet another version of the manual of logical style I have been working on for many years This semester, instead of posting
More information(ÀB Ä (A Â C)) (A Ä ÀC) Á B. This is our sample argument. Formal Proofs
(ÀB Ä (A Â C)) (A Ä ÀC) Á B This is our sample argument. Formal Proofs From now on, formal proofs will be our main way to test arguments. We ll begin with easier proofs. Our initial strategy for constructing
More informationNatural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms)
More informationLECTURE 15: SIMPLE LINEAR REGRESSION I
David Youngberg BSAD 20 Montgomery College LECTURE 5: SIMPLE LINEAR REGRESSION I I. From Correlation to Regression a. Recall last class when we discussed two basic types of correlation (positive and negative).
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationLogarithms and Exponentials
Logarithms and Exponentials Steven Kaplan Department of Physics and Astronomy, Rutgers University The Basic Idea log b x =? Whoa...that looks scary. What does that mean? I m glad you asked. Let s analyze
More informationA. Incorrect! This inequality is a disjunction and has a solution set shaded outside the boundary points.
Problem Solving Drill 11: Absolute Value Inequalities Question No. 1 of 10 Question 1. Which inequality has the solution set shown in the graph? Question #01 (A) x + 6 > 1 (B) x + 6 < 1 (C) x + 6 1 (D)
More informationElla failed to drop the class. Ella dropped the class.
Propositional logic In many cases, a sentence is built up from one or more simpler sentences. To see what follows from such a complicated sentence, it is helpful to distinguish the simpler sentences from
More informationLogical Agents. September 14, 2004
Logical Agents September 14, 2004 The aim of AI is to develop intelligent agents that can reason about actions and their effects and about the environment, create plans to achieve a goal, execute the plans,
More informationPropositional Logic Review
Propositional Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane The task of describing a logical system comes in three parts: Grammar Describing what counts as a formula Semantics Defining
More informationKrivine s Intuitionistic Proof of Classical Completeness (for countable languages)
Krivine s Intuitionistic Proof of Classical Completeness (for countable languages) Berardi Stefano Valentini Silvio Dip. Informatica Dip. Mat. Pura ed Applicata Univ. Torino Univ. Padova c.so Svizzera
More informationThe paradox of knowability, the knower, and the believer
The paradox of knowability, the knower, and the believer Last time, when discussing the surprise exam paradox, we discussed the possibility that some claims could be true, but not knowable by certain individuals
More informationMITOCW MITRES_18-007_Part5_lec3_300k.mp4
MITOCW MITRES_18-007_Part5_lec3_300k.mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources
More information2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
More information10/5/2012. Logic? What is logic? Propositional Logic. Propositional Logic (Rosen, Chapter ) Logic is a truth-preserving system of inference
Logic? Propositional Logic (Rosen, Chapter 1.1 1.3) TOPICS Propositional Logic Truth Tables Implication Logical Proofs 10/1/12 CS160 Fall Semester 2012 2 What is logic? Logic is a truth-preserving system
More informationCONSTRUCTION OF sequence of rational approximations to sets of rational approximating sequences, all with the same tail behaviour Definition 1.
CONSTRUCTION OF R 1. MOTIVATION We are used to thinking of real numbers as successive approximations. For example, we write π = 3.14159... to mean that π is a real number which, accurate to 5 decimal places,
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationIt is not the case that ϕ. p = It is not the case that it is snowing = It is not. r = It is not the case that Mary will go to the party =
Introduction to Propositional Logic Propositional Logic (PL) is a logical system that is built around the two values TRUE and FALSE, called the TRUTH VALUES. true = 1; false = 0 1. Syntax of Propositional
More informationTo factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x
Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression
More informationCSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
More informationThe Conjunction and Disjunction Theses
The Conjunction and Disjunction Theses Abstract Rodriguez-Pereyra (2006) argues for the disjunction thesis but against the conjunction thesis. I argue that accepting the disjunction thesis undermines his
More informationMAT2345 Discrete Math
Fall 2013 General Syllabus Schedule (note exam dates) Homework, Worksheets, Quizzes, and possibly Programs & Reports Academic Integrity Do Your Own Work Course Web Site: www.eiu.edu/~mathcs Course Overview
More information2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic
CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares
More informationHonors Advanced Mathematics Determinants page 1
Determinants page 1 Determinants For every square matrix A, there is a number called the determinant of the matrix, denoted as det(a) or A. Sometimes the bars are written just around the numbers of the
More informationLogic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
More informationConnectives Name Symbol OR Disjunction And Conjunction If then Implication/ conditional If and only if Bi-implication / biconditional
Class XI Mathematics Ch. 14 Mathematical Reasoning 1. Statement: A sentence which is either TRUE or FALSE but not both is known as a statement. eg. i) 2 + 2 = 4 ( it is a statement which is true) ii) 2
More informationBig-oh stuff. You should know this definition by heart and be able to give it,
Big-oh stuff Definition. if asked. You should know this definition by heart and be able to give it, Let f and g both be functions from R + to R +. Then f is O(g) (pronounced big-oh ) if and only if there
More informationHANDOUT AND SET THEORY. Ariyadi Wijaya
HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics
More informationDiscrete Mathematics for CS Fall 2003 Wagner Lecture 3. Strong induction
CS 70 Discrete Mathematics for CS Fall 2003 Wagner Lecture 3 This lecture covers further variants of induction, including strong induction and the closely related wellordering axiom. We then apply these
More informationMA103 STATEMENTS, PROOF, LOGIC
MA103 STATEMENTS, PROOF, LOGIC Abstract Mathematics is about making precise mathematical statements and establishing, by proof or disproof, whether these statements are true or false. We start by looking
More informationMethods of Mathematics
Methods of Mathematics Kenneth A. Ribet UC Berkeley Math 10B April 19, 2016 There is a new version of the online textbook file Matrix_Algebra.pdf. The next breakfast will be two days from today, April
More informationDescription Logics. Deduction in Propositional Logic. franconi. Enrico Franconi
(1/20) Description Logics Deduction in Propositional Logic Enrico Franconi franconi@cs.man.ac.uk http://www.cs.man.ac.uk/ franconi Department of Computer Science, University of Manchester (2/20) Decision
More information