Changes

The concept of '''logical implication''' encompasses, depending on the context of use, a specific logical [[function (mathematics)|function]], a specific logical [[relation (mathematics)|relation]], and the various symbols that are used to denote this function and this relation.  In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.

The concept of '''logical implication''' encompasses a specific logical [[function (mathematics)|function]], a specific logical [[relation (mathematics)|relation]], and the various symbols that are used to denote this function and this relation.  In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.

A close approximation to the concept of logical implication or material conditional is expressed in ordinary language by means of the following conditional form:

Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:

:* If ''p'' then ''q''.

:* <math>p</math> implies <math>q</math>.

Here ''p'' and ''q'' are propositional variables that stand for any propositions in a given language.  In a statement of the form "if ''p'' then ''q''", the first term, ''p'', is called the ''[[antecedent (logic)|antecedent]]'' and the second term, ''q'', is called the ''[[consequent]]'', while the statement as a whole is called either the ''conditional'' or the ''consequence''.  Assuming that the conditional is true, then the truth of the antecedent is a [[sufficient condition]] for the truth of the consequent, while the truth of the consequent is a [[necessary condition]] for the truth of the antecedent.

 

Here <math>p</math> and <math>q</math> are propositional variables that stand for any propositions in a given language.  In a statement of the form "if <math>p</math> then <math>q</math>", the first term, <math>p</math>, is called the ''[[antecedent (logic)|antecedent]]'' and the second term, <math>q</math>, is called the ''[[consequent]]'', while the statement as a whole is called either the ''[[conditional]]'' or the ''[[consequence]]''.  Assuming that the conditional statement is true, then the truth of the antecedent is a ''[[sufficient condition]]'' for the truth of the consequent, while the truth of the consequent is a ''[[necessary condition]]'' for the truth of the antecedent.

 

'''Note.'''  Many writers draw a technical distinction between the form "<math>p</math> implies <math>q</math>" and the form "if <math>p</math> then <math>q</math>".  In this usage, writing "<math>p</math> implies <math>q</math>" asserts the existence of a certain relation between the logical value of <math>p</math> and the logical value of <math>q</math> while writing "if <math>p</math> then <math>q</math>" merely forms a compound statement whose logical value is a function of the logical values of <math>p</math> and <math>q</math>.  This will be discussed in detail below.

==Definition==

==Definition==

The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false.

The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of <math>\mathrm{false}</math> just in case the first operand is true and the second operand is false.

The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p&nbsp;→&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;⇒&nbsp;q''') is as follows:

In the interpretation where <math>0 = \mathrm{false}</math> and <math>1 = \mathrm{true}</math>, the truth table associated with the statement "<math>p</math> implies <math>q</math>", symbolized as <math>p \Rightarrow q</math>, is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"

! style="width:15%" | p ⇒ q

! style="width:15%" | p ⇒ q

|-

|-

| F || F || T

| 0 || 0 || 1

|-

|-

| F || T || T

| 0 || 1 || 1

|-

|-

| T || F || F

| 1 || 0 || 0

|-

|-

| T || T || T

| 1 || 1 || 1

|}

|}

<br>

<br>