Implication
The implication is a logical connective that establishes an “if-then” relationship between two statements. We will use an arrow to symbolize this relationship.
a → c
The statement to the left of the arrow is called the antecedent. It is also commonly referred to as the “sufficient condition”.
The statement to the right of the arrow is called the consequent. It is also commonly referred to as the “necessary condition”.
Given an implication, if the sufficient condition is true, then the necessary condition must be true. However, if the necessary condition is false, then the sufficient condition must also be false.
Let us whip up an example to tie our definitions to: “If you are in Los Angeles (L), then you are in California (C).
L → C
If the antecedent is true, that is, you are in Los Angeles, then it is sufficient to conclude that you are in California. This is because being in California is necessary for being in Los Angeles.
That said, if the necessary condition is false, then the antecedent must also be false. In this example, if you are not in California, then you cannot be in Los Angeles.
An implication then allows us to make these assertions outright:
- If the antecedent is true, the consequent must be true.
- If the consequent is false, the antecedent must be false.
Note: The first assertion is essentially the implication itself restated: a → c. The second assertion is known as the contrapositive. For any implication: a → c, its contrapositive is:
¬c → ¬a
Note: The contrapositive is a two-step process: flip the order of the variables, then negate both variables.
So, what if all you knew was that the consequent is true, can you infer the antecedent? Ask yourself, if all you knew was that you were in California, can you infer that you are in Los Angeles? The answer is no. You can potentially be in any place in California. If you said yes, you would have committed the fallacy of the converse.
Secondly, what if all we knew was that the antecedent is false, can we infer anything about the consequent? Ask yourself, if all you knew was that you were NOT in Los Angeles, can you infer anything else about your location? The answer is no, you can potentially be anywhere else, including places outside of California. If you said yes, particularly that if the antecedent is false, then the consequent must also be false, you would have committed the fallacy of the inverse.
The converse of an implication, a → c (mistaken sufficiency):
c → a
The inverse of an implication, a → c (mistaken necessity):
¬a → ¬c
We will revisit the contrapositive as well as another valid inference that can be made using an implication alone in a later lesson, but before we get to them, we need to learn about the other logical connectives.
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The implication is a logical connective that establishes an “if-then” relationship between two statements. We will use an arrow to symbolize this relationship.
a → c
The statement to the left of the arrow is called the antecedent. It is also commonly referred to as the “sufficient condition”.
The statement to the right of the arrow is called the consequent. It is also commonly referred to as the “necessary condition”.
Given an implication, if the sufficient condition is true, then the necessary condition must be true. However, if the necessary condition is false, then the sufficient condition must also be false.
Let us whip up an example to tie our definitions to: “If you are in Los Angeles (L), then you are in California (C).
L → C
If the antecedent is true, that is, you are in Los Angeles, then it is sufficient to conclude that you are in California. This is because being in California is necessary for being in Los Angeles.
That said, if the necessary condition is false, then the antecedent must also be false. In this example, if you are not in California, then you cannot be in Los Angeles.
An implication then allows us to make these assertions outright:
- If the antecedent is true, the consequent must be true.
- If the consequent is false, the antecedent must be false.
Note: The first assertion is essentially the implication itself restated: a → c. The second assertion is known as the contrapositive. For any implication: a → c, its contrapositive is:
¬c → ¬a
Note: The contrapositive is a two-step process: flip the order of the variables, then negate both variables.
So, what if all you knew was that the consequent is true, can you infer the antecedent? Ask yourself, if all you knew was that you were in California, can you infer that you are in Los Angeles? The answer is no. You can potentially be in any place in California. If you said yes, you would have committed the fallacy of the converse.
Secondly, what if all we knew was that the antecedent is false, can we infer anything about the consequent? Ask yourself, if all you knew was that you were NOT in Los Angeles, can you infer anything else about your location? The answer is no, you can potentially be anywhere else, including places outside of California. If you said yes, particularly that if the antecedent is false, then the consequent must also be false, you would have committed the fallacy of the inverse.
The converse of an implication, a → c (mistaken sufficiency):
c → a
The inverse of an implication, a → c (mistaken necessity):
¬a → ¬c
We will revisit the contrapositive as well as another valid inference that can be made using an implication alone in a later lesson, but before we get to them, we need to learn about the other logical connectives.
Want an interactive way to learn logic?
Enhance your critical thinking skills with Symbols Logic. Unlock access to the first studycard and the first level of all activities for free. Get the app now: