Advanced Critical reasoning: Deductive Logic

CAT Exam
In a deductively “valid” argument, if all the premises are true, the conclusion must also be true, with 100% certainty. Luckily, on the CAT, 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 AT. For shorthand, the arguments are labelled with a “P” for premise and a “C” for conclusion: 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 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 For example: “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. TOUGH STUFF: 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.” 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 For example: “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 CAT. 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 Many 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.

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