ADVANCED CRITICAL REASONING, Part II: Deductive Logic
My last article discussed the difference between inductive and deductive arguments. Today’s article will focus mostly on the rules of deductive arguments. I promise to nerd out on inductive reasoning in later articles.
Here’s a quick quiz on the difference between inductive and deductive logic: //www.thatquiz.org/tq/previewtest?F/Z/J/V/O3UL1355243858
To review: In a deductively “valid” argument, if all the premises are true, the conclusion must also be true, with 100% certainty. Luckily, on the GMAT, we should usually act as if the premises of an argument are true, especially when the question specifies, “the statements above are true.”
Deductive reasoning shows up most often on inference (aka “draw a conclusion”) questions and “mimic the reasoning” questions, but it often appears on other types of questions, and even on reading comprehension!
On inference questions, the correct answer will usually be deductively valid (or very very strong, inductively). An incorrect answer will be deductively invalid, with some significant probability that it could be false.
What follows are most of the formal rules of deductive reasoning (from a stack of logic textbooks I have on my shelf), with examples from the GMAT. For shorthand, I’ll label the arguments with a “P” for premise and a “C” for conclusion:
P) premise
P) premise
C) conclusion
Remember: these are not the same kind of conclusions (opinions) you’ll see on strengthen and weaken questions. Deductive conclusions are deductively “valid” facts that you can derive with 100% certainty from given premises.
EASY STUFF: Simplification/conjunction (“and” statements)
This is kind of a “duh” conclusion, but here goes: If two things are linked with an “and,” then you know each of them exist. Conversely, if two things exist, you can link them with an “and.”
Simplification:
P) A and B
C) Therefore, A
Conjunction:
P) A
P) B
C) Therefore, A and B
P) Bill is tall and was born in Texas.
P) Bill rides a motorcycle.
C) Therefore, Bill was born in Texas (simplification).
C) Therefore, at least one tall person named Bill was born in Texas and rides a motorcycle (conjunction).
CAUTION: Fallacies ahead!!
Don’t confuse “and” with “or.” (More about this later.) More importantly, don’t confuse “and” with causality, condition, or representativeness. Bill’s tallness probably has nothing to do with Texas, so keep an eye out for wrong answers that say, “Bill is tall because he was born in Texas” or “Most people from Texas ride motorcycles.”
MEDIUM STUFF: Disjunctive syllogism (“or” statements)
With “or” statements, if one thing is missing, the other must be true.
Valid conclusions:
P) A or B
P) not B (shorthand: ~B)
C) Therefore, A
P) We will go to the truck rally or to a Shakespeare play
P) We won’t go to the Shakespeare play.
C) Therefore, we will go to the truck rally.
CAUTION: Fallacies ahead!!
Unlike in the real world, “or” statements do not always imply mutual exclusivity, unless the argument explicitly says so. For example, in the above arguments, A and B might both be true; we might go to a play and go to the movies. Yes, really. A wrong answer might say “We went to a play, so we won’t go to the movies.” This error is called “affirming the disjunct.”
Invalid:
P) A or B
P) B
C) Not A
GMAT example:
To see this in action, check out your The Official Guide for GMAT Review 13th Edition, by GMAC®*, question 41. This argument opens with an implied “or” statement:
“Installing scrubbers in smokestacks and switching to cleaner-burning fuel are the two methods available to Northern Power…”
The author here incorrectly assumes that by using one method, Northern Power can’t use both methods at the same time. Question 51 does the same thing; discuss it in the comments below?
TOUGH STUFF: Fun with conditional statements
This is important! Keep a sharp eye out for statements that can be expressed conditionally and practice diagramming them. Look for key words such as “if,” “when,” “only,” and “require.”
I use the symbol “–>” to express an if/then relationship, and a “~” to express the word “not.” Use single letters or abbreviations to stand in for your elements.
If/then statements:
If you jump into that mud, you will get dirty: J –> D
If you don’t stop, I will faint: ~S –> F
I will scream if I hear that Bieber song again: B –>S
I will go only if you buy me dinner: Go –> Din
(Hint, replace the words “only if” with the arrow. See necessary/sufficient below.)
Extreme categorical statements (all, none, every, each, only, always, never):
I always go bowling on Tuesdays: T –> B
Every dog has ears: D –> E
Only teenagers listen to Bieber: B –> T (notice that “only” is backwards from “every”)
No Librarians are Constructivists: L –> ~C
None of my friends eat sushi: F –> ~S
“or” statements:
I will order the cake or the pie: ~C –> P (and ~P –> C)
If you run across the word “unless,” it might help to replace it with “if not”:
I will show up to the barbecue unless its raining.
(“If not” raining, then BBQ): ~R –> B
Necessary/Sufficient statements (need, required, guarantee)
Remember this: Sufficient (guarantee, enough) goes on the left; Necessary (need, requirement) on the right
Sufficient –> Necessary
A good party needs beer: P –> B
A Katy Perry album guarantees a good time: KP –> GT
GMAT example:
Check out CR question 60 from the Official Guide for GMAT Review 13th Edition, by GMAC®. Brackets mine:
“Neither a rising standard of living [RSL] nor balanced trade [BT], by itself establishes a country’s ability to compete[C] in the international marketplace. Both are required simultaneously…”
Diagram this: C –> RSL & BT (both are necessary)
DON’T diagram this: RSL or BT –> C (each is sufficient)
Now, answering the question should be easy. Go for it.
VALID CONCLUSIONS FROM CONDITIONAL STATEMENTS
There are only a few valid deductions one can make from conditionals, and MANY invalid ones. Obviously, you won’t be tested on the Latin names, so worry more about the rules themselves and how they apply
Modus Tollens
Latin for “method that affirms by affirming,” this one more or less repeats the conditional statement as given:
P) A –> B
P) A
C) Therefore, B
If you think that’s too easy, check out Official Guide Question 60 again. It uses Modus Tollens!
P) A –> B & C
P) A
C) Therefore, B & C
Modus Ponens (the “contrapositive”)
EXTREMELY COMMON! Latin for “method that denies by denying,” this shows up all over the GMAT.
P) A –> B
P) ~B
C) Therefore, ~A
P) If you’re in Auckland, you’re in New Zealand
P) You’re not in New Zealand
C) Therefore, you’re not in Auckland
I (and many others) call this the contrapositive. To find the contrapositive, “flip and negate.” Just swap the elements and change negatives to positives:
X –> Y
~Y –> ~X
If you’re a libertarian, you’re not a communist: L –> ~C
Therefore: C –> ~L (If you’re a communist, you’re not a libertarian)
If you jump into that mud, you will get dirty: J –> D
~D –> ~J
If you don’t stop, I will faint: ~S –> F
~F –> S
Try diagramming the contrapositive for all the examples you’ve seen so far.
Advanced note: If a conditional contains an “and” or an “or,” change “and” to “or” and vice versa in the contrapositive. Remember to negate everything.
A –> B or C
(If I get a raise, I’ll go on vacation or buy a car.)
~B and ~C –> ~A
(I didn’t buy a car AND I didn’t go on vacation, so you know I didn’t get a raise.)
This works well with necessary/sufficient reasoning:
A good party needs beer and chips (remember, necessary elements go on the right):
P –> B and C
Therefore, ~B or ~C –> ~P
No beer? Not a good party. No chips? Not a good party.
GMAT example:
The Official Guide for GMAT Review 13th Edition, by GMAC®, question 103. Brackets mine:
“For a trade embargo [TE]…to succeed, a high degree of both international accord [IA] and ability to prevent goods [PG]…must be sustained.”
I diagram it like this: TE –> IA and PG
Then I do the contrapositive: ~IA or ~PG –> ~TE
If one of those elements is missing, you can’t have a trade embargo–much like our party without chips above. Work out the rest of the question for yourself.
Hypothetical Syllogism
Easy enough. You can chain if/then statements if they work out left to right:
P) A –> B
P) B –> C
C) Therefore, A –> C
GMAT Example:
This is a free question from the GMATPrep® software v.2.1*:
Increases in the level of high-density lipoprotein (HDL) in the human bloodstream lower bloodstream-cholesterol levels by increasing the body’s capacity to rid itself of excess cholesterol. Levels of HDL in the bloodstream of some individuals are significantly increased by a program of regular exercise and weight reduction.
Which of the following can be correctly inferred from the statements above?
(A) Individuals who are underweight do not run any risk of developing high levels of cholesterol in the bloodstream.
(B) Individuals who do not exercise regularly have a high risk of developing high levels of cholesterol in the bloodstream late in life.
(C) Exercise and weight reduction are the most effective methods of lowering bloodstream cholesterol levels in humans.
(D) A program of regular exercise and weight reduction lowers cholesterol levels in the bloodstream of some individuals.
(E) Only regular exercise is necessary to decrease cholesterol levels in the bloodstream of individuals of average weight.
You can chain the premises using conditionals as follows
Premise) incHDL –> lowBCL (B –> C)
Premise) someEX –> incHDL (A –> B)
Conclusion) Therefore, someEX –> lowBCL (A –> C)
Which is, more or less, the correct answer a nutshell. Work it out for yourself.
(Keep an eye out for words like “some” and “most” by the way)
CAUTION: Fallacies ahead!!
One common way the GMAT constructs wrong answers (and incorrect assumptions) is to mess up conditional logic in some way or another. Wrong answers will flip without negating, negate without flipping, confuse necessary with sufficient, mess up syllogisms, and make a series of either/or mistakes.
Flipping without negating (Affirming the consequent)
Invalid:
P) A –> B
P) B
C) Therefore, A
If you’re in Auckland, you’re in New Zealand. You’re in New Zealand. Therefore, you must be in Auckland.
(Nope, you might be in Auckland, but there are lots of other places you could be in New Zealand other than Auckland: Wellington, Nelson, Hobbittown, etc.)
This one is common in politics as well as on the GMAT:
No democrats are republicans (D –> ~R). You’re not a republican, so you must be a democrat (R –>~D).
(Nope, there are a lot of other political parties in the world…)
Negating without flipping (Denying the antecedent)
Invalid:
P) A –> B
P) ~A
C) ~B
If you’re in Auckland, you’re in New Zealand. You’re not in Auckland, so you can’t be in New Zealand.
No democrats are republicans (D –> ~R). You’re not a democrat, so you must be a republican (~D –> R).
Confusing necessary with sufficient or sufficient with necessary:
A good party needs chips and beer. We have chips and beer, so it’s going to be a good party.
(Nope, there may be other necessary requirements to a good party, such as music, a place to have the party, actual other people…)
Chopping off your leg is a guarantee that you’ll lose 30 pounds. Bill lost 30 pounds, so he must have chopped off his leg.
(There are other sufficient ways to lose 30 pounds.)
GMAT examples:
The Official Guide for GMAT Review 13th Edition, by GMAC®, question number 103: Remember this?
“For a trade embargo [TE]…to succeed, a high degree of both international accord [IA] and ability to prevent goods [PG]…must be sustained.”
One of the wrong answers says:
(B) As long as international opinion is unanimously against Patria, a trade embargo is likely to succeed.
Which is a whole lot like saying: “I have chips, so it’s going to be a good party!”
Even on other kinds of questions, the GMAT will confuse necessary/sufficient in wrong answers:
From the GMATPrep® 2.1 CAT exam practice test*, on one assumption question, the argument states:
“The interview is an essential part of a successful hiring program”
(Interview is necessary: SHP –> Int.)
Whereas one of the wrong answers states:
“A hiring program will be successful if it includes interviews.”
(An interview is sufficient: Int. –> SHP)
Syllogism Fallacies:
Be careful how you link syllogisms. Make sure they chain up correctly.
Invalid:
P) A –> B
P) A –> C
C) B –> C
Many wrong answers do this in weird ways. See Official Guide question 103:
“(E) For a blockade of Patria’s ports to be successful, international opinion must be unanimous.”
Either/Or fallacies (Affirming the disjunct)
We’ve covered this already, but to sum up: When you see “or” statements on the GMAT, pay attention to the precise phrasing.
Are the two things mutually exclusive (Cats and Dogs)? If so, do those two categories account for everything in the universe? Or are there possibilities of being other things? If so, you might want to diagram it as follows:
C –> ~D
D –> ~C
So it would be invalid to say: “That’s not a dog, so it must be a cat.” (~D –> C). You never know, it might be a wombat or the Empire State Building.
Seems obvious, but people do it all the time: (You’re not a democrat, so you must be a republican.)
On the other hand, is it a simple “or” statement that leaves the possibility of both things being true? (See disjunctive syllogism way earlier). “A or B” should be diagramed as:
~A –> B
~B –> A
Premise: My light doesn’t work. Either the power is out or the bulb is blown.
(~P or ~B)
Valid conclusion: The power is working, so the bulb must be blown. (P, so ~B)
Invalid conclusion: The bulb is blown, so the power must be working. (~B, so P) (Both things could be true)
Again, check out Official Guide questions 41 and 45!
IN CONCLUSION
Overwhelmed? Don’t be. The most important rule to remember is this:
Premise) A –> B
Conclusion) ~B –> ~A
Wrong) B –> A, ~A –> B
Otherwise, I just wanted to expand your mind a little, make you aware of the ways in which the GMAT construct right and wrong answers, and to give you some tools to deeply analyze tough Critical Reasoning questions.
From now on when you struggle with a CR question, try to figure out which logical fallacy the test writers used to construct each wrong answer (during the review process, NOT on the test itself).
For example: take the HDL problem we discussed earlier. You now know how the correct answer was written. What about the incorrect ones? Can you spot the fallacies at work? Please discuss in the comments section.
Increases in the level of high-density lipoprotein (HDL) in the human bloodstream lower bloodstream-cholesterol levels by increasing the body’s capacity to rid itself of excess cholesterol. Levels of HDL in the bloodstream of some individuals are significantly increased by a program of regular exercise and weight reduction.
Which of the following can be correctly inferred from the statements above?
(A) Individuals who are underweight do not run any risk of developing high levels of cholesterol in the bloodstream.
(B) Individuals who do not exercise regularly have a high risk of developing high levels of cholesterol in the bloodstream late in life.
(C) Exercise and weight reduction are the most effective methods of lowering bloodstream cholesterol levels in humans.
(D) A program of regular exercise and weight reduction lowers cholesterol levels in the bloodstream of some individuals.
(E) Only regular exercise is necessary to decrease cholesterol levels in the bloodstream of individuals of average weight.
Have fun! There’s a cool quiz here: //www.think-logically.co.uk/lt.htm
More to come. If there are specific issues or questions you want me to cover, let me know.
* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.
* The text excerpted above from The Official Guide for GMAT Review 13th Edition is copyright GMAC (the Graduate Management Admissions Council). The short excerpts are quoted under fair-use statutes for scholarly or journalistic work; use of these excerpts does not imply endorsement of this article by GMAC.
Primary source: Critical Thinking, by Jamie Carlin Watson and Robert Arp. Continuum International Publishing.