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In philosophy, logic, and mathematics, a strict conditional (also called strict implication and conditional statement) is a proposition that can be written in the form "If p, then q," where p and q are propositions. The proposition immediately following the word "if" is called the hypothesis (also called antecedent). The proposition immediately following the word "then" is called the conclusion (also called consequence). In the aforementioned form for strict conditionals, p is the hypothesis and q is the conclusion. A strict conditional is often called simply a conditional (also called an implication). Contrary to popular misconception, a strict conditional is not necessarily a material conditional in that a strict conditional is not necessarily truth-functional, while a material conditional is always truth-functional. Neither is a strict conditional the same as a logical implication in that there is no requirement for a strict conditional, as there is for a logical implication, that p and not q be logically inconsistent. Strict conditionals are often symbolized using an arrow (→) as p → q (read "p implies q"). The arrow is also used to denote material implication, which has made the arrow a rather ambiguous form of denotation. Since the arrow can have different meanings in different contexts, its meaning should always be made clear within the context. The strict conditional in symbolic form is as follows:

1. ${\displaystyle p\rightarrow q}$, where "${\displaystyle \rightarrow }$" represents strict implication. Or, in the original notation of Clarence Irving Lewis,
2. ${\displaystyle p\prec q\equiv \Box (p\rightarrow q)}$, where "${\displaystyle \prec }$" represents strict implication and "${\displaystyle \rightarrow }$" represents material implication.

As a proposition, a strict conditional is either true or false. A strict conditional is true if and only if the conclusion is true in every case that the hypothesis is true. A strict conditional is false if and only if a counterexample to the strict conditional exists. A counterexample to a strict conditional exists if and only if there is a case in which the hypothesis is true, but the conclusion is false.

A strict conditional p → q is logically equivalent to the modal claim "It is necessary that it is not the case that: p and not q." The conditional p → q is false if and only if it is not necessary that it is not the case that: both p is true and q is false. In other words, p → q is true if and only if it is necessary that: p is false or q is true (or both). Yet another way of describing the conditional is that it is equivalent to: "It is necessary that: not p, or q (or both)."

Examples of strict conditionals include:

1. If I am running, then my legs are moving.
2. If a person makes lots of jokes, then the person is funny.
3. If the Sun is out, then it is midnight.
4. If you locked your car keys in your car, then 7 + 6 = 2.

Variations of the strict conditional

The strict conditional "If p, then q" can be expressed in many ways; among these ways include:

1. If p, then q. (called "if-then" form)
2. If p, q.
3. p implies q.
4. p only if q. (called "only-if" form)
5. p is sufficient for q.
6. A sufficient condition for q is p.
7. q if p.
8. q whenever p.
9. q when p.
10. q every time that p.
11. q is necessary for p.
12. A necessary condition for p is q.
13. q follows from p.
14. q unless ¬p.

The converse, inverse, contrapositive, and biconditional of a strict conditional

The strict conditional "If p, then q" is related to several other strict conditionals and propositions involving propositions p and q.

The converse

The converse of a strict conditional is the strict conditional produced when the hypothesis and conclusion are interchanged with each other. The resulting conditional is as follows:

• ${\displaystyle q\rightarrow p}$

The inverse

The inverse of a strict conditional is the strict conditional produced when both the hypothesis and the conclusion are negated. The resulting conditional is as follows:

• ${\displaystyle \lnot p\rightarrow \lnot q}$

The contrapositive

The contrapositive of a strict conditional is the strict conditional produced when the hypothesis and conclusion are interchanged with each other and then both negated. The resulting conditional is as follows:

• ${\displaystyle \lnot q\rightarrow \lnot p}$

The biconditional

The biconditional of a strict conditional is the proposition produced out of the conjunction of the strict conditional and its converse. When written in its standard English form, the hypothsis and conclusion are joined by the words "if and only if." The biconditional of a strict conditional is equivalent to the conjunction of the strict conditional and its converse. The resulting proposition is as follows:

• ${\displaystyle p\leftrightarrow q}$; or equivalently,
• ${\displaystyle (p\rightarrow q)\land (q\rightarrow p)}$

Logical equivalencies of the strict conditional

The strict conditional is a modal claim, and as such it requires the use of modal operators. Namely, it requires the use of the necessary operator (□). The following are some logical equivalencies to the strict conditional "If p, then q." In this section, the arrow (${\displaystyle \rightarrow }$) represents only the strict conditional, and not the material conditional.

1. ${\displaystyle p\rightarrow q\equiv \Box \lnot (p\land \lnot q)}$
2. ${\displaystyle p\rightarrow q\equiv \Box (\lnot p\lor q)}$
3. ${\displaystyle p\rightarrow q\equiv \lnot q\rightarrow \lnot p}$; The contrapositive of a strict conditional is equivalent to the strict conditional itself.
4. ${\displaystyle q\rightarrow p\equiv \lnot p\rightarrow \lnot q}$; The converse of a strict conditional is equivalent to the inverse of the strict conditional.

Distinction between strict conditionals, material conditionals, and logical implications

The terms "conditional statement," "material conditional," and "logical implication" are often used interchangeably. Since, in logic, these terms have nonequivalent definitions, using them interchangeably often creates strong ambiguities. In fact, these ambiguities are so deeply rooted that they have generated a very popular misconception that the terms "conditional statement," "material conditional," and "material implication" all mean the same thing. This is far from true; the three terms are not all equivalent. Only the latter two have equivalent meanings.

The truth table

Strict conditionals and material conditionals are associated with the same truth table, given below. How exactly each is related to this truth table, however, is different.

The strict conditional vs. the material conditional

The difference between a strict conditional p → q and a material conditional p → q is that a strict conditional need not be truth-functional. While the truth of a material conditional is determined directly by the truth table, the truth of a strict conditional is not. The truth of a strict conditonal cannot in general be determined merely through classical logic. The strict conditional is a modal claim, and as such it requires the use of the branch of logic known as modal logic. A strict conditional p → q is equivalent to "It is necessary that it is not the case that: p and not q. A material conditional, on the other hand, is equivalent to "It is not the case that: p and not q. Note the lack of the clause "It is necessary that" in the latter equivalency. In general, a strict conditional is a necessary version of its corresponding material conditional. C.I. Lewis was the first to develop modern modal logic in order to express the general conditional statement properly.

The strict conditional vs. the logical implication

The difference between a strict conditional p → q and a logical implication p → q is that a strict conditonal need not have a valid logical form. Once again, a strict conditional statement is a modal claim equivalent to "It is necessary that it is not the case that: p and not q. A logical implication, on the other hand, is equivalent to "p and not q are logically inconsistent," which would be due to their abstract logical form. This requirement does not exist for strict conditionals.

Example

To show clearly the difference between the strict conditional p → q, the material conditional p → q, and the logical implication p → q, consider the following ambiguous statement in which hypothesis p is "Today is Tuesday," and conclusion q is "5 + 5 = 4":

(1) p → q

The strict conditional expressed by (1) is false: a counterexample exists. It can be Tuesday, but 5 + 5 still does not equal 4. In fact, 5 + 5 never equals 4. The material conditional expressed by (1) is true every day execpt Tuesday. This is because on every day except Tuesday, both the hypothesis and the conclusion are false, hence the material implication is true. This corresponds to the last row on the truth table for material conditionals. On Tuesday, however, the hypothesis is true, but the conclusion is false, hence the material conditional is false. This corresponds to the second row on the truth table for material conditionals. The logical implication expressed by (1) is false: Γ = {"Today is Tuesday."} does not entail "5 + 5 = 4," since "Today is Tuesday" and "5 + 5 ≠ 4" are not logically inconsistent. Both of the former statements could (in theory) be true when considering only their abstract logical form; their logical form being p and not q. As can be seen, the same syntactic statement p → q can have different truth values, depending on whether the statement is expressing a strict conditional, a material conditional, or a logical implication. Therefore, there is a fundamental difference between a strict conditional, a material conditional, and a logical implication.

C.I. Lewis and his unprecedented approach to strict conditionals

C.I. Lewis took a slightly different approach to expressing strict conditionals. C.I. Lewis was interested in creating a symbolic version of the strict conditional. He knew there was a difference between strict conditionals and material conditionals, and he wanted to make that difference clear. Instead of using the arrow (${\displaystyle \rightarrow }$) for both strict conditionals and material conditionals (as many logicians do today), C.I. Lewis decided to devise a series of logical systems in which use of the arrow was restricted to only material conditionals. He used a different symbol to represent strict conditionals, a hook (${\displaystyle \prec }$). C.I. Lewis' unambiguous approach to setting apart strict conditionals from material conditionals makes the difference between the two clear and straightforward. His approach, however, has not found its way into mainstream popular literature very well, as many authors fail to make any syntactic difference between the two nonequivalent concepts.

As stated earlier, a strict conditional is a necessary version of its corresponding material conditional. Using C.I. Lewis' notation, this gives us the following useful equivalency:

• ${\displaystyle p\prec q\equiv \Box (p\rightarrow q)}$

This equivalency clearly shows the difference between the strict conditional "If p, then q," and the material conditional "p → q."

Paradoxes of strict implication

Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems. The following sentence, for example, is not correctly formalized by a strict conditional:

If Bill Gates graduated in Medicine, then 2 + 2 = 4.

Using strict conditionals, this sentence is expressed as:

${\displaystyle \Box }$ (Bill Gates graduated in Medicine ${\displaystyle \rightarrow }$ 2 + 2 = 4), where "${\displaystyle \rightarrow }$" represents material implication and not strict implication.

In modal logic, this formula means that, in every possible world where Bill Gates graduated in medicine, it holds that 2 + 2 = 4. Since 2 + 2 is equal to 4 in all possible worlds, this formula is true.

Some logicians view this situation as paradoxical, and to avoid it they have created counterfactual conditionals. Others, such as Paul Grice, have used conversational implicature to argue that, despite apparent difficulties, the material conditional is just fine as a translation for the natural language 'if...then...'. Others still have turned to relevant logic to supply a connection between the antecedent and consequent of provable conditionals.

Notes

1. Larson, Boswell, et al. 2007, p. 79
2. Hardegree 2009, p. I-9 and III-18
3. Larson, Boswell, et al. 2007, p. 79
4. Rosen 2007, p. 6
5. Larson, Boswell, et al. 2007, p. 79
6. Hardegree 1994, p. 42
7. Larson, Boswell, et al. 2007, p. 79
8. Rosen 2007, p. 6
9. Larson, Boswell, et al. 2007, p.95
10. Rosen 2007, p. 6
11. Hardegree 1994, p. 41-44
12. Barwise and Etchemendy 2008, p. 178-181
13. Larson, Boswell, et al. 2007 p.79-80
14. Larson, Boswell, et al. 2007 p. 94
15. Larson, Boswell, et al. 2007, p. 94-95
16. Rosen 2007, p. 6
17. Barwise and Etchemendy 2008, p. 178-181
18. Larson, Boswell, et al. 2007 p. 94
19. Hardegree 2009, p. I-9 and III-18
20. Larson, Boswell, et al. 2007, p. 80
21. Larson, Boswell, et al. 2007, p. 80
22. Larson, Boswell, et al. 2007, p. 80
23. Larson, Boswell, et al. 2007, p. 80
24. Hardegree 2009, p. I-9
25. Rosen 2007, p. 25
26. Hardegree 2009, p. I-9
27. Rosen 2007, p. 25
28. Hardegree 2009, p. I-9
29. Rosen 2007, p. 25
30. Hardegree 2009, p. I-9
31. Rosen 2007, p. 25
32. Rosen 2007, p. 6
33. Larson, Boswell, et al. 2001, p. 80
34. Larson, Boswell, et al. 2007, p. 79
35. Larson, Boswell, et al. 2001, p.80
36. Larson, Boswell, et al. 2007, p. 80
37. Rosen 2007, p. 8
38. Larson, Boswell, et al. 2007, p. 80
39. Rosen 2007, p. 8
40. Larson, Boswell, et al. 2007, p. 80
41. Rosen 2007, p. 8
42. Larson, Boswell, et al. 2007, p. 80
43. Rosen 2007, p. 8
44. Larson, Boswell, et al. 2007, p. 82
45. Rosen 2007, p. 9
46. Hardegree 2009, p. I-9
47. Hardegree 2009, p. I-9
48. Hardegree 2009, p. I-9
49. Rosen 2007, p. 25
50. Rosen 2007, p. 6
51. Barwise and Etchemendy 2008, p. 178-181
52. Larson, Boswell, et al. 2007, p. 79-80
53. Rosen 2007, p. 6
54. Hardegree 1994, p. 41-44
55. Larson, Boswell, et al. 2001, p. 71
56. Barwise and Etchemendy 2008, p. 176-223
57. Hardegree 2009, p. I-9
58. Larson, Boswell, et al. 2007, p. 79-80, 94-95
59. Rosen 2007, p. 6
60. Hardegree 1994, p. 41-44
61. Larson, Boswell, et al. 2001, p. 71
62. Barwise and Etchemendy 2008, p. 176-223
63. Hardegree 2009, p. I-9
64. Larson, Boswell, et al. 2007, p. 79-80, 94-95
65. Rosen 2007, p. 6
66. Hardegree 1994, p. 41-44
67. Larson, Boswell, et al. 2001, p. 71
68. Barwise and Etchemendy 2008, p. 176-223
69. Rosen 2007, p. 6
70. Larson, Boswell, et al. 2007, p. 94-95
71. Rosen 2007, p. 6
72. Hardegree 1994, p. 44
73. Barwise and Etchemendy 2008, p. 178
74. Hardegree 2009, p. I-9
75. Larson, Boswell, et al. 2007, p. 79-80, 94-95
76. Rosen 2007, p. 6
77. Hardegree 1994, p.44
78. Barwise and Etchemendy 2008, p. 178
79. Hardegree 2009, p. I-9
80. Larson, Boswell, et al. 2007, p. 80
81. Hardegree 2009, p. I-9
82. Hardegree 2009, p. I-9
83. Hardegree 2009, p. I-9
84. Barwise and Etchemendy 2008, p. 178
85. Hardegree 2009, p. I-9
86. Hardegree 2009, p. I-9
87. Hardegree 1994, p. 41-44
88. Larson, Boswell, et al. 2007, p. 79-80
89. Rosen 2007, p. 6
90. Barwise and Etchemendy 2008, p. 94-113
91. Hardegree 2009, p. I-9
92. Barwise and Etchemendy 2008, p. 94-113
93. Larson, Boswell, et al. 2007, p. 79-80
94. Rosen 2007, p. 6
95. Hardegree 2009, p. I-9
96. Hardegree 2009, p. I-9
97. Hardegree 2009, p. I-9
98. Larson, Boswell, et al. 2007, p. 94-95
99. Larson, Boswell, et al. 2001, p. 87
100. Barwise and Etchemendy 2008, p. 178
101. Rosen 2007, p.6
102. Hardegree 2009, p. I-9
103. Hardegree 2009, p. I-9
104. Larson, Boswell, et al. 2007, p. 79-80, 94-95
105. Rosen 2007, p.6
106. Hardegree 1994, p. 41-44
107. Barwise and Etchemendy 2008, p. 179

References

• Hardegree, Gary. Introduction to Modal Logic. UMass Amherst Department of Philosophy, 2009. Web. 18 December 2011 <http://people.umass.edu/gmhwww/511/text.htm>
• Larson, Boswell, et al. Geometry. Boston: McDougal Littell, 2007. Print.
• Rosen, Kenneth H. Discrete Mathematics and Its Applications, Sixth Edition. Boston: McGraw-Hill, 2007. Print.
• Hardegree, Gary. Symbolic Logic: A First Course (2nd Edition). UMass Amherst Department of Philosophy, 1994. Web. 18 December 2011 <http://courses.umass.edu/phil110-gmh/text.htm>
• Larson, Boswell, et al. Geometry. Boston: McDougal Littell, 2001. Print.
• Barwise, Jon, and John Etchemendy. Language, Proof and Logic. Stanford: CSLI (Center for the Study of Language and Information) Publications, 2008. Print.
• Edgington, Dorothy, 2001, "Conditionals," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.

For an introduction to non-classical logic as an attempt to find a better translation of the conditional, see:

• Priest, Graham, 2001. An Introduction to Non-Classical Logic. Cambridge Univ. Press.

For an extended philosophical discussion of the issues mentioned in this article, see:

• Mark Sainsbury, 2001. Logical Forms. Blackwell Publishers.
• Jonathan Bennett, 2003. A Philosophical Guide to Conditionals. Oxford Univ. Press.

See also

• Material conditional
• Logical implication
• Modal logic
• C.I. Lewis
• Counterfactual conditional
• Indicative conditional
• Propositional logic

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• Conditionals
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• Necessity
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