Logical implication

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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:

Failed to parse (Missing texvc executable; please see math/README to configure.):

Here and are propositional variables that stand for any propositions in a given language. In a statement of the form Failed to parse (Missing texvc executable; please see math/README to configure.):

the first term, Failed to parse (Missing texvc executable; please see math/README to configure.): 
is called the antecedent and the second term, Failed to parse (Missing texvc executable; please see math/README to configure.): 
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 Failed to parse (Missing texvc executable; please see math/README to configure.):

and the form Failed to parse (Missing texvc executable; please see math/README to configure.): 
 In this usage, writing Failed to parse (Missing texvc executable; please see math/README to configure.): 
asserts the existence of a certain relation between the logical value of  and the logical value of Failed to parse (Missing texvc executable; please see math/README to configure.): 
whereas writing Failed to parse (Missing texvc executable; please see math/README to configure.): 
merely forms a compound statement whose logical value is a function of the logical values of  and Failed to parse (Missing texvc executable; please see math/README to configure.): 
 This will be discussed in detail below.

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 and Failed to parse (Missing texvc executable; please see math/README to configure.): , the truth table associated with the statement Failed to parse (Missing texvc executable; please see math/README to configure.):

symbolized as Failed to parse (Missing texvc executable; please see math/README to configure.): 
appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


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 Failed to parse (Missing texvc executable; please see math/README to configure.):

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 Failed to parse (Missing texvc executable; please see math/README to configure.): 
where  and  are logical values.  The truth table associated with this operation appears below:


Failed to parse (Missing texvc executable; please see math/README to configure.):
Failed to parse (Missing texvc executable; please see math/README to configure.):


Some logicians draw a firm distinction between the conditional connective, the symbol Failed to parse (Missing texvc executable; please see math/README to configure.):

and the implication relation, the object denoted by the symbol Failed to parse (Missing texvc executable; please see math/README to configure.): 
 These logicians use the phrase if–then for the conditional connective and the term implies for the implication relation.  Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation.  In most fields of mathematics, it is treated as a variation in the usage of the single sign Failed to parse (Missing texvc executable; please see math/README to configure.): 
not requiring two separate signs.  Not all of those who use the sign Failed to parse (Missing texvc executable; please see math/README to configure.): 
for the conditional connective regard it as a sign that denotes any kind of 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 
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.

Failed to parse (Missing texvc executable; please see math/README to configure.):

Let Failed to parse (Missing texvc executable; please see math/README to configure.):

be the boolean domain consisting of two logical values.  The truth table shows the ordered triples of a triadic relation Failed to parse (Missing texvc executable; please see math/README to configure.): 
that is defined as follows:
Failed to parse (Missing texvc executable; please see math/README to configure.):

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

Failed to parse (Missing texvc executable; please see math/README to configure.):

The relationship between Failed to parse (Missing texvc executable; please see math/README to configure.):

and  exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.

The conditional sign Failed to parse (Missing texvc executable; please see math/README to configure.):

denotes the same formal object as the function name Failed to parse (Missing texvc executable; please see math/README to configure.): 
the only difference being that the first is written infix while the second is written prefix.  Thus we have the following equation:
Failed to parse (Missing texvc executable; please see math/README to configure.):

Consider once again the triadic relation Failed to parse (Missing texvc executable; please see math/README to configure.):

that is defined in the following equivalent fashion:
Failed to parse (Missing texvc executable; please see math/README to configure.):

Associated with the triadic relation is a binary relation Failed to parse (Missing texvc executable; please see math/README to configure.):

that is called the fiber of  with  in the third place.  This object is defined as follows:
Failed to parse (Missing texvc executable; please see math/README to configure.):

The same object is achieved in the following way. Begin with the binary operation:

Failed to parse (Missing texvc executable; please see math/README to configure.):

Form the binary relation that is called the fiber of Failed to parse (Missing texvc executable; please see math/README to configure.):

at Failed to parse (Missing texvc executable; please see math/README to configure.): 
notated as follows:
Failed to parse (Missing texvc executable; please see math/README to configure.):

This object is defined as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.):

The implication sign Failed to parse (Missing texvc executable; please see math/README to configure.):

denotes the same formal object as the relation names Failed to parse (Missing texvc executable; please see math/README to configure.): 
and Failed to parse (Missing texvc executable; please see math/README to configure.): 
the only differences being purely syntactic.  Thus we have the following logical equivalence:
Failed to parse (Missing texvc executable; please see math/README to configure.):

This completes the derivation of the mathematical objects that are denoted by the signs Failed to parse (Missing texvc executable; please see math/README to configure.):

and Failed to parse (Missing texvc executable; please see math/README to configure.): 
in this discussion.  It needs to be remembered, though, that not all writers observe this distinction in every context.  Especially in mathematics, where the single arrow sign Failed to parse (Missing texvc executable; please see math/README to configure.): 
is reserved for function notation, it is common to see the double arrow sign Failed to parse (Missing texvc executable; please see math/README to configure.): 
being used for both concepts.

References

  • Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
  • Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.
  • Edgington, Dorothy (2006), "Conditionals", in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Eprint.
  • Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.

Syllabus

Focal nodes

Peer nodes

Logical operators

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Relational concepts

Information, Inquiry

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Document history

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