DOCSLIB.ORG

Set Theory & Predicate Calculus Applied Discrete Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Set Theory & Predicate Calculus Applied Discrete Mathematics CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus Information Pages 2-9 First Hour Exam Study Sheet Page 10 Some First Hour Exam review material Pages 11-16 Set Theory Information Pages 17-26 Arithmetic Proofs Information Page 27 CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Design of Full Adder A full binary adder has 3 inputs 1. bit A (X) 2. bit B (Y) 3. carry in (C) and 2 outputs 1. bit out (S) 2. carry to next digit (Cout ) Start by making truth table assuming all inputs and specifying all outputs. Inputs Outputs X Y C S Cout 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 then write "and" condition for all "1" outputs then "or" all "and" conditions. S = X'Y'C + X'YC' + XY'C' + XYC Cout = X'YC + XY'C + XYC' + XYC then simplify. 1 Predicate Calculus Outline I. Predicates - binding of variables to form propositions. II. General Quantifiers A. ∀ - Universal - means "for all" B. ∃ - Existential - means "there exists" C. Combinations of two variables using quantifiers D. Distribution of negation using De Morgan's rules III. Compound propositions of a single variable and proving set theory problems (next topic that will be covered) 2 Predicate - function which becomes a proposition when the variables are assigned values from an appropriate Universe of Discourse (U). QUANTIFIERS P(x): x2 > x (not a proposition) But these are: - For all x in ℜ, x2 > x - For some x in ℜ, x2 > x - There exists x in ℜ for which x2 > x Predicates 1. P(x): x2 > x P(x) is not a proposition Choose value for x from ℜ. P(5): 52 > 5 pin variable at 5 P(5) is a proposition 2. P(x): x ⊕ x = x P(x) is not a proposition Choose value for x from P({0,1}). P(0): 0 ⊕ 0 = 0 pin variable at 0 P(0) is a proposition 3. Q(x,y): (x < y) ∧ (x2 < y2 ) Q(x,y) is not a proposition Choose x,y from ℜ. Q(-3,2): (-3 < 2) ∧ ((-3)2 < 22 ) Q(-3,2) is a proposition From predicates (P(x), Q(x)) one can form compound predicates: ¬P(x) (P(x) → Q(x)) (P(x) ∨ Q(x)) (P(x) ↔ Q(x)) (P(x) ∧ Q(x)) ∃x P(x) ∀x Q(x) Example: [ ∃x (P(x) ∧ Q(x)) → ( ∃x P(x) ∧ ∃x Q(x))] 3 Propositional Logic Equivalences and Implications Still Hold Example: ( ∃x P(x) → ∀x Q(x)) ⇔ ( ¬(∃x P(x)) ∨ ∀x Q(x)) (implication equivalence) Example: ∀x (P(x) → Q(x)) ∧ ∃x P(x) ⇒ ∃x Q(x) (modus ponens implication) Also Have New Relationships: De Morgan's Rules in Predicate Calculus ¬(∀x P(x)) ⇔ ∃x (¬P(x)) ¬(∀x P(x)) P(x) ⇔ ¬P(x) ∃x (¬P(x)) 1 at least one 1 (at least 1 P(x) false one true) 0 all P(x) true 1 (all false) 0 ¬(∃x P(x)) ⇔ ∀x (¬P(x)) P(x) ¬(∃x P(x)) ⇔ ∀x (¬P(x)) ¬P(x) all false 1 1 1 (all true) at least 0 1 0 (at least one true one false) 4 Predicate Calculus Truth Table Example 1: ∀x P(x) ∧ ∀x Q(x) ⇔ ∀x [P(x) ∧ Q(x)] We have two propositions on the left side which form the compound proposition [∀x P(x) ∧ ∀x Q(x)]. We must consider all combinations of the truth values: ∀x P(x) ∀x Q(x) ∀x P(x) ∧ ∀x Q(x) ∀x [P(x) ∧ Q(x)] 0 0 0 Row 1 0 1 0 Row 2 1 0 0 Row 3 1 1 1 Row 4 Row 1: ∀x P(x) = 0 → P(x) has at least one 0 (false for some value of x) ∀x Q(x) = 0 → Q(x) has at least one 0 Therefore, P(x) ∧ Q(x) has at least one 0 Therefore, ∀x(P(x) ∧ Q(x)) = 0 Row 2: ∀x P(x) = 0 → P(x) has at least one 0 ∀x Q(x) = 1 → Q(x) = 1 for all x P(x) ∧ Q(x) = 0 since P(x) = 0 for some x Therefore, ∀x(P(x) ∧ Q(x)) = 0 Row 3: ∀x P(x) = 1 ∀x Q(x) = 0 Work same way and show Case 3 = 0 Row 4: ∀x P(x) = 1 ∀x Q(x) = 1 Work same way and show Case 4 = 1 ∀x P(x) ∧ ∀x Q(x) ⇔ ∀x [P(x) ∧ Q(x)] is the same on both sides of the truth table and therefore the statement is true. 5 Predicate Calculus Conversion Example: We can prove that: ∃x P(x) ∨ ∃x Q(x) ⇔ ∃x [P(x) ∨ Q(x)] from our previous statement without using truth tables: Start: ∀x P(x) ∧ ∀x Q(x) ⇔ ∀x [P(x) ∧ Q(x)] negate both sides ¬[∀x P(x) ∧ ∀x Q(x)] ⇔ ¬[∀x [P(x) ∧ Q(x)]] ∃x [¬P(x)] ∨ ∃x [¬Q(x)] ⇔ ∃x [¬P(x) ∨ ¬Q(x)] substitute ¬P(x) = A(x) ¬Q(x) = B(x) ∃x [A(x)] ∨ ∃x [B(x)] ⇔ ∃x [A(x) ∨ B(x)] Predicate Calculus Truth Table Example 2: Is ∀x P(x) ∨ ∀x Q(x) ⇔ ∀x [P(x) ∨ Q(x)] true? ∀x P(x) ∀x Q(x) ∀x P(x) ∨ ∀x Q(x) ∀x [P(x) ∨ Q(x)] 0 0 0 Row 1: Can be true for 0 1 1 some P(x) ∨ Q(x) and 1 0 1 therefore not logically 1 1 1 equivalent. (See example below.) Row 1: ∀x P(x) = 0 → P(x) has at least one 0 ∀x Q(x) = 0 → Q(x) has at least one 0 Does P(x) ∨ Q(x) always have at least one "0" for any P(x) and Q(x)? ∀x P(x) ∀x Q(x) P(x) Q(x) P(x) ∨ Q(x) ∀x [P(x) ∨ Q(x)] 0 0 0 1 1 1 0 0 1 0 1 1 6 Predicate Calculus Truth Table Example 3 (Independent Variables): ∀x P(x) ∨ ∃y Q(y) ⇔ ∀x ∃y [P(x) ∨ Q(y)] We have two propositions on the left side which form the compound proposition ∀x ∃y [P(x) ∨ Q(y)].We must consider all combinations of the truth values: ∀x P(x) ∃y Q(y) ∀x P(x) ∨ ∃y Q(y) ∀x ∃y [P(x) ∨ Q(y)] 0 0 0 Row 1 0 1 1 Row 2 1 0 1 Row 3 1 1 1 Row 4 Row 1: ∃y Q(y) = 0 → Q(y)=0 for all y values Therefore, sub in Q(y)=0 to get ∀x ∃y [P(x) ∨ 0 ] = ∀x ∃y P(x) = ∀x P(x) Therefore, since ∀x P(x) is false, then ∀x ∃y [P(x) ∨ Q(y)] = 0 Row 2: ∃y Q(y) = 1 → Q(c)=1 for some c Therefore, sub in Q(y)=1 to get ∀x ∃y [P(x) ∨ 1 ] = ∀x ∃y 1 = 1 Therefore, ∀x ∃y [P(x) ∨ Q(y)] = 1 Row 1: ∃y Q(y) = 0 → Q(y)=0 for all y values Therefore, sub in Q(y)=0 to get ∀x ∃y [P(x) ∨ 0 ] = ∀x ∃y P(x) = ∀x P(x) Therefore, since ∀x P(x) is true, then ∀x ∃y [P(x) ∨ Q(y)] = 1 Row 2: ∃y Q(y) = 1 → Q(c)=1 for some c Therefore, sub in Q(y)=1 to get ∀x ∃y [P(x) ∨ 1 ] = ∀x ∃y 1 = 1 Therefore, ∀x ∃y [P(x) ∨ Q(y)] = 1 ∀x P(x) ∨ ∃y Q(y) ⇔ ∀x ∃y [P(x) ∨ Q(y)] is the same on both sides of the truth table and therefore the statement is true. In this case, the variables in each predicate are independent, so the order of the quantifiers DOES NOT MATTER. 7 Example: Definition of Continuity A function of f is continuous on the real numbers if and only if ∀ε [(ε > 0) → ∃δ ((δ > 0) ∧ ∀x∀y [x - y < δ → f(x) - f(y) < ε])] Universe of Discourse = ℜ One can negate an expression. For example, consider the expression ∀x [P(x) → ∃y Q(y)], negating gives the following: ¬ ∀x [P(x) → ∃y Q(y)] ⇔ ∃x ¬ [P(x) → ∃y Q(y)] ⇔ ∃x ¬ [¬P(x) ∨ ∃y Q(y)] ⇔ ∃x [P(x) ∧ ¬(∃y Q(y))] ⇔ ∃x [P(x) ∧ ∀y ¬(Q(y))] **Recommended: Negate the definition of continuity: i.e., f is not continuous on ℜ ⇔ … 8 Given the following predicates, evaluate the propositions generated by using the indicated quantifiers with those predicates. P(x,y) x ∈ ℜ, y ∈ ℜ P(x,y) x ∈ {0,1}, y ∈ {0,1} Quantifiers x2 + y = y2 + 2x x → y ∀x∀y = ∀y∀x 0 (x = 1 and y = 2 is false) 0 (x = 1 and y = 0 is false) ∃x∃y = ∃y∃x 1 (x = 0 and y = 0 is true) 1 (x = 0 and y = 0 is true) 1 x y 0: (2x - x2 ) < , for x = 1 is imag 1: y = 1 ∀ ∃ 4 See ** below. ∃y∀x 0 1: y = 1 ∀y∃x 1: y - y2 < 1, is real for all y 1: x = 0 See * below. ∃x∀y 0 1: x = 0 *: ∀y∃x Case: x2 + y = y2 + 2x **: ∀x∃y Case: x2 + y = y2 + 2x x2 - 2x - (y2 - y) = 0 y2 - y - (x2 - 2x) = 0 2 ± 4 + 4(y2 - y) 1 ± 1 + 4(x2 - 2x) x = y = 2 2 (4 + 4(y2 - y)) > 0 (1 + 4(x2 - 2x)) > 0 4(y2 - y) > -4 4(x2 - 2x) > -1 1 y2 - y > -1 x2 - 2x > - 4 1 y - y2 < 1 2x - x2 < 4 is true for all values is false when x = 1 of y 9 First Hour Exam Study Sheet I.
Load more

Recommended publications
  • Classifying Material Implications Over Minimal Logic
    Classifying Material Implications over Minimal Logic Hannes Diener and Maarten McKubre-Jordens March 28, 2018 Abstract The so-called paradoxes of material implication have motivated the development of many non- classical logics over the years [2–5, 11]. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. Keywords: reverse mathematics; minimal logic; ex falso quodlibet; implication; paraconsistent logic; Peirce’s principle. 1 Introduction The project of constructive reverse mathematics [6] has given rise to a wide literature where various the- orems of mathematics and principles of logic have been classified over intuitionistic logic. What is less well-known is that the subtle difference that arises when the principle of explosion, ex falso quodlibet, is dropped from intuitionistic logic (thus giving (Johansson’s) minimal logic) enables the distinction of many more principles. The focus of the present paper are a range of principles known collectively (but not exhaustively) as the paradoxes of material implication; paradoxes because they illustrate that the usual interpretation of formal statements of the form “. → . .” as informal statements of the form “if. then. ” produces counter-intuitive results. Some of these principles were hinted at in [9]. Here we present a carefully worked-out chart, classifying a number of such principles over minimal logic.
    [Show full text]
  • On Basic Probability Logic Inequalities †
    mathematics Article On Basic Probability Logic Inequalities † Marija Boriˇci´cJoksimovi´c Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca154, 11000 Belgrade, Serbia; [email protected] † The conclusions given in this paper were partially presented at the European Summer Meetings of the Association for Symbolic Logic, Logic Colloquium 2012, held in Manchester on 12–18 July 2012. Abstract: We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A ! B, and B ! C have probabilities a, b, c, r, and s, respectively, then for probability p of A ! C, we have f (a, b, c, r, s) ≤ p ≤ g(a, b, c, r, s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner. Keywords: inequality; probability logic; inference rule MSC: 03B48; 03B05; 60E15; 26D20; 60A05 1. Introduction The main part of probabilization of logical inference rules is defining the correspond- Citation: Boriˇci´cJoksimovi´c,M. On ing best possible bounds for probabilities of propositions. Some of them, connected with Basic Probability Logic Inequalities. conjunction and disjunction, can be obtained immediately from the well-known Boole’s Mathematics 2021, 9, 1409.
    [Show full text]
  • 7.1 Rules of Implication I
    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) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification.
    [Show full text]
  • Tables of Implications and Tautologies from Symbolic Logic
    Tables of Implications and Tautologies from Symbolic Logic Dr. Robert B. Heckendorn Computer Science Department, University of Idaho March 17, 2021 Here are some tables of logical equivalents and implications that I have found useful over the years. Where there are classical names for things I have included them. \Isolation by Parts" is my own invention. By tautology I mean equivalent left and right hand side and by implication I mean the left hand expression implies the right hand. I use the tilde and overbar interchangeably to represent negation e.g. ∼x is the same as x. Enjoy! Table 1: Properties of All Two-bit Operators. The Comm. is short for commutative and Assoc. is short for associative. Iff is short for \if and only if". Truth Name Comm./ Binary And/Or/Not Nands Only Table Assoc. Op 0000 False CA 0 0 (a " (a " a)) " (a " (a " a)) 0001 And CA a ^ b a ^ b (a " b) " (a " b) 0010 Minus b − a a ^ b (b " (a " a)) " (a " (a " a)) 0011 B A b b b 0100 Minus a − b a ^ b (a " (a " a)) " (a " (a " b)) 0101 A A a a a 0110 Xor/NotEqual CA a ⊕ b (a ^ b) _ (a ^ b)(b " (a " a)) " (a " (a " b)) (a _ b) ^ (a _ b) 0111 Or CA a _ b a _ b (a " a) " (b " b) 1000 Nor C a # b a ^ b ((a " a) " (b " b)) " ((a " a) " a) 1001 Iff/Equal CA a $ b (a _ b) ^ (a _ b) ((a " a) " (b " b)) " (a " b) (a ^ b) _ (a ^ b) 1010 Not A a a a " a 1011 Imply a ! b a _ b (a " (a " b)) 1100 Not B b b b " b 1101 Imply b ! a a _ b (b " (a " a)) 1110 Nand C a " b a _ b a " b 1111 True CA 1 1 (a " a) " a 1 Table 2: Tautologies (Logical Identities) Commutative Property: p ^ q $ q
    [Show full text]
  • Propositional Logic (PDF)
    Mathematics for Computer Science Proving Validity 6.042J/18.062J Instead of truth tables, The Logic of can try to prove valid formulas symbolically using Propositions axioms and deduction rules Albert R Meyer February 14, 2014 propositional logic.1 Albert R Meyer February 14, 2014 propositional logic.2 Proving Validity Algebra for Equivalence The text describes a for example, bunch of algebraic rules to the distributive law prove that propositional P AND (Q OR R) ≡ formulas are equivalent (P AND Q) OR (P AND R) Albert R Meyer February 14, 2014 propositional logic.3 Albert R Meyer February 14, 2014 propositional logic.4 1 Algebra for Equivalence Algebra for Equivalence for example, The set of rules for ≡ in DeMorgan’s law the text are complete: ≡ NOT(P AND Q) ≡ if two formulas are , these rules can prove it. NOT(P) OR NOT(Q) Albert R Meyer February 14, 2014 propositional logic.5 Albert R Meyer February 14, 2014 propositional logic.6 A Proof System A Proof System Another approach is to Lukasiewicz’ proof system is a start with some valid particularly elegant example of this idea. formulas (axioms) and deduce more valid formulas using proof rules Albert R Meyer February 14, 2014 propositional logic.7 Albert R Meyer February 14, 2014 propositional logic.8 2 A Proof System Lukasiewicz’ Proof System Lukasiewicz’ proof system is a Axioms: particularly elegant example of 1) (¬P → P) → P this idea. It covers formulas 2) P → (¬P → Q) whose only logical operators are 3) (P → Q) → ((Q → R) → (P → R)) IMPLIES (→) and NOT. The only rule: modus ponens Albert R Meyer February 14, 2014 propositional logic.9 Albert R Meyer February 14, 2014 propositional logic.10 Lukasiewicz’ Proof System Lukasiewicz’ Proof System Prove formulas by starting with Prove formulas by starting with axioms and repeatedly applying axioms and repeatedly applying the inference rule.
    [Show full text]
  • Chapter 9: Answers and Comments Step 1 Exercises 1. Simplification. 2. Absorption. 3. See Textbook. 4. Modus Tollens. 5. Modus P
    Chapter 9: Answers and Comments Step 1 Exercises 1. Simplification. 2. Absorption. 3. See textbook. 4. Modus Tollens. 5. Modus Ponens. 6. Simplification. 7. X -- A very common student mistake; can't use Simplification unless the major con- nective of the premise is a conjunction. 8. Disjunctive Syllogism. 9. X -- Fallacy of Denying the Antecedent. 10. X 11. Constructive Dilemma. 12. See textbook. 13. Hypothetical Syllogism. 14. Hypothetical Syllogism. 15. Conjunction. 16. See textbook. 17. Addition. 18. Modus Ponens. 19. X -- Fallacy of Affirming the Consequent. 20. Disjunctive Syllogism. 21. X -- not HS, the (D v G) does not match (D v C). This is deliberate to make sure you don't just focus on generalities, and make sure the entire form fits. 22. Constructive Dilemma. 23. See textbook. 24. Simplification. 25. Modus Ponens. 26. Modus Tollens. 27. See textbook. 28. Disjunctive Syllogism. 29. Modus Ponens. 30. Disjunctive Syllogism. Step 2 Exercises #1 1 Z A 2. (Z v B) C / Z C 3. Z (1)Simp. 4. Z v B (3) Add. 5. C (2)(4)MP 6. Z C (3)(5) Conj. For line 4 it is easy to get locked into line 2 and strategy 1. But they do not work. #2 1. K (B v I) 2. K 3. ~B 4. I (~T N) 5. N T / ~N 6. B v I (1)(2) MP 7. I (6)(3) DS 8. ~T N (4)(7) MP 9. ~T (8) Simp. 10. ~N (5)(9) MT #3 See textbook. #4 1. H I 2. I J 3.
    [Show full text]
  • Chapter 10: Symbolic Trails and Formal Proofs of Validity, Part 2
    Essential Logic Ronald C. Pine CHAPTER 10: SYMBOLIC TRAILS AND FORMAL PROOFS OF VALIDITY, PART 2 Introduction In the previous chapter there were many frustrating signs that something was wrong with our formal proof method that relied on only nine elementary rules of validity. Very simple, intuitive valid arguments could not be shown to be valid. For instance, the following intuitively valid arguments cannot be shown to be valid using only the nine rules. Somalia and Iran are both foreign policy risks. Therefore, Iran is a foreign policy risk. S I / I Either Obama or McCain was President of the United States in 2009.1 McCain was not President in 2010. So, Obama was President of the United States in 2010. (O v C) ~(O C) ~C / O If the computer networking system works, then Johnson and Kaneshiro will both be connected to the home office. Therefore, if the networking system works, Johnson will be connected to the home office. N (J K) / N J Either the Start II treaty is ratified or this landmark treaty will not be worth the paper it is written on. Therefore, if the Start II treaty is not ratified, this landmark treaty will not be worth the paper it is written on. R v ~W / ~R ~W 1 This or statement is obviously exclusive, so note the translation. 427 If the light is on, then the light switch must be on. So, if the light switch in not on, then the light is not on. L S / ~S ~L Thus, the nine elementary rules of validity covered in the previous chapter must be only part of a complete system for constructing formal proofs of validity.
    [Show full text]
  • Argument Forms and Fallacies
    6.6 Common Argument Forms and Fallacies 1. Common Valid Argument Forms: In the previous section (6.4), we learned how to determine whether or not an argument is valid using truth tables. There are certain forms of valid and invalid argument that are extremely common. If we memorize some of these common argument forms, it will save us time because we will be able to immediately recognize whether or not certain arguments are valid or invalid without having to draw out a truth table. Let’s begin: 1. Disjunctive Syllogism: The following argument is valid: “The coin is either in my right hand or my left hand. It’s not in my right hand. So, it must be in my left hand.” Let “R”=”The coin is in my right hand” and let “L”=”The coin is in my left hand”. The argument in symbolic form is this: R ˅ L ~R __________________________________________________ L Any argument with the form just stated is valid. This form of argument is called a disjunctive syllogism. Basically, the argument gives you two options and says that, since one option is FALSE, the other option must be TRUE. 2. Pure Hypothetical Syllogism: The following argument is valid: “If you hit the ball in on this turn, you’ll get a hole in one; and if you get a hole in one you’ll win the game. So, if you hit the ball in on this turn, you’ll win the game.” Let “B”=”You hit the ball in on this turn”, “H”=”You get a hole in one”, and “W”=”you win the game”.
    [Show full text]
  • Philosophy 109, Modern Logic Russell Marcus
    Philosophy 240: Symbolic Logic Hamilton College Fall 2014 Russell Marcus Reference Sheeet for What Follows Names of Languages PL: Propositional Logic M: Monadic (First-Order) Predicate Logic F: Full (First-Order) Predicate Logic FF: Full (First-Order) Predicate Logic with functors S: Second-Order Predicate Logic Basic Truth Tables - á á @ â á w â á e â á / â 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 Rules of Inference Modus Ponens (MP) Conjunction (Conj) á e â á á / â â / á A â Modus Tollens (MT) Addition (Add) á e â á / á w â -â / -á Simplification (Simp) Disjunctive Syllogism (DS) á A â / á á w â -á / â Constructive Dilemma (CD) (á e â) Hypothetical Syllogism (HS) (ã e ä) á e â á w ã / â w ä â e ã / á e ã Philosophy 240: Symbolic Logic, Prof. Marcus; Reference Sheet for What Follows, page 2 Rules of Equivalence DeMorgan’s Laws (DM) Contraposition (Cont) -(á A â) W -á w -â á e â W -â e -á -(á w â) W -á A -â Material Implication (Impl) Association (Assoc) á e â W -á w â á w (â w ã) W (á w â) w ã á A (â A ã) W (á A â) A ã Material Equivalence (Equiv) á / â W (á e â) A (â e á) Distribution (Dist) á / â W (á A â) w (-á A -â) á A (â w ã) W (á A â) w (á A ã) á w (â A ã) W (á w â) A (á w ã) Exportation (Exp) á e (â e ã) W (á A â) e ã Commutativity (Com) á w â W â w á Tautology (Taut) á A â W â A á á W á A á á W á w á Double Negation (DN) á W --á Six Derived Rules for the Biconditional Rules of Inference Rules of Equivalence Biconditional Modus Ponens (BMP) Biconditional DeMorgan’s Law (BDM) á / â -(á / â) W -á / â á / â Biconditional Modus Tollens (BMT) Biconditional Commutativity (BCom) á / â á / â W â / á -á / -â Biconditional Hypothetical Syllogism (BHS) Biconditional Contraposition (BCont) á / â á / â W -á / -â â / ã / á / ã Philosophy 240: Symbolic Logic, Prof.
    [Show full text]
  • CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction Theorem
    CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction Theorem The Hilbert proof systems are systems based on a language with implication and contain a Modus Ponens rule as a rule of inference. They are usually called Hilbert style formalizations. We will call them here Hilbert style proof systems, or Hilbert systems, for short. Modus Ponens is probably the oldest of all known rules of inference as it was already known to the Stoics (3rd century B.C.). It is also considered as the most "natural" to our intuitive thinking and the proof systems containing it as the inference rule play a special role in logic. The Hilbert proof systems put major emphasis on logical axioms, keeping the rules of inference to minimum, often in propositional case, admitting only Modus Ponens, as the sole inference rule. 1 Hilbert System H1 Hilbert proof system H1 is a simple proof system based on a language with implication as the only connective, with two axioms (axiom schemas) which characterize the implication, and with Modus Ponens as a sole rule of inference. We de¯ne H1 as follows. H1 = ( Lf)g; F fA1;A2g MP ) (1) where A1;A2 are axioms of the system, MP is its rule of inference, called Modus Ponens, de¯ned as follows: A1 (A ) (B ) A)); A2 ((A ) (B ) C)) ) ((A ) B) ) (A ) C))); MP A ;(A ) B) (MP ) ; B 1 and A; B; C are any formulas of the propositional language Lf)g. Finding formal proofs in this system requires some ingenuity. Let's construct, as an example, the formal proof of such a simple formula as A ) A.
    [Show full text]
  • Proposi'onal Logic Proofs Proof Method #1: Truth Table Example
    9/9/14 Proposi'onal Logic Proofs • An argument is a sequence of proposi'ons: Inference Rules ² Premises (Axioms) are the first n proposi'ons (Rosen, Section 1.5) ² Conclusion is the final proposi'on. p p ... p q TOPICS • An argument is valid if is a ( 1 ∧ 2 ∧ ∧ n ) → tautology, given that pi are the premises • Logic Proofs (axioms) and q is the conclusion. ² via Truth Tables € ² via Inference Rules 2 Proof Method #1: Truth Table Example Proof By Truth Table s = ((p v q) ∧ (¬p v r)) → (q v r) n If the conclusion is true in the truth table whenever the premises are true, it is p q r ¬p p v q ¬p v r q v r (p v q)∧ (¬p v r) s proved 0 0 0 1 0 1 0 0 1 n Warning: when the premises are false, the 0 0 1 1 0 1 1 0 1 conclusion my be true or false 0 1 0 1 1 1 1 1 1 n Problem: given n propositions, the truth 0 1 1 1 1 1 1 1 1 n table has 2 rows 1 0 0 0 1 0 0 0 1 n Proof by truth table quickly becomes 1 0 1 0 1 1 1 1 1 infeasible 1 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 3 4 1 9/9/14 Proof Method #2: Rules of Inference Inference proper'es n A rule of inference is a pre-proved relaon: n Inference rules are truth preserving any 'me the leJ hand side (LHS) is true, the n If the LHS is true, so is the RHS right hand side (RHS) is also true.
    [Show full text]
  • An Introduction to Critical Thinking and Symbolic Logic: Volume 1 Formal Logic
    An Introduction to Critical Thinking and Symbolic Logic: Volume 1 Formal Logic Rebeka Ferreira and Anthony Ferrucci 1 1An Introduction to Critical Thinking and Symbolic Logic: Volume 1 Formal Logic is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. 1 Preface This textbook has developed over the last few years of teaching introductory symbolic logic and critical thinking courses. It has been truly a pleasure to have beneted from such great students and colleagues over the years. As we have become increasingly frustrated with the costs of traditional logic textbooks (though many of them deserve high praise for their accuracy and depth), the move to open source has become more and more attractive. We're happy to provide it free of charge for educational use. With that being said, there are always improvements to be made here and we would be most grateful for constructive feedback and criticism. We have chosen to write this text in LaTex and have adopted certain conventions with symbols. Certainly many important aspects of critical thinking and logic have been omitted here, including historical developments and key logicians, and for that we apologize. Our goal was to create a textbook that could be provided to students free of charge and still contain some of the more important elements of critical thinking and introductory logic. To that end, an additional benet of providing this textbook as a Open Education Resource (OER) is that we will be able to provide newer updated versions of this text more frequently, and without any concern about increased charges each time.
    [Show full text]