Document
... Predicates can be quantified using the universal quantifier (“for all”) or the existential quantifier (“there exists”) Quantified predicates can be negated as follows x P(x) x P(x) x P(x) x P(x) Quantified variables are called “bound” Variables that are not quantified are called ...
... Predicates can be quantified using the universal quantifier (“for all”) or the existential quantifier (“there exists”) Quantified predicates can be negated as follows x P(x) x P(x) x P(x) x P(x) Quantified variables are called “bound” Variables that are not quantified are called ...
Propositional Logic What is logic? Propositions Negation
... • Essentially, logic formalizes our reasoning process. – It provides a common language through which we can demonstrate to each other that our reasoning is valid. ...
... • Essentially, logic formalizes our reasoning process. – It provides a common language through which we can demonstrate to each other that our reasoning is valid. ...
Mathematics for Computer Science/Software Engineering
... Heuristic justification for this definition: if p is true and q is false, then the statement ‘if p is true then q is true’ obviously cannot be true, and therefore must be false. On the other hand, if p is false, then the statement ‘if p is true then ...’ is an empty statement—it is saying nothing at ...
... Heuristic justification for this definition: if p is true and q is false, then the statement ‘if p is true then q is true’ obviously cannot be true, and therefore must be false. On the other hand, if p is false, then the statement ‘if p is true then ...’ is an empty statement—it is saying nothing at ...
Gresham Ideas - Gresham College
... John Buridan’s study of self-reference seems to me to be remarkably modern, and I am going to show you a couple of his examples. First of all, consider the following statement: I say that I am the greatest mathematician in the world. Now, let it be clearly understood that I am not really the greates ...
... John Buridan’s study of self-reference seems to me to be remarkably modern, and I am going to show you a couple of his examples. First of all, consider the following statement: I say that I am the greatest mathematician in the world. Now, let it be clearly understood that I am not really the greates ...
ch1_1
... 1. Assume p is true and q is false 2. Show that ~p is also true. 3. Then we have that p ^ (~p) is true. 4. But this is impossible, since the statement p ^ (~p) is always false. There is a contradiction! 5. So, q cannot be false and therefore it is true. ...
... 1. Assume p is true and q is false 2. Show that ~p is also true. 3. Then we have that p ^ (~p) is true. 4. But this is impossible, since the statement p ^ (~p) is always false. There is a contradiction! 5. So, q cannot be false and therefore it is true. ...
Discrete Mathematics
... Mathematical logic spells out these rules in complete detail, defining what constitutes a formal proof. Learning mathematical logic is a good way to learn logic because it puts you on a firm foundation. Writing formal proofs in mathematical logic is a lot like computer programming. The rules of the ...
... Mathematical logic spells out these rules in complete detail, defining what constitutes a formal proof. Learning mathematical logic is a good way to learn logic because it puts you on a firm foundation. Writing formal proofs in mathematical logic is a lot like computer programming. The rules of the ...
Proofs 1 What is a Proof?
... This proposition is also false, but the smallest counterexample has more than 1000 digits! Proposition 2.5. Every map can be colored with 4 colors so that adjacent1 regions have different colors. This proposition is true and is known as the “fourcolor theorem”. However, there have been many incorre ...
... This proposition is also false, but the smallest counterexample has more than 1000 digits! Proposition 2.5. Every map can be colored with 4 colors so that adjacent1 regions have different colors. This proposition is true and is known as the “fourcolor theorem”. However, there have been many incorre ...
Unit-1-B - WordPress.com
... q: Raju will eat fruit-salad containing mangoes. If p then q, when p and q are propositions can be written as p → q. The above sentence (p → q) states only that Raju will eat fruitsalad containing mangoes. It does not, however, rule out the possibility that Raju will eat fruit-salad containing apple ...
... q: Raju will eat fruit-salad containing mangoes. If p then q, when p and q are propositions can be written as p → q. The above sentence (p → q) states only that Raju will eat fruitsalad containing mangoes. It does not, however, rule out the possibility that Raju will eat fruit-salad containing apple ...
Basic Concepts of Formal Logic
... is false. The negation of a proposition, in turn, can be shown to be false if, from it, another proposition that is known to be false can be validly derived. If the conclusion of a valid argument is a false proposition, then at least one of the premises from which that conclusion has been derived mu ...
... is false. The negation of a proposition, in turn, can be shown to be false if, from it, another proposition that is known to be false can be validly derived. If the conclusion of a valid argument is a false proposition, then at least one of the premises from which that conclusion has been derived mu ...
Document
... • 1 and 2 are not Propositions because they are not declarative sentences. 3 and 4 are not Propositions because they are neither True nor False. • Truth value of proposition is true, denoted by T and truth value of proposition is false denoted F. • The area of logic that deals with propositions is c ...
... • 1 and 2 are not Propositions because they are not declarative sentences. 3 and 4 are not Propositions because they are neither True nor False. • Truth value of proposition is true, denoted by T and truth value of proposition is false denoted F. • The area of logic that deals with propositions is c ...
2/TRUTH-FUNCTIONS
... Expressions: it is not true that/ it is false that/ it is not the case that Definition: p = p is true/ -p = p is false Example: it is false that `he who has a why to live for can bear with almost any how'(Nietzsche). 9b. Conjunction: And (.) dot -> p.q Interpretation: p.q = both conjuncts are true - ...
... Expressions: it is not true that/ it is false that/ it is not the case that Definition: p = p is true/ -p = p is false Example: it is false that `he who has a why to live for can bear with almost any how'(Nietzsche). 9b. Conjunction: And (.) dot -> p.q Interpretation: p.q = both conjuncts are true - ...
p q
... How can both p and q be false, and pq be true? •Think of p as a “contract” and q as its “obligation” that is only carried out if the contract is valid. •Example: “If you make more than $25,000, then you must file a tax return.” This says nothing about someone who makes less than $25,000. So the imp ...
... How can both p and q be false, and pq be true? •Think of p as a “contract” and q as its “obligation” that is only carried out if the contract is valid. •Example: “If you make more than $25,000, then you must file a tax return.” This says nothing about someone who makes less than $25,000. So the imp ...
Discrete Structures & Algorithms Propositional Logic
... 2(3k+1). Thus 3n+2 is even, because it equals 2j for integer j = 3k+1. So 3n+2 is not odd. We have shown that ¬(n is odd)→¬(3n+2 is odd), thus its contra-positive (3n+2 is odd) → (n is odd) is also true. □ ...
... 2(3k+1). Thus 3n+2 is even, because it equals 2j for integer j = 3k+1. So 3n+2 is not odd. We have shown that ¬(n is odd)→¬(3n+2 is odd), thus its contra-positive (3n+2 is odd) → (n is odd) is also true. □ ...
p and q
... • Basic idea is to assume that the opposite of what you are trying to prove is true and show that it results in a violation of one of your initial assumptions. • In the previous proof we showed that assuming that the sum of a rational number and an irrational number is rational and showed that it re ...
... • Basic idea is to assume that the opposite of what you are trying to prove is true and show that it results in a violation of one of your initial assumptions. • In the previous proof we showed that assuming that the sum of a rational number and an irrational number is rational and showed that it re ...
Logic for Gottlob Frege and Bertrand Russell:
... I. Frege: formal logic can answer this question by developing a logical notation (Begriffsschrift) that allows for the expression of two things: 1. all propositions (i.e., everything true or false); and 2. general logical laws governing all inferential relations among propositions. These general log ...
... I. Frege: formal logic can answer this question by developing a logical notation (Begriffsschrift) that allows for the expression of two things: 1. all propositions (i.e., everything true or false); and 2. general logical laws governing all inferential relations among propositions. These general log ...
Chapter1_Parts2
... Example: Show that every compound proposition can be put in disjunctive normal form. ! Solution: Construct the truth table for the proposition. Then an equivalent proposition is the disjunction with n disjuncts (where n is the number of rows for which the formula evaluates to T). Each disjunct has m ...
... Example: Show that every compound proposition can be put in disjunctive normal form. ! Solution: Construct the truth table for the proposition. Then an equivalent proposition is the disjunction with n disjuncts (where n is the number of rows for which the formula evaluates to T). Each disjunct has m ...
The Foundations: Logic and Proofs - UTH e
... their relations. Later we’ll see how to draw inferences. ...
... their relations. Later we’ll see how to draw inferences. ...
Logic
... In grammatical terminology, this consists of a transitive verb and two open places, one to be filled by the name of a subject, such as “Jane”, the other of an object, such as “John”. Binary relation ...
... In grammatical terminology, this consists of a transitive verb and two open places, one to be filled by the name of a subject, such as “Jane”, the other of an object, such as “John”. Binary relation ...
6.042J Chapter 1: Propositions
... one knows if it is true or false. But that doesn’t prevent you from answering the question! This proposition has the form P IMPLIES Q where the hypothesis, P , is “the Riemann Hypothesis is true” and the conclusion, Q, is “x 2 0 for every real number x”. Since the conclusion is definitely true, we ...
... one knows if it is true or false. But that doesn’t prevent you from answering the question! This proposition has the form P IMPLIES Q where the hypothesis, P , is “the Riemann Hypothesis is true” and the conclusion, Q, is “x 2 0 for every real number x”. Since the conclusion is definitely true, we ...
slides - Computer and Information Science
... • So we have a method for determining whether consequence of we use a truth table to see whether If it is, then ...
... • So we have a method for determining whether consequence of we use a truth table to see whether If it is, then ...
CA320 - Computability & Complexity Overview
... have the same truth value for every possible combination of base propositions. Hence, in any expression where P is used we can substitute Q and the entire expression remains unchanged. A proposition P logically implies a proposition Q, P ⇒ Q, if in every case P is true then Q is also true. Beware of ...
... have the same truth value for every possible combination of base propositions. Hence, in any expression where P is used we can substitute Q and the entire expression remains unchanged. A proposition P logically implies a proposition Q, P ⇒ Q, if in every case P is true then Q is also true. Beware of ...
Chapter 1: The Foundations: Logic and Proofs
... The biconditional statement pq is true when p and q have the same truth value, and is false otherwise. Biconditional statements are also called bi-implications. ...
... The biconditional statement pq is true when p and q have the same truth value, and is false otherwise. Biconditional statements are also called bi-implications. ...
PPT
... Logic Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and formalized in the 19th and early 20th century. George Boole (Boolean algebra) introduced Math ...
... Logic Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and formalized in the 19th and early 20th century. George Boole (Boolean algebra) introduced Math ...
6. Truth and Possible Worlds
... members, separated by commas, between braces. For example, {2, 4, 5} is the set whose members are 2, 4 and 5. There is one set with no members, written {} or ∅. If two sets have the same members, then they are actually the same set. We assume that all objects are members of the universal set U. Here ...
... members, separated by commas, between braces. For example, {2, 4, 5} is the set whose members are 2, 4 and 5. There is one set with no members, written {} or ∅. If two sets have the same members, then they are actually the same set. We assume that all objects are members of the universal set U. Here ...
January 12
... representations – whereas we can in fact count anything, including physical objects. 5. Psychologism cannot account for our ability to do arithmetical operations with large numbers. For it is surely impossible for our imagination to distinguish 13,543 sensations or mental pictures from 13,544 sensat ...
... representations – whereas we can in fact count anything, including physical objects. 5. Psychologism cannot account for our ability to do arithmetical operations with large numbers. For it is surely impossible for our imagination to distinguish 13,543 sensations or mental pictures from 13,544 sensat ...
Bernard Bolzano
Bernhard Placidus Johann Nepomuk Bolzano (Bernard Bolzano in English; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his antimilitarist views. Bolzano wrote in German, his mother tongue. For the most part, his work came to prominence posthumously.