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Assumptions
Necessary Assumptions vs. Sufficient Assumptions
- Definition of an assumption: An unstated premise or condition that must be true for the conclusion to hold.
- Importance of assumptions in arguments: Assumptions form the backbone of logical reasoning and argument construction.
- Brief explanation of necessary and sufficient assumptions: These are two types of assumptions that serve different roles in arguments.
Necessary Assumptions – Definition
- Explanation of necessary assumptions: These are assumptions that must be true for the conclusion to be valid. Without them, the argument falls apart.
- Key features of necessary assumptions: They are essential for the argument but alone may not be enough to guarantee the conclusion.
Necessary Assumptions – Examples
- Argument 1: All dogs bark. My pet, Rover, is a dog.
- Necessary Assumption: Rover is capable of barking.
- Argument 2: If it rains, the picnic will be canceled. The picnic was canceled.
- Necessary Assumption: It rained.
- Argument 3: Every employee who works overtime gets a bonus. Jane got a bonus.
- Necessary Assumption: Jane worked overtime.
- Argument 4: Vegetarians do not eat meat. Sam does not eat meat.
- Necessary Assumption: Sam is a vegetarian.
- Argument 5: All citizens must pay taxes. John pays taxes.
- Necessary Assumption: John is a citizen.
Necessary Assumptions – Discussion
- Recap of necessary assumptions: Necessary assumptions are crucial for the validity of the argument.
- Common scenarios where necessary assumptions are made: Everyday reasoning, hypothesis testing in research, legal arguments, etc.
Sufficient Assumptions – Definition
- Explanation of sufficient assumptions: These are conditions that, if met, guarantee the truth of the conclusion.
- Key features of sufficient assumptions: They are enough to ensure the conclusion but aren’t always necessary for the conclusion to be true.
Sufficient Assumptions – Examples
- Argument 1: John studied hard for the math test. Therefore, John will pass the math test.
- Conclusion: John will pass the math test.
- Sufficient Assumption: Studying hard for the math test guarantees passing the test.
- Argument 2: It snowed heavily last night. Therefore, school will be canceled today.
- Conclusion: School will be canceled today.
- Sufficient Assumption: Heavy snowfall the night before results in school cancellation.
- Argument 3: Sarah eats a balanced diet and exercises regularly. Therefore, Sarah is healthy.
- Conclusion: Sarah is healthy.
- Sufficient Assumption: Regular exercise and a balanced diet ensure good health.
- Argument 4: Our team won the final game of the season. Therefore, we will top the league.
- Conclusion: We top the league.
- Sufficient Assumption: Winning the final game of the season ensures topping the league.
- Argument 5: I can save $500 a month. Therefore, I can buy a car next year.
- Conclusion: I can buy a car next year.
- Sufficient Assumption: Saving $500 a month is enough to buy a car next year.
In each of these arguments, the premise leads to a conclusion. The sufficient assumption in each case is the unstated premise that the condition described in the premise leads directly and unerringly to the conclusion.
Sufficient Assumptions – Discussion
- Recap of sufficient assumptions: Sufficient assumptions provide enough conditions to ensure the conclusion.
- Common scenarios where sufficient assumptions are made: Planning, risk analysis, scientific predictions, etc.
Necessary vs. Sufficient Assumptions
Necessary assumptions are premises that must be true for the conclusion to be valid. If these assumptions are false or not met, the argument would fall apart.
Sufficient assumptions, on the other hand, if true, are enough to guarantee the conclusion, but they are not the only way the conclusion could be true.
Consider the examples:
Argument 1: John studied hard for the math test. Therefore, John will pass the math test.
- Sufficient Assumption: Studying hard for the math test guarantees passing the test.
- However, studying hard is not a necessary condition for passing. John might also pass if, for example, he is already highly proficient in math, he gets lucky in guessing some answers, or the test is easier than expected.
Argument 2: It snowed heavily last night. Therefore, school will be canceled today.
- Sufficient Assumption: Heavy snowfall the night before results in school cancellation.
- However, heavy snowfall is not a necessary condition for a school cancellation. School could be canceled for other reasons, like a power outage, a water main break, or other emergencies.
Argument 3: Sarah eats a balanced diet and exercises regularly. Therefore, Sarah is healthy.
- Sufficient Assumption: Regular exercise and a balanced diet ensure good health.
- However, a balanced diet and regular exercise aren’t necessary conditions for being healthy. Sarah could be healthy due to other factors such as genetics or perhaps she takes a medication that keeps her healthy.
Argument 4: Our team won the final game of the season. Therefore, we will top the league.
- Sufficient Assumption: Winning the final game of the season ensures topping the league.
- However, winning the final game is not a necessary condition for topping the league. The team could also top the league if the other leading teams lost their final games.
Argument 5: I can save $500 a month. Therefore, I can buy a car next year.
- Sufficient Assumption: Saving $500 a month is enough to buy a car next year.
- However, saving $500 a month isn’t a necessary condition for buying a car next year. The person could come into an inheritance, win the lottery, or earn a substantial raise at work.
These are just a few examples to illustrate why the assumptions listed are sufficient but not necessary. There are often multiple paths to a given outcome (conclusion), and a sufficient assumption represents just one of those paths.
Consider however the necessary assumption examples provided earlier:
- Argument 1: All dogs bark. My pet, Rover, is a dog.
- Necessary Assumption: Rover is capable of barking.
This assumption is necessary but not sufficient. Just because Rover is capable of barking (assuming this to be true), it doesn’t guarantee that Rover will bark.
- Necessary Assumption: Rover is capable of barking.
- Argument 2: If it rains, the picnic will be canceled. The picnic was canceled.
- Necessary Assumption: It rained.
The assumption that it rained is necessary but not sufficient. The picnic could have been canceled for a number of other reasons as well (e.g., a lack of attendees, food safety concerns, etc.).
- Necessary Assumption: It rained.
- Argument 3: Every employee who works overtime gets a bonus. Jane got a bonus.
- Necessary Assumption: Jane worked overtime.
The assumption that Jane worked overtime is necessary but not sufficient. Jane might have received a bonus for reasons other than working overtime, such as excellent performance or reaching a sales target.
- Necessary Assumption: Jane worked overtime.
- Argument 4: Vegetarians do not eat meat. Sam does not eat meat.
- Necessary Assumption: Sam is a vegetarian.
This assumption is necessary but not sufficient. Sam might not eat meat for reasons other than being a vegetarian, like health issues or personal preference.
- Necessary Assumption: Sam is a vegetarian.
- Argument 5: All citizens must pay taxes. John pays taxes.
- Necessary Assumption: John is a citizen.
The assumption that John is a citizen is necessary but not sufficient. John might be paying taxes for reasons other than being a citizen, such as owning property or conducting business in the country.
- Necessary Assumption: John is a citizen.
In each of these cases, while the assumption is necessary for the conclusion to be valid, it is not sufficient to guarantee the conclusion because there could be other contributing factors or reasons.
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Gmat anxiety, the lizard brain and common sense
Gmat anxiety is a main cause of test day under performance. It is common to tell oneself to ‘be less stressed’ or to ‘get over your anxiety’. But, there is little value in forcing yourself to calm down.
Over the course of millions of years the human brain evolved into a complex structure capable of high order thinking. But there is still a building block of the brain, deep at its core, that is primitive and reactionary that is called the ‘lizard brain’. Naturally reactionary because it was necessary for our survival: to dodge a saber tooth tiger or duck as a rival tribe member takes a swing at you with their club. We had just enough time to react but not to think.
This lizard brain still ‘exists’, and kicks in whenever we feel threatened. So, you can think of ‘gmat anxiety’ as a defense mechanism. Hardwired into our brain, it is almost impossible to consciously dominate. That is why we are unable to reason our way out of anxiousness. It is a survival mechanism. This is important to remember because on the test, there is no use telling yourself ‘not to be anxious’. The lizard brain is still at work. Sensing the threat that you’re running out of time, it will sabotage your common sense and coerce you into looking for answers. Of course, it is trying to help you, but, the lizard brain cannot see the big picture and does not have any common sense. It wants to save you and will therefore try to push you into choosing the right answer fast. That can be a good thing, but more often it is not. Especially when you still have not understood a wordy quant problem or a complex critical reasoning passage. Your first objective should be to understand problems. This is hard when you’re running out of time and the natural reflex is to look for answers instead of to understand problems.
You should not fight gmat anxiety because you can’t. Instead, learn to accept your anxiousness and work with it. Visualize your test many times during the days and weeks leading up to the test. Brainwash your brain into a procedure that will kick in once your anxiousness kicks in. Learn to accept your anxiousness and when you do, you will be more likely to remember the most important thing you have to do on the test: Understand the problem.
The lizard brain will try and force you to rush, look for answers and think about formulas before taking your time to think. Back in the day, there was no need to ponder because the problem was clear: survival. Now the problems aren’t always very clear and you must spend time to understand what they are first.
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Logical thinking on the GMAT and the Wason selection test
Logical thinking on the GMAT brings great returns. How good is your logical thinking?
Situation: In a deck of cards, any card has a letter on one side and a number on the other. The rule is that if a card has X one side then it has a 2 on the other side.
Question: What is the minimum number of cards you must turn to verify that rule?
The answer choices are:
A. Card X only
B. Cards X and 2
C. Cards X and 5
D. Cards X, 2, and 5
E. All cards
The answer is:
C. Cards X and 5.
This may seem counterintuitive but in fact is logically the correct answer.
A common response is B: Cards X and 2.
Remember however that the rule is that ‘if X then 2’, and not ‘if 2 then X’. An equivalent but more familiar argument is the following: if I am in Barcelona then I am in Europe. However, that does not mean that if I am in Europe I am in Barcelona. From this, we can conclude that turning card X is indeed necessary to prove the rule, but not turning card 2.
The trickier part is why we have to turn the card with the number 5 on it. Note that the question asks us for the movements we need to make to prove the statement ‘if X then 2’ given that every card has a letter on one side and a number on the other. What if we turn card 5 and find the letter X on it? Then the rule ‘if X then 2’ is false.
Hence, we must turn the card with number 5 on it to prove the rule.
This question is based on the Wason Selection test that is designed to test your logical thinking. The relevance to logical thinking on the GMAT is obvious. On a very abstract level, the Quant section tests logical thinking. The critical reasoning section is also a test of logical thinking, especially analysis of arguments and assumption questions. The good news is that logic is learnable skill. Remember to turn it on and improve it.
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Andre Agassi, Boris Becker, and the GMAT
The first 20 seconds of this video tell you everything you need to know to beat the GMAT and to beat anything in life really: Problem Solving.
How do you solve problems? You draw in information from your surroundings. Anything and everything available in your surroundings to understand problems you are faced with.
To follow a map, you must understand the landscape.
To fix a machine, you must know what is broken.
To beat Boris Becker’s serve, well, you have to look at his tongue.
Failure to solve problems is most often a consequence of not understanding the problem. And to understand the problem, you must observe the information available and infer as much as you can. If you begin to solve without trying to infer from the data available, you may get lost.
Try this approach on this data sufficiency problem. Infer as much as you can from whatever data is available. (Remember you have 2 min on average to solve such questions)
Is y > -1 ?
(1) |a – 2| < y + 1
(2) a < 1
Start by reminding yourself that an absolute value describes distance and thus will always be positive or 0.
So for statement (1) the worst case scenario is that |a – 2| = 0. Now let’s rearrange the inequality:
“something at least 0” – 1 < y
So, y must be greater than -1 and hence statement (1) is sufficient.
Statement (2) itself is insufficient and we do not need that information to make our inference about y.
In the case of Andre Agassi, he discovered a connection between Boris Becker’s serve and tongue to solve the serve return problem. Of course, that will not be helpful for the GMAT, but your takeaway should be to always start by understanding the problem by inferring as much as you can from your surroundings.
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The coolest concept you can learn for the GMAT
But first, try solving this question:
If you mix a 200 liter solution of 7% alcohol and a 300 liter solution of 12% alcohol, what is the resulting concentration?
A. 6 %
B. 6.5%
C. 8%
D. 10%
E. 12.5%Think about it before you scroll down
This is a weighted average GMAT type question. Traditionally you can determine the answer by using the simple formula: (200•7% + 300•12%)/(200+300) and get 10%, or you can quickly solve this problem remembering something very fundamental: the average of two quantities can never be more or less than either of these quantities, and must be between them. The new concentration we are trying to calculate is like the weighted average of either concentration. Like the average, the weighted average must lie between the two quantities, except that it will be closer to the concentration in higher quantity.
For example, the average of 123 and 134 cannot be more or less than either one and must be between them. This may be obvious in a simple situation, but harder to notice when mixtures or ratios are thrown into the problem.
In the case of the problem above, you can quickly eliminate all answer choices except C and D because only C and D lie within the range. Because there is more of the 12% solution, we know that the answer must be closer to 12% than 7%, leaving D as the possible answer. In this case, no calculations were even necessary.
Think of the GMAT as the simulation of small business situations in which you must consider all available information, including the possible solutions, and make a decision in a limited amount of time.
A more challenging application can be in more abstract situations in which one or two extra inference steps are required. Consider this problem:
In a classroom, some students are signed up for extra-curricular activities and some are not. Is the ratio of boys signed up to the extra-curricular activities to the boys who are not less than the ratio of the students signed up to extra-curricular activities to the students who are not?
(1) More than 2/3 of the boys are signed up for extra-curricular activities and more than 3/4 of the girls are not signed up for extra-curricular activities.
(2) The ratio of girls who are signed up for extra-curricular activities to girls who are not is less than that ratio for the whole class.
Think about it before you scroll down
This can also be thought of as the weighted average problem above. The ratio of students taking extra-curricular activities to students who are not can be thought of as a “concentration”, say 10% alcohol, only in this case it is an x% taking extra-curricular activities. This case is exactly the same.
From statement (1) we know that this concentration is less for the girls than for the boys (more than 3/4 not signed up therefore less than 1/4 signed up), hence, the average concentration for the whole class must be less than that for the boys. – This statement on its on is Sufficient.
From statement (2) we know that this concentration for the girls is the less than that for the the whole class, therefore, the one for the boys must be higher than that of the whole class, which must lie in the middle. – This statement on its on is Sufficient.
Weighted average is perhaps the most useful concept you can use on the GMAT. Even though you are more likely to use it on the Quant section, you may see it appear on a few verbal questions in Critical reasoning as well.
Remember that the “concentration” of the whole can never be more or less than that of any of the parts that make up the whole, and must lie in between.