Homework How-to » logic http://homeworkhowto.com Homework. Easy. Wed, 06 Jan 2010 21:14:39 +0000 http://wordpress.org/?v=2.8.4 en hourly 1 Contradiction http://homeworkhowto.com/contradiction/ http://homeworkhowto.com/contradiction/#comments Tue, 13 Oct 2009 22:16:21 +0000 Christine http://homeworkhowto.com/contradiction/ How to Identify Contradictions

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Description

A detailed tutorial on identifying contradictions. Step by step tutorial including several examples of how to identify contradictions for reference.

Overview

A contradiction is a statement of only false values – one that is false no matter how you look at it. In terms of mathematical logic, it is defined as a propositional form that is false for every assignment of truth values to its components. In order for a statement to be a contradiction, when the proposition is on a truth table it must be false for every possible combination of P and Q.

]]> http://homeworkhowto.com/contradiction/feed/ 0 Tautology http://homeworkhowto.com/tautology/ http://homeworkhowto.com/tautology/#comments Tue, 13 Oct 2009 22:13:50 +0000 Christine http://homeworkhowto.com/tautology/ How to Identify Tautologies

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Description

A detailed tutorial on identifying tautologies. Step by step tutorial including several examples of how to identify tautologies for reference.

Overview

A tautology is a statement of truth – one that is true no matter how you look at it. In terms of mathematical logic, it is defined as a propositional form that is true for every assignment of truth values to its components. In order for a statement to be a tautology, when the proposition is on a truth table it must be true for every possible combination of P and Q.

]]> http://homeworkhowto.com/tautology/feed/ 0 Existential and Universal Quantification http://homeworkhowto.com/existential-and-universal-quantification/ http://homeworkhowto.com/existential-and-universal-quantification/#comments Wed, 07 Oct 2009 00:13:55 +0000 Christine http://homeworkhowto.com/existential-and-universal-quantification/ Overview of Existential and Universal Quantification

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Description

A detailed tutoral on existential and universal quantification. Step by step tutorial including several examples of existential and universal quantification for reference.

Overview

Existential and universal quantifiers give us different ways to write expressions and mathematical equations. The existential quanitfier looks like a backwards capital E, and basically means “some”. The universal quantifier looks like an upside down A, and basically means “all”. For example, take the sentence “Some children don’t like clowns.” In the mathematical form of quantifiers, this would be written as (Ex) (x is a child) ^ (Ay) (y is a clown) –> (x does not like y). “Some children” indicates that you would use an existential quantifier, not a universal quantifier. Since clowns in not specific, based on context we must assume that the statement refers to all clown, and therefore we use the universal quantifier. The ^ is the symbol for “and”, implying that both of these statements are true, and the arrow is an implication stating that if there is a clown, some children will not like it based on the previous statement.

]]> http://homeworkhowto.com/existential-and-universal-quantification/feed/ 0 Modus Tollens http://homeworkhowto.com/modus-tollens/ http://homeworkhowto.com/modus-tollens/#comments Thu, 24 Sep 2009 18:31:17 +0000 Christine http://homeworkhowto.com/modus-tollens/ The Modus Tollens Rule Explained

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A detailed tutorial on the modus tollens rule. Step by step tutorial including several example problems of the modus tollens rule for reference.

 
Overview
Modus tollendo tollens, often simply referred to as modus tollens, is an argument in logic that states if P, then Q.  Negation of Q, therefore negation of P. This is sometimes called denying the consequent, and is often confused with the indirect proof of proving by contraposition. There are several forms that the modus tollens rule can take, depending on when and how you are using it.

 

 

Logical Operator Notation: P\to Q, \neg Q \vdash \neg P

 

Basic Form: \frac{P\to Q ~,~~ \neg Q}{\neg P}

 

With Assumptions: \frac{\Gamma \vdash P\to Q ~~~ \Gamma \vdash\neg Q}{\Gamma \vdash \neg P}

 

Set Theory:

P\subseteq Q
x\notin Q
\therefore x\notin P

 

Predicate Logic:

\forall x.~P(x) \to Q(x)
\exists x.~\neg Q(x)
\therefore \exists x.~\neg P(x) ]]> http://homeworkhowto.com/modus-tollens/feed/ 0 Modus Ponens http://homeworkhowto.com/modus-ponens/ http://homeworkhowto.com/modus-ponens/#comments Thu, 24 Sep 2009 18:08:58 +0000 Christine http://homeworkhowto.com/modus-ponens/ The Modus Ponens Rule Explained

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Description

 

A detailed tutorial on the modus ponens rule. Step by step tutorial including several examples of the modus ponens rule for reference.

 

Overview

 

Modus ponendo ponens, typically shortened to just modus ponens, is an argument in logic. It is closely related to the argument modus tollens. Modus ponens states that if P, then Q. P, therefore Q. This can be expressed in either sequent form or rule form for formal notation.

Sequent Form: P \to Q, P \vdash Q

Rule Form: \frac{P \rightarrow Q, P}{Q}.

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