# logical implication

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## 1 Short version

The truth table  for the logical implication operation that is written as $p\Rightarrow q$ and read as $p\ \operatorname{implies}\ q",$ also written as $p\rightarrow q$ and read as $\operatorname{if}\ p\ \operatorname{then}\ q",$ is as follows:

 Logical Implication $p$ $q$ $p\Rightarrow q$ F F T F T T T F F T T T

## 2 Long version

The mathematical objects that inform our capacity for logical reasoning are easier to describe in a straightforward way than it is to reconcile their traditional accounts. Still, some discussion of the language  that occurs in the literature cannot be avoided.

The concept  of logical implication encompasses a specific logical function, a specific logical 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.

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

• $p$ implies $q$.

• If $p$ then $q$.

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 and the second term, $q$, is called the consequent, while the statement as a whole is called either the 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.

Many writers draw a technical distinction between the form “$p$ implies $q$” and the form “if $p$ then $q$”. In this view, writing “$p$ implies $q$” asserts the existence of a certain relation between the logical value of $p$ and the logical value of $q$ while writing “if $p$ then $q$” simply forms a compound sentence  whose logical value is a function of the logical values of $p$ and $q$. Notice that a relation is a mathematical object while a sentence, whether open or closed, is a syntactic form that exists in the domain of signs (http://planetmath.org/SignRelation).

Two factors enter at cross-purposes to our understanding at this point, and they have historically been the source of many divergent thinkers talking skew to each other on this score.

### 2.1 Definition

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 false just in case the first operand is true and the second operand is false.

In the interpretation where $0=\mathrm{false}$ and $1=\mathrm{true}$, the truth table associated with the statement “$p$ implies $q$”, symbolized as $p\Rightarrow q$, is as follows:

 Logical Implication $p$ $q$ $p\Rightarrow q$ 0 0 1 0 1 1 1 0 0 1 1 1

### 2.2 Discussion

The usage of the terms logical implication and material conditional varies from field to field and even across different contexts of discussion. One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.

The main formal object under discussion is a logical operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false. By way of a temporary name, the logical operation in question may be written as $\operatorname{cond}(p,q)$, where $p$ and $q$ are logical values. The truth table associated with this operation is as follows:

 $\operatorname{cond}:\mathbb{B}\times\mathbb{B}\to\mathbb{B}$ $p$ $q$ $\operatorname{cond}(p,q)$ 0 0 1 0 1 1 1 0 0 1 1 1

Some writers draw a firm distinction between the (the syntactic sign $\rightarrow"$) and the implication  relation (the formal object denoted by the sign $\Rightarrow"$). These writers use the phrase “if–then” for the conditional connective and the term “implies” for the implication relation. The difference  is sometimes explained by saying that the conditional is the “contemplated” relation while the implication is the “asserted” relation. In most areas of mathematics, the distinction is treated as a variation in the usage of the single sign $\Rightarrow"$, not requiring two separate signs. Not all of those who use the sign $\rightarrow"$ for the conditional connective regard it as a sign that denotes any kind of formal object, but treat it as a so-called syncategorematic sign, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign $\rightarrow"$ to denote the boolean function that is associated with the truth table of the material conditional. These considerations result in the following scheme of notation.

$\begin{matrix}p\rightarrow q&&&p\Rightarrow q\\ \mathrm{if}\ p\ \mathrm{then}\ q&&&p\ \mathrm{implies}\ q\\ \end{matrix}$

Let $\mathbb{B}=\{0,1\}$, where 0 is interpreted as the logical value $\mathrm{false}$ and 1 is interpreted as the logical value $\mathrm{true}$. The truth table shows the ordered triples of a triadic relation $L\subseteq\mathbb{B}\times\mathbb{B}\times\mathbb{B}$ that is defined as follows:

$L=\{(p,q,r)\in\mathbb{B}\times\mathbb{B}\times\mathbb{B}:\operatorname{cond}(p% ,q)=r\}$.

Regarded as a set, this triadic relation is the same thing as the binary operation:

$\operatorname{cond}:\mathbb{B}\times\mathbb{B}\to\mathbb{B}$.

The relationship between $\operatorname{cond}$ and $L$ exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.

The conditional sign $\rightarrow"$ denotes the same formal object as the function name $\operatorname{cond}$, the only difference being that the first is written infix while the second is written prefix. Thus we have the following equation:

$(p\rightarrow q)=\operatorname{cond}(p,q)$.

$L=\{(p,q,\operatorname{cond}(p,q)):(p,q)\in\mathbb{B}\times\mathbb{B}\}$.

Associated with the 3-adic relation $L$ is a 2-adic relation $L_{..1}\subseteq\mathbb{B}\times\mathbb{B}$ that is called the fiber of $L$ with 1 in the third place. This object is defined as follows:

$L_{..1}:=\{(p,q)\in\mathbb{B}\times\mathbb{B}:(p,q,1)\in L\}$.

The same object is achieved in the following way. Begin with the 2-adic operation:

$\operatorname{cond}:\mathbb{B}\times\mathbb{B}\to\mathbb{B}$.

Form the 2-adic relation that is called the fiber of $\operatorname{cond}$ at 1, notated as follows:

$\operatorname{cond}^{-1}(1)\subseteq\mathbb{B}\times\mathbb{B}$.

This object is defined as follows:

$\operatorname{cond}^{-1}(1)=\{(p,q)\in\mathbb{B}\times\mathbb{B}:\operatorname% {cond}(p,q)=1\}$.

The implication sign $\Rightarrow"$ denotes the same formal object as the relation names $L_{..1}"$ and $\operatorname{cond}^{-1}(1)"$, the only differences being purely syntactic. Thus we have the following logical equivalence:

$(p\Rightarrow q)\iff(p,q)\in L_{..1}\iff(p,q)\in\operatorname{cond}^{-1}(1)$.

This completes the derivation of the mathematical objects that are denoted by the signs $\rightarrow"$ and $\Rightarrow"$ in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the sign $\rightarrow"$ is reserved for function notation, it is common to see the sign $\Rightarrow"$ being used for both concepts.

## 4 Document history

Portions of the above article are adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

• http://www.mywikibiz.com/Logical_implicationLogical implication, http://www.mywikibiz.com/MyWikiBiz.

• http://www.getwiki.net/-Logical_ImplicationLogical implication, http://www.getwiki.net/GetWiki.

• http://www.wikinfo.org/index.php/Logical_implicationLogical implication, http://www.wikinfo.org/Wikinfo.

• http://en.wikipedia.org/w/index.php?title=Logical_implication&oldid=77109738Logical implication, http://en.wikipedia.org/Wikipedia.

 Title logical implication Canonical name LogicalImplication Date of creation 2013-03-22 17:55:15 Last modified on 2013-03-22 17:55:15 Owner Jon Awbrey (15246) Last modified by Jon Awbrey (15246) Numerical id 24 Author Jon Awbrey (15246) Entry type Definition Classification msc 03F03 Classification msc 03C05 Classification msc 03B22 Classification msc 03B05 Synonym material implication Related topic Implication Related topic LogicalConnective Defines material conditional Defines antecedent Defines consequent Defines conditional Defines consequence