Laws of logic examples
[PDF File] Logic and Proofs - School of Engineering & Applied Science
http://5y1.org/file/12683/logic-and-proofs-school-of-engineering-applied-science.pdf
exists”, and thereby increase the expressive and reasoning power of logic. Universal quantifier : It is the phrase “for every” (or “for all”), and it is denoted ∀. Existential quantifier : It is the phrase “there exists” or “there is” (at least), and is denoted ∃.
[PDF File] Chapter 3 Syllogistic Reasoning - Stanford University
http://5y1.org/file/12683/chapter-3-syllogistic-reasoning-stanford-university.pdf
This chapter ‘opens the box’ of propositional logic, and looks further inside the statements that we make when we describe the world. Very often, these statements are about objects and their properties, and we will now show you a first logical system that deals with these. Syllogistics has been a standard of logical reasoning since Greek ...
[PDF File] Boolean Algebra 1 A Boolean algebra B - Virginia Tech
http://5y1.org/file/12683/boolean-algebra-1-a-boolean-algebra-b-virginia-tech.pdf
Boolean algebra is a set B of values together with: two binary operations, commonly denoted by + and ∙ , a unary operation, usually denoted by ˉ or ~ or ’, two elements usually called zero and one, such that for every element x of B: x 1 and x x 0. In addition, certain axioms must be satisfied: closure properties for both binary operations ...
[PDF File] Mathematical Logic - Stanford University
http://5y1.org/file/12683/mathematical-logic-stanford-university.pdf
If two propositional logic statements φ and ψ always have the same truth values as one another, they are called logically equivalent. We denote this by φ ≡ ψ. ≡ is not a connective. Connectives are a part of logic statements; ≡ is something used to describe logic statements. It is part of the metalanguage rather than the language.
[PDF File] Laws and Theorems of Boolean Logic - Harvard University
http://5y1.org/file/12683/laws-and-theorems-of-boolean-logic-harvard-university.pdf
Axioms of Boolean Algebra (4 of 4) •Axiom 6 –Distributive laws. •For every a, b, and c in B, •a + (b · c) = (a + b) · (a + c) •a · (b + c) = (a · b) + (a · c) •Axiom 7 –Complement. •For each a in B, there exists an element a' in B (the complement of a) s.t. •a + a' = 1 •a · a' = 0. 5. Order of Evaluation of Boolean ...
[PDF File] Discrete Mathematics, Chapter 1.1.-1.3: Propositional Logic
http://5y1.org/file/12683/discrete-mathematics-chapter-1-1-1-3-propositional-logic.pdf
Propositions. proposition is a declarative sentence that is either true or false. Examples of propositions: The Moon is made of green cheese. Trenton is the capital of New Jersey. Toronto is the capital of Canada. + 0 = 1. 0 + 0 = 2 Examples that …
[PDF File] The Foundations: Logic and Proofs
http://5y1.org/file/12683/the-foundations-logic-and-proofs.pdf
Translating from English to Logic Example 1: translate the following sentence into predicate logic: Every student in this class has taken a course in Java. Solution: first decide on the domain U Solution 1: if U is all students in this class, define a propositional function J(x) denoting “x has taken a course in Java” and translate as x J(x)
[PDF File] Standard Logical Arguments - Minnesota State University …
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Common invalid arguments are often called Logical Fallacies or just Fallacies. The names of these fallacies come from the fact that their reasoning is based an accidentally assuming that the converse and the inverse are logically equivalent to the related conditional statement (which is false). The Fallacy of the Converse.
[PDF File] Sets and Logic - University of Cambridge
http://5y1.org/file/12683/sets-and-logic-university-of-cambridge.pdf
top level. With the help of the Idempotence and Complements laws we can remove redundant occurrences of basic sets. The standard forms for set expressions reappear in propositional logic as disjunctive and conjunctive normal forms for propositions. Exercise 2.4 Using the set laws transform (A\B)c\(A[C) to a standard form as a union of ...
[PDF File] Propositional Logic: Syntax and Semantics - University of …
http://5y1.org/file/12683/propositional-logic-syntax-and-semantics-university-of.pdf
It is called propositional or sentential logic. This logic is a symbolic language to reason about propositions. Propositions are declarative sentences that are either true or false. Examples of such include \Spring eld is the capital of Illinois", \1+1 = 2", \2+2 = 3". Notice that propositions don’t need to be true facts (like
[PDF File] LECTURE #3 APPLYING LAWS OF LOGIC - Virtual University …
http://5y1.org/file/12683/lecture-3-applying-laws-of-logic-virtual-university.pdf
SIMPLIFYING A STATEMENT: “You will get an A if you are hardworking and the sun shines, or you are hardworking and it rains.”. Rephrase the condition more simply. Solution: Let. = “You are hardworking’. = “The sun shines”. r = “It rains” .The condition is then (p ∧ q) ∨ (p ∧ r) And using distributive law in reverse,
[PDF File] The Laws of Logic - University of Victoria
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It is apparent that the Laws of Logic come in pairs. The dual of a statement is obtained by replacing _by ^; ^by _; 0 by 1; and 1 by 0, wherever they occur. It is a theorem of logic that if s 1 is logically equivalent to s 2, then the dual of …
[PDF File] Predicates and Quantifiers
http://5y1.org/file/12683/predicates-and-quantifiers.pdf
Part 1: Use quantifiers to express the statement that “There does not exist a woman who has taken a flight on every airline in the world.”. Solution: w a f ( P ( w , f ) Q ( f , a ) ) Part 2: Now use De Morgan’s Laws to move the negation as far inward as possible. Solution:
[PDF File] The Foundations: Logic and Proofs - Kent
http://5y1.org/file/12683/the-foundations-logic-and-proofs-kent.pdf
Propositional Functions. Propositional functions become propositions (and have truth values) when their variables are each replaced by a value from the domain (or boundby a quantifier, as we will see later). The statement P(x) is said to be the value of the propositional function Pat x. For example, letP(x)denote “x> 0” and the domain be ...
[PDF File] Propositional Logic - University at Buffalo
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The rules of logic specify the precise meanings of mathematical statements. It is the basis of the correct mathematical arguments, that is, the proofs. It also has important applications in computer science: to verify that computer programs produce the correct output for all possible input values.
[PDF File] BASIC CONCEPTS OF LOGIC - UMass
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Logic may be defined reasoning. as the science Howev suggest that logic is an empirical physics, biology, or psychology. mathematics. Also, in saying that that it is concerned with the actual thinking being when it is reasoning. proc …
[PDF File] Chapter 2 Propositional Logic
http://5y1.org/file/12683/chapter-2-propositional-logic.pdf
Propositional Logic Overview The most basic logical inferences are about combinations of sentences, ex-pressed by such frequent expressions as ‘not’, ‘and’, ‘or’, ‘if, then’. ... an inference is valid if it has no ‘counter-examples’: that is, situations where the premises are all true while the conclusion is false. This is a ...
[PDF File] MATH 213: Logical Equivalences, Rules of Inference and …
http://5y1.org/file/12683/math-213-logical-equivalences-rules-of-inference-and.pdf
Examples Tables of Logical Equivalences Note: In this handout the symbol is used the tables instead of ()to help clarify where one statement ends and the other begins, particularly in those that have a biconditional as part of the statement. The abbreviations are not universal. Equivalence Name Abbr. p^T p Identity / Idempotent (Conjunction) IdC
[PDF File] Discrete Mathematics, Chapter 1.1.-1.3: Propositional Logic
http://5y1.org/file/12683/discrete-mathematics-chapter-1-1-1-3-propositional-logic.pdf
Chapter 1.1-1.3 6 / 21. Implication. If p and q are propositions, then p !q is a conditional statement or implication which is read as “if p, then q” and has this truth table: In p !q, p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). Implication can be expressed by disjunction and negation: p !q :p _q.
[PDF File] Laws of Logic - UMD
http://5y1.org/file/12683/laws-of-logic-umd.pdf
Laws of Logic. Given any statement variables p, q, and r , a tautology t and a contradiction c, the following logical equivalences hold: 1. Commutative laws: p ^ q ⌘ q ^ p p _ q ⌘ q p _. 2.
[PDF File] Computability of 0-1 Laws - Harvard University
http://5y1.org/file/12683/computability-of-0-1-laws-harvard-university.pdf
0-1 Laws Subsets of Models Computability of the 0-1 Law Operator Examples Collections of Finite Structures Graphs Since the discovery of the 0-1 law for rst order logic there have been many attempts to generalize it. One of the most successful of which has come from limiting the collection of nite models we consider.
[PDF File] Aristotle’s Logic - UW Faculty Web Server
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Aristotle’s logic compared to contemporary logic To one trained in post-Fregean first-order logic (quantification theory), Aristotle’s syllogistic may seem a narrow, barren, and stultifying theory. But this is not so. To think this would be to wrongly blame Aristotle for the authority his teachings subsequently had bestowed upon them.
[PDF File] B Exercises Exercise Sheet 1: Propositional Logic
http://5y1.org/file/12683/b-exercises-exercise-sheet-1-propositional-logic.pdf
Exercise Sheet 2: Predicate Logic 1. Formalise the following statements in predicate logic, making clear what your atomic predicate symbols stand for and what the domains of any variables are. (a) Anyone who has forgiven at least one person is a saint. (b) Nobody in the calculus class is smarter than everybody in the discrete maths class.
[PDF File] Proofs in Propositional Logic - Department of Computer …
http://5y1.org/file/12683/proofs-in-propositional-logic-department-of-computer.pdf
Proofs in Propositional Logic 1. Pierre Cast ́eran. Beijing, August 2010. This lecture corresponds mainly to Chapter 3 :“Propositions and Proofs” and part of Chapter 5 :“Everyday Logic”of the book. In this class, we introduce the reasoning techniques used in Coq, starting with a very reduced fragment of logic, propositional ...
[PDF File] TruthTables,Tautologies,andLogicalEquivalences
http://5y1.org/file/12683/truthtables-tautologies-andlogicalequivalences.pdf
Truth Tables, Tautologies, and Logical Equivalences. Mathematicians normally use a two-valued logic: Every statement is either True or False. This is called the Law of the Excluded Middle. A statement in sentential logic is built from simple statements using the logical connectives ¬, ∧, ∨, →, and ↔. The truth or falsity of a statement ...
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