# 10 tips for sharpening your logical thinking

Logical thinking helps you discern the truth, solve problems, and make good decisions -- unless your logic is flawed. Here are a few principles that will help ensure correct reasoning.

Logical thinking is critical for IT professionals, managers, and executives. You must be able to diagnose problems end users are having. You must be able to evaluate vendor claims. You must be able to refute your boss when he or she turns down your request for a raise or promotion. The following concepts will help you hone your logical thinking skills.

## 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 is true: "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.

Calvin Sun is an attorney who writes about technology and legal issues for TechRepublic.

bigjude

Back in the '60s, having never even seen a waterproof camera housing, I was overcome by a wave while filming with a Bolex H16 (windup clockwork powered 16mm movie camera circa 1936-47.) Shortly afterwards, I dropped it into a bucket of fresh water, swished it around a lot, changed the water several times and then put it in the sun to drain and dry. Unbelievably, when dry, it worked perfectly. More recently, an acquaintance had a similar experience with a modern hi-specced camcorder. It was ruined. When dropped in the sea, clockwork powered movie cameras survive and work again. When dropped in the sea, camcorders don't. Ergo, clockwork is a more reliable power option for movie cameras than electronics. Funny, and not really logical, but probably true.

sparent

I do remember having a hard time with the following: if p is FALSE, then p → q is TRUE.

mig25jet

You could have warned readers that it was written by an Attorney!

Especially in the light of this article If I read Techrepublic, I will become educated But: Reading Techrepublic is neither a necessary or sufficient condition to becoming educated. But it is a much more interesting means of refreshing my knowledge of many subjects, and I appreciate everybody's comments.

ayambeng

Thanks for the precision and simplicity in your explanations. Very interesting

Thatmanstu

I was not aware of the origin of "mind your p's and q's" .

Charles Bundy

logic must not be a necessary survival trait in modern times. Good article tho, thanks!

Tony Hopkinson

Most logic errors, come from mixing proposition and conclusion....

AnsuGisalas

logic only performs correctly when supplied with exactly true premises. How many exactly true and non-trivial statements can you come up with? Try it.

To enable one to easier gets one's head around the relationship between sufficient and necessary conditions it is better to define a necessary condition like this: A necessary condition is one that, if NOT met, absolutely guarantees the ABSENCE of a certain result. Try it, you will see that it works.

Spitfire_Sysop

Did you recently have a run in with an illogical cellphone repair tech? P.S. I used to work with a guy who dropped a Palm Treo 650 in to the ocean while it was on and it still worked flawlessly when he dried it out.

kashyap.bikram

Was the camera labeled as waterproof. Usually the environment/conditions are assumed. We need to be careful whether the assumptions are valid. Example: P -> Q. If watch is waterproof, and you go swimming wearing it, it will not be spoiled. Assumption: the watch is waterproof. If you mix the two thing, it becomes, If you go swimming wearing the watch, it will not be spoiled. But it the assumption is wrong, then the logic is flawed. Always think about as many variables as possible while forming the logic, and leave less to assumptions. Else someone may contradict using an assumption which invalidates your logic.

mike.laing

All men like fish. (assumption, demonstrabley false as I don't like fish) Jack is a man. Therefore Jack likes fish. P always causes q if p is FALSE, then p -> q is logically VALID, or true.

Dogcatcher

... it is irrelevant that Mr Sun is an attorney. Classic logic, or symbolic logic, is most likely to be in the curriculum of the philosophy or mathematics department. Law schools should teach it, but most don't. Classic logic is a great tool for dissecting sloppy arguments. Well worth studying.

mike.laing

, you will become educated. You read tech republic Therefore, you will become educated. It IS a sufficient condition, as defined in the premise.

hippiekarl

...but that phrase came from typesetters, who had to literally 'mind them' as they looked so similar. 'P' and 'Q' were the movable-type letters p and q before they were initials in 'logic'; hence "minding one's p's and q's" means 'paying attention to detail', not (necessarily) 'keeping one's premises and conclusions in order'.

Kent Lion

...this is on the right track, but try "many errors of logic come from improper understanding/use of logic, and many more come from applying logic to assumptions". Another rule required for this article to do any good would be "check your assumptions": The application of logic is the application of a set of tools. Alone, that accomplishes nothing; you must know how to use tools (what this article's tips are about), and you must use them on the proper "materials" (data, including assumptions, are the materials to which logic is applied). You can have the best set of carpenter's tools available, but if you don't know how to use them, you'll fail as a carpenter; whereas a really good carpenter can work wonders with only adequate tools. At the same time the best carpenter on earth will not be able to build a good structure of inferior materials or on an inadequate foundation. It's the same with everything, including computers. The most efficient program of ironclad logic will produce faulty results if applied to faulty data (garbage in - garbage out). Where human thought most often fails is in the selection of the data that thought is applied to. Humans appear to be the only creature with the ability to imagine things that were not, are not, and cannot be (i.e., unrelated to anything in reality, defined as "that which doesn't go away when you stop believing in it"). For some reason, this ability comes with the ability to mentally replace reality with those imaginings; and perhaps this is why people do not always adequately discriminate among all the variations from testable truth to blind faith in their "data".

santeewelding

When you turn the thing on itself. I don't recommend it. Doing so would get you banished from here with record-breaking negative votes.

Michael Jay

it would follow that if it actually worked to start with, a dip in the ocean should not be a problem. Sorry, we deployed a large number of Treo's and had nothing but trouble, my bias is showing.

Gandolfwzrd

That's because there is little correlation between the practice of law and truth or logic.

rocketmouse

I butter my bread on the other side...

Jtempys

That moveable type-setting existed before the mathematical representations of logic, though I can concede that the origin of the catch phrase might not be related to logic.

Tony Hopkinson

int Foo(int i) { try { return numbers[i]; } catch { return -1; } } No need to test i so see if it's in bounds..... 99% of robust coding which is ineffiicient performance wise, is not assuming i is in bounds.

seanferd

If I read you correctly, I believe you may be in error, which is the reason for my uncertainty. I see plenty of the involute here. And everywhere else.

AnsuGisalas

the word Treo is most acutely associated with headaches...

Michael Jay

that the user got so mad at the unit that they punched it out, or threw it across the room where it met an untimely demise with the next wall. There were a few that worked fine but I never found the cause of the unreliability factor but most failed in a matter of weeks. We then switched to Blackberry and in support of your original post there were some that said don't take my Treo, I love it. Yes it was a fine product but the reliability was just not there, we spent too much time with the failed units, so it was on to Blackberry. Blackberry is not without problems as well, but usually battery out/battery in fixes most, or wipe and re-provision fixes the rest. I think the lack of a good web experience is what kills the Berry and we are looking at the i-phone and droid for our next change, time will tell.

Spitfire_Sysop

It was a different time. I remember the Treo 650 being the best thing out there. Everyone wanted a Treo and a bluetooth. Especially suits and image concious "players". The only problems I remember them having is they would often come back with shattered screens. If you knew how to use it you had the most powerful device available at the time.