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Difference between revisions of "Logic"

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'''Logic''' is the systematic use of symbolic and mathematical techniques to determine the forms of valid deductive or inductive argument. it is sometimes considered a branch of [[abstract algebra]].
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'''Logic''' is the systematic use of symbolic and mathematical techniques to determine the forms of valid deductive or inductive argument.  
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== Statements ==
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{{main|statement}}
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A statement is either true or false, but it will never be both or neither. An example of a statement is "A duck is a bird" which is true. Another example is "A pencil does not exist" which is false.
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===Conditional===
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{{main|conditional}}
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If <math>P</math> then <math>Q</math>. For example, "If it is a duck then it is a bird."
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===Inverse===
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{{main|inverse}}
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The inverse of the conditional statement is: If not <math>P</math> then not <math>Q</math>.
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===Converse===
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{{main|converse}}
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The converse of the conditional statement is: If <math>Q</math> then <math>P</math>.
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===Contrapositive===
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{{main|contrapositive}}
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The contrapositive of the conditional statement is: If not <math>Q</math> then not <math>P</math>.
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The conditional is equivalent to the contrapositive. The inverse is equivalent to the converse. When both the conditional and the converse are true at the same time, this is equivalent to an [[if and only if]] statement.
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==Logical Notations==
  
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==Logical Notation==
 
 
{{main|Logical notation}}
 
{{main|Logical notation}}
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'''Logical notation''' is a special syntax that is shorthand for logical statements.
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=== Implication ===
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The implication between two statements is <math>p \imples q</math>
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===Negations===
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The negation of <math>p</math>, denoted by <math>\neg p</math>, is the statement that is true when <math>p</math> is false and is false when <math>p</math> is true. This means simply "it is not the case that <math>p</math>."
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===Conjunction===
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The conjunction of two statements basically means "<math>p</math> and <math>q</math>" and is denoted by <math>p \land q</math>.
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===Disjunction===
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The disjunction of two statements basically means "<math>p</math> or <math>q</math>" and is denoted by <math>p \vee q</math>.
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===Implication, Conditional===
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The statement "If <math>p</math> then <math>q</math>" is denoted by <math>p\implies q</math>. For example, <math>x+3=5\implies x=2</math> means "If  <math>x+3=5</math> then <math>x=2</math>."
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===Converse===
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The converse of the statement <math>p \implies q</math> is <math>q \implies p</math>.
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===Inverse===
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The inverse of the statement <math>p \implies q</math> is <math>\neg p \implies \neg q</math>.
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===Contrapositive===
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The contrapositive of the statement <math>p \implies q</math> is <math>\neg q \implies \neg p</math>. These statements are logically equivalent.
  
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'''Logical notation''' is a special syntax that is shorthand for logical statements.
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==Truth Tables==
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A truth table is the list of all possible values of a compound statement.
  
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For example, both <math>p\to q</math> and <math>p \subset q</math> mean that p ''implies'' q, or "If ''p'', then ''q''."
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==Quantifiers==
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Note that this can be also written <math>p \cup \neg q</math>, or "''p'' or not ''q''".
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There are two types of quantifiers: A universal Quantifier: "for all" and an existential Quantifier: "there exists". A universal quantifier is denoted by <math>\forall</math> and an existential quantifier is denoted by <math>\exists</math>.
  
 
==See Also==
 
==See Also==
 
*[[Dual]]
 
*[[Dual]]
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*[[Abstract algebra]]
 
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{{stub}}
 
 
[[Category:Definition]]
 
[[Category:Definition]]
 
[[Category:Logic]]
 
[[Category:Logic]]
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{{stub}}

Latest revision as of 17:13, 14 February 2025

Logic is the systematic use of symbolic and mathematical techniques to determine the forms of valid deductive or inductive argument.

Statements

Main article: statement

A statement is either true or false, but it will never be both or neither. An example of a statement is "A duck is a bird" which is true. Another example is "A pencil does not exist" which is false.

Conditional

Main article: conditional

If $P$ then $Q$. For example, "If it is a duck then it is a bird."

Inverse

Main article: inverse

The inverse of the conditional statement is: If not $P$ then not $Q$.

Converse

Main article: converse

The converse of the conditional statement is: If $Q$ then $P$.

Contrapositive

Main article: contrapositive

The contrapositive of the conditional statement is: If not $Q$ then not $P$.

The conditional is equivalent to the contrapositive. The inverse is equivalent to the converse. When both the conditional and the converse are true at the same time, this is equivalent to an if and only if statement.

Logical Notations

Main article: Logical notation

Logical notation is a special syntax that is shorthand for logical statements.

Implication

The implication between two statements is $p \imples q$ (Error compiling LaTeX. Unknown error_msg)

Negations

The negation of $p$, denoted by $\neg p$, is the statement that is true when $p$ is false and is false when $p$ is true. This means simply "it is not the case that $p$."

Conjunction

The conjunction of two statements basically means "$p$ and $q$" and is denoted by $p \land q$.

Disjunction

The disjunction of two statements basically means "$p$ or $q$" and is denoted by $p \vee q$.

Implication, Conditional

The statement "If $p$ then $q$" is denoted by $p\implies q$. For example, $x+3=5\implies x=2$ means "If $x+3=5$ then $x=2$."

Converse

The converse of the statement $p \implies q$ is $q \implies p$.

Inverse

The inverse of the statement $p \implies q$ is $\neg p \implies \neg q$.

Contrapositive

The contrapositive of the statement $p \implies q$ is $\neg q \implies \neg p$. These statements are logically equivalent.

Truth Tables

A truth table is the list of all possible values of a compound statement.

Quantifiers

There are two types of quantifiers: A universal Quantifier: "for all" and an existential Quantifier: "there exists". A universal quantifier is denoted by $\forall$ and an existential quantifier is denoted by $\exists$.

See Also

This article is a stub. Help us out by expanding it.

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