In propositional logic, a conditional statement is an "if-then" sentence construction that links together an independent (p) and dependent (q) proposition. For any conditional statement, there are three possible permutations of it: the inverse, the converse and the contrapositive. The inverse has the same syntax as the original statement, except that both the p and q propositions are negated.
From a logician's standpoint, a "proposition" is a statement that defines an idea or concept. This statement can refer to an idea that is tangible (e.g. the dog is hungry) or abstract (e.g. the dog is noble). The only requirement is that the proposition must be either "true" or "not true." Propositional logic does not allow for an ambiguous third value.
A conditional statement consists of two propositions: a hypothesis and a conclusion. The syntax of the statement establishes a specific relationship between the propositions; namely, which proposition is the hypothesis and which is the conclusion. Generally, the syntax of a conditional statement uses an "if-then" construction, i.e. "If [hypothesis], then [conclusion]."
Translated, this statement is claiming that whenever the hypothesis has a certain value (e.g., "true"), the conclusion must have a particular value (e.g. "true"). For example, if the proposition "it is raining" is true, then the proposition "the ground is wet" must be true.
The Hypothesis, Or "P"
In textbooks, logicians use the letter "P" to represent the hypothesis of a conditional statement. Within this context, the hypothesis can be considered "independent" because its value is not predicated on the value of the other proposition.
The Conclusion, Or "Q"
Meanwhile, the letter "Q" is reserved for the conclusion of a conditional statement. The conclusion is dependent on the hypothesis, but only for certain values of the latter. For example, if "it is raining" is true, then "the ground is wet" must also be true. However, if "it is raining" is false (i.e. there's no water falling from the sky), then "the ground is wet" could be either true or false (i.e. maybe a dam broke and flooded the area). The original conditional statement only defines the relationship for one value of the hypothesis.
In the inverse of a conditional statement, the values of both the hypothesis and conclusion are inverted. For example, if the original statement was "if it is raining, then the ground is wet," the inverse of that statement would be "if it is not raining, then the ground will not be dry."
Note: Constructing the inverse isn't about switching all of the propositions' values to "false." Rather, the essence of the inverse statement is "inversion," which meaning switching "true" values to "false" as well as "false" values to "true." For example, the inverse of the statement, "If I do not have a coat, then I will get hypothermia," would be "If I do have a coat, then I will not get hypothermia."
- Photo Credit diversitÃ image by UBE from Fotolia.com
What Is a Conditional Statement?
Conditional statements are a part of any programming language. Conditional statements execute lines of code only if the condition results to true....
What Is a Conditional Statement in Math?
A conditional statement in math is a statement in the if-then form. Conditional statements, often called conditionals for short, are used extensively...
What Is the Converse of a Conditional Statement?
In traditional logic, mathematicians use capital letters as placeholders for “simple statements,” which are simple declarations of fact. For example, "P" could...
What Is a Propositional Statement?
In logic and philosophy, a propositional statement is a sentence or expression that is either true or false. Generally speaking, a statement...
How to Make a Hypothesis Statement
A hypothesis statement predicts a relationship between two variables. Writing a hypothesis should always precede any actual experiments and is an important...
Difference Between Inverse & Converse
It is not uncommon to hear the words "inverse" and "converse" used interchangeably, especially in their noun and adverb forms. You may...
Examples of Inverse Relationships in Math
Inverse relationships are the mathematical equivalent of a see-saw. In an inverse relationship, when one number goes up, the other goes down....
What Is the Multiplicative Inverse Property?
The multiplicative inverse property states that for every "a" there exists a number such that "a X 1/a = 1." Practice math...
How to Find the Inverse of a Function
An inverse function changes the operations of what the original function will do. Replace every X and Y with the other when...