Section I(e)
... You might think there is a misprint in TABLE 11, with regards to the bottom two rows, which is read as ‘if P is false’ then ‘ P Q ’ is true, independent of the truth value of Q . There is no misprint, this is correct. How can we justify these statements? Let’s consider an example where P and Q are ...
... You might think there is a misprint in TABLE 11, with regards to the bottom two rows, which is read as ‘if P is false’ then ‘ P Q ’ is true, independent of the truth value of Q . There is no misprint, this is correct. How can we justify these statements? Let’s consider an example where P and Q are ...
A puzzle about de rebus beliefs
... are critics who admire only one another’ as asserting that Ralph believes one of the de rebus or plural propositions that can be obtained by supplying some critics as argument to the propositional function described by the English open sentence ‘that they admire only one another’; there are some ind ...
... are critics who admire only one another’ as asserting that Ralph believes one of the de rebus or plural propositions that can be obtained by supplying some critics as argument to the propositional function described by the English open sentence ‘that they admire only one another’; there are some ind ...
Overview of proposition and predicate logic Introduction
... The subject of logic is to examine human reasoning and to formulate rules to ensure that such reasoning is correct. Modern logic does so in a formal mathematical way, hence names like “symbolic logic”, “formal logic”, “mathematical logic”. The logical approach includes the expression of human knowle ...
... The subject of logic is to examine human reasoning and to formulate rules to ensure that such reasoning is correct. Modern logic does so in a formal mathematical way, hence names like “symbolic logic”, “formal logic”, “mathematical logic”. The logical approach includes the expression of human knowle ...
Section 1
... Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is Proposition 3 1. An implication and its contrapositive are logically equivalent 2. The converse a ...
... Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is Proposition 3 1. An implication and its contrapositive are logically equivalent 2. The converse a ...
Braun Defended
... propositions (10) and (11) differ and hence differ in truth-value. This objection, however, ignores the distinction between the proposition expressed by a sentence, its assertoric content, and what contribution a sentence makes to complex sentences of which it is a part, its ingredient sense. If we ...
... propositions (10) and (11) differ and hence differ in truth-value. This objection, however, ignores the distinction between the proposition expressed by a sentence, its assertoric content, and what contribution a sentence makes to complex sentences of which it is a part, its ingredient sense. If we ...
Lecture01 - Mathematics
... Set Theory: Informally we define a set as a collection of objects. The resulting theory of how one can operate on sets is known as naïve set theory. It is naïve because the informal definition leads to subtle paradoxes. A more careful definition of set removes these paradoxes and leaves the conclusi ...
... Set Theory: Informally we define a set as a collection of objects. The resulting theory of how one can operate on sets is known as naïve set theory. It is naïve because the informal definition leads to subtle paradoxes. A more careful definition of set removes these paradoxes and leaves the conclusi ...
Slide 1
... proof – Corollary: a theorem that can be directly established from a theorem that has been proved – Conjecture: a statement that is being proposed to be true ...
... proof – Corollary: a theorem that can be directly established from a theorem that has been proved – Conjecture: a statement that is being proposed to be true ...
p q
... sentence that may be assigned a ‘true’ or ‘false’ value, but not both. This value is the truth value of the proposition. Propositions: “1+2=3”, “Peter is a programmer”, “It is snowing”. Not Propositions: “Is 1+2=3?”, “What a beautiful evening!”, “The number x is an integer”. Also Propositions: “Ther ...
... sentence that may be assigned a ‘true’ or ‘false’ value, but not both. This value is the truth value of the proposition. Propositions: “1+2=3”, “Peter is a programmer”, “It is snowing”. Not Propositions: “Is 1+2=3?”, “What a beautiful evening!”, “The number x is an integer”. Also Propositions: “Ther ...
ch1_Logic_and_proofs
... 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. ...
Propositional Logic
... Why study propositional logic? • A formal mathematical “language” for precise reasoning. • Start with propositions. • Add other constructs like negation, conjunction, disjunction, implication etc. • All of these are based on ideas we use daily to reason about things. ...
... Why study propositional logic? • A formal mathematical “language” for precise reasoning. • Start with propositions. • Add other constructs like negation, conjunction, disjunction, implication etc. • All of these are based on ideas we use daily to reason about things. ...
Lecture 6 Induction
... • A proposition denoted by symbols P(n) are propositions having to do with all numbers of value n. • Terminology and notation: If we say that proposition P(n) is true for all natural numbers n, then we mean that: P(1) ∧ P(2) ∧ P(3) ∧ ... is true. That is the logical “and” of all of these proposition ...
... • A proposition denoted by symbols P(n) are propositions having to do with all numbers of value n. • Terminology and notation: If we say that proposition P(n) is true for all natural numbers n, then we mean that: P(1) ∧ P(2) ∧ P(3) ∧ ... is true. That is the logical “and” of all of these proposition ...
Document
... the truth, and knaves, who always lie. You go to the island and meet A and B. A says “B is a knight.” B says “The two of us are of opposite types.” Example: What are the types of A and B? Solution: Let p and q be the statements that A is a knight and B is a knight, respectively. So, then p re ...
... the truth, and knaves, who always lie. You go to the island and meet A and B. A says “B is a knight.” B says “The two of us are of opposite types.” Example: What are the types of A and B? Solution: Let p and q be the statements that A is a knight and B is a knight, respectively. So, then p re ...
Solutions to Workbook Exercises Unit 16: Categorical Propositions
... U.D.: animals Bx: x barks Cx: x is a cat Lx: x likes to walk Dx: x is a dog Mx: x meows Fx: x likes canned food Wx: x wags its tail (a) Some dogs howl. ∃x (Dx • Hx) x ...
... U.D.: animals Bx: x barks Cx: x is a cat Lx: x likes to walk Dx: x is a dog Mx: x meows Fx: x likes canned food Wx: x wags its tail (a) Some dogs howl. ∃x (Dx • Hx) x ...
notes
... Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there exists i ∈ {1, . ...
... Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there exists i ∈ {1, . ...
Section I(c)
... The statement x 5 2 is an example of a predicate. What does predicate mean? It is a statement with a unknown such as x . Once we substitute values for all the unknowns then it becomes a proposition. For example if we substitute x 3 then ...
... The statement x 5 2 is an example of a predicate. What does predicate mean? It is a statement with a unknown such as x . Once we substitute values for all the unknowns then it becomes a proposition. For example if we substitute x 3 then ...
Discrete Structure
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in module 2) for any branch of mathematics. • Predicate logic with function symbols, the “=” operator, and a few proof-buil ...
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in module 2) for any branch of mathematics. • Predicate logic with function symbols, the “=” operator, and a few proof-buil ...
Introduction to proposition
... “You can access the Internet from campus only if you are a computer science major or you are not a freshman.” Solution: There are many ways to translate this sentence into a logical expression. Although it is possible to represent the sentence by a single propositional variable, such as p, this woul ...
... “You can access the Internet from campus only if you are a computer science major or you are not a freshman.” Solution: There are many ways to translate this sentence into a logical expression. Although it is possible to represent the sentence by a single propositional variable, such as p, this woul ...
The Foundations: Logic and Proofs
... In English “or” has two distinct meanings. “Inclusive Or” - In the sentence “Students who have taken CS202 or Math120 may take this class,” we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p ∨q to be true, either ...
... In English “or” has two distinct meanings. “Inclusive Or” - In the sentence “Students who have taken CS202 or Math120 may take this class,” we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p ∨q to be true, either ...
Horseshoe and Turnstiles
... that is, if φ’s truth is not dependent on any other proposition(s). It is always true. We can express this as, (2) ⊧ φ. In this case, φ is also sometimes called a ‘logical truth’. More controversially, we could say that (2) says that φ is an axiom or a self-evident truth. Now, (3) Γ ⊧ φ iff there is ...
... that is, if φ’s truth is not dependent on any other proposition(s). It is always true. We can express this as, (2) ⊧ φ. In this case, φ is also sometimes called a ‘logical truth’. More controversially, we could say that (2) says that φ is an axiom or a self-evident truth. Now, (3) Γ ⊧ φ iff there is ...
Chapter 1 - National Taiwan University
... 1.2. A Famous Problem in Computer Science. In Example 2, we show how to use propositional logic to write system specifications. Additionally, we mentioned that system specifications should be consistent. That is, we should be able to assign truth values to propositions such that all requirements are s ...
... 1.2. A Famous Problem in Computer Science. In Example 2, we show how to use propositional logic to write system specifications. Additionally, we mentioned that system specifications should be consistent. That is, we should be able to assign truth values to propositions such that all requirements are s ...
CHAPTER 0: WELCOME TO MATHEMATICS A Preface of Logic
... is being defined and italics to emphasize text for other reasons. For example, we will often use italics to emphasize terms that have not yet been defined. These terms may be undefinable, defined later in the notes, or you may need to look elsewhere for their definition. We use the term theorem in r ...
... is being defined and italics to emphasize text for other reasons. For example, we will often use italics to emphasize terms that have not yet been defined. These terms may be undefinable, defined later in the notes, or you may need to look elsewhere for their definition. We use the term theorem in r ...
Document
... one is a notorious liar one is a pokerface, sometimes liar sometimes honest They make the following statements: A says: "I love mathematics." B says: "C always tells the truth." C says: "A hates math." Who is most likely the honest one? ...
... one is a notorious liar one is a pokerface, sometimes liar sometimes honest They make the following statements: A says: "I love mathematics." B says: "C always tells the truth." C says: "A hates math." Who is most likely the honest one? ...
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.