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| Logical Thinking |
1: The conditional statement
Have you ever dropped your smartphone into water? Not good, correct? Let’s assume, for purposes of this article, that every time it happens, without exception, that phone is ruined. In other words, this statement istrue: “If you drop your smartphone into water, then it will become ruined.”
This statement, in logic, is known as a conditional statement. The first part of the sentence states a condition or requirement. The second part of the sentence states the result of that condition. If the condition is fulfilled, the result will occur. If you’ve done any application programming, you doubtless have worked with conditional statements. The principles of conditional statements are the same for logical thinking.
2: Understanding premise and conclusion shorthand
The two parts of a conditional statement have specific terms with respect to logic. The first part is called a premise, and the second part is called a conclusion. Within a conditional statement, if a premise is true, the conclusion will be too, because it follows, or results from, the truth of the premise.
Sometimes, in shorthand, you will see the abbreviations “p” and “q” for “premise” and “conclusion,” respectively. The causal relationship (the “then”) is indicated by an arrow: →. Here, “p” would represent “If you drop your smartphone into water,” “q” would represent “the smartphone will become ruined,” and → would represent the “then.” The general nature of a conditional statement can be represented as p → q.
Once we understand the structure of an original conditional statement in terms of p and q, we can understand three other statements related to it. They are the converse, the inverse, and the contrapositive. Knowing these three is important to avoid faulty reasoning and to detect faulty reasoning by others.
3: The converse statement
The converse of the original conditional statement simply reverses the premise and the conclusion. In shorthand terms, therefore, the converse is q → p. In our smartphone example, the converse statement would be: “If your smartphone is ruined, then it was because you dropped it into water.”
As you can see, in this case the converse is not true, because a smartphone can be ruined in many other ways besides dropping it into water. Similarly, though someone who lives in Florida lives in the United States, not everyone who lives in the United States lives in Florida. Assuming that the converse is true, in fact, leads to the fallacy of the “false syllogism”:
- If a phone is dropped into water, it is ruined.
- John’s phone is ruined.
- Therefore, John’s phone must have been dropped into water.
An example of similar potentially faulty reasoning is the following:
- Every computer that has virus x has symptom y.
- Joe’s computer has symptom y.
- Therefore, Joe’s computer has virus x.
This reasoning is faulty for the same reason — namely, that a computer could have symptom y for other reasons. A correct analysis would be the following:
- If a computer has virus x, then it has symptom y.
- Joe’s computer has virus x.
- Therefore, Joe’s computer has symptom y.
The false syllogism is better illustrated this classic way:
- Dogs have four legs.
- Cats have four legs.
- Therefore, dogs are cats.
4: The inverse statement
The inverse of the original statement keeps the original premise and original conclusion but negates each one. In shorthand, the inverse is ~p → ~q.
The inverse of the smartphone statement would be: “If you do not drop your smartphone into water, your smartphone will not become ruined.” Sometimes, the inverse is true. But other times, such as with our example, it isn’t. A smartphone can be ruined in many ways. Therefore, even if we refrain from dropping the phone into water, it doesn’t prevent other bad things from happening to it. The inverse of the virus statement would be: “If a computer does not have virus x, it will not have symptom y.” This statement might not be true if symptom y can result from reasons other than virus x.
Be careful of inverse reasoning.
5: The contrapositive statement
The contrapositive is either the converse of the inverse or the inverse of the converse. That is, it involves a negation of both the premise and the conclusion, along with their reversal. Our smartphone contrapositive would be: “If your smartphone is not ruined, then you did not drop it into water.” The virus contrapositive would be “If a computer does not have symptom y, then it does not have virus x.” In shorthand, the contrapositive is ~q → ~p.
Assuming the truth of the original conditional statement, the contrapositive is the only alternative statement that will always be true.
6: Necessary conditions
Closely related to the conditional and related statements are the ideas of necessary conditions and sufficient conditions.
A necessary condition is one that must be met for a certain result to be achieved. For a smartphone not to be ruined, it must be kept out of water. Therefore “keeping a smartphone out of water” is necessary to prevent it from being ruined. The absence of virus x is necessary to have assurance that a computer does not have symptom y.
I know the objections you are raising right now, but keep reading for my further points.
7: Sufficient conditions
A sufficient condition is one that, if met, absolutely guarantees the occurrence of a certain result — that is, a result that is dependent on that condition. Dropping a smartphone into water is sufficient for ruining that phone.
Doing so guarantees that the phone is ruined. The presence of virus x is a sufficient condition for a computer to exhibit symptom y.
8: Necessary but not sufficient
A condition can be necessary but not sufficient. Keeping your smartphone out of water is necessary for preventing its ruin. However, even if you do so, your smartphone could be ruined in other ways, such as being crushed by a car or dropped from a height. In the same way, even if virus x is absent from the computer, they system could still display symptom y for some other reason. Therefore, keeping a smartphone out of water, and keeping virus x off a computer are necessary but not sufficient conditions for preventing smartphone ruin or the presence of symptom y.
9: Sufficient but not necessary
Similarly, a condition can be sufficient but not necessary. Dropping the smartphone into water is a sufficient condition for ruining it. However, it is not a necessary condition for ruining it. Having virus x is a sufficient condition for symptom y. However, if symptom y can arise from other causes, having virus x is not a necessary condition.
10: Neither necessary nor sufficient
A condition can be neither necessary nor sufficient with respect to a result. To prevent the ruin of your smartphone, it is neither necessary nor sufficient that its area code begin with an even number. To prevent virus x, it is neither necessary nor sufficient that the system unit have a property tag.
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