Formal logic basics

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gilchristh
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Formal logic basics

Postby gilchristh » Fri Jul 31, 2009 1:05 am

***I think these are good to go, but if anyone spots anything that's unclear or in need of edit, please feel free to post here or PM me and I will fix and/or clarify it. G'luck, TLSers!***
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I created these notes quite some time ago for my students. I know formal logic is something that many of you grapple with in your prep, so I thought I'd post these notes in hopes that they help some of you gain some clarity on some formal logic basics. They're not exhaustive by any means, and they're written in conversational language so you don't need a background in logic or philosophy to understand them (just as you don't need a background in logic or philosophy to kick butt on the LSAT). Hopefully they'll serve as a good primer for those of you new to formal logic or stuck in the conceptualization phase. Here goes:

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Formal Logic can be used to make INFERENCES, identify FLAWS, express rules in logic games rules, and more. Here’s a primer (not an exhaustive guide) of what you need to know about Formal Logic on the LSAT:

• Formal Logic is about certainty.
Many things get in the way of certainty, including time. Logic doesn’t care about time; time is irrelevant to logic. To simplify difficult translating, just ask yourself what you know FOR SURE and don’t write down anything that isn’t certain.

• Formal Logic makes the LSAT easier.
With practice, Formal Logic makes several LR questions easier for you on Test Day. In fact, if it doesn’t, you can still use other strategies to eliminate choices and select the correct answer.

• Formal Logic deals with TRIGGERS and RESULTS.
The trigger is what comes before the arrow in a formal logic statement; the result is what comes after the arrow in a formal logic statement.
Every time the trigger happens, the result happens. If TRIGGER → RESULT

• Think in complete sentences.
Always translate and abbreviate the statement into its simplest form, but be sure you understand what you mean. Think in complete sentences, but write in shorthand. Practice paraphrasing arguments as often as possible; this will help you break down arguments more easily and correctly (read: more efficiently). Using the most obvious representation for a concept can be helpful--i.e., “If I go to the park” should become “If P -->” or “If park -->” or whatever is concise, unambiguous, and habitual for you.

• Contrapositives
There are always two things we can declare with certainty in any given formal logic statement: the "original" statement, and its "equal but opposite" truth to it. We call this "equal but opposite" truth its “contrapositive." [Note: there's never one clear "original" and one "contrapositive"--they're interchangeable based on how you translate, and distinguishing the "original" from the "contrapositive" is unnecessary.]

· · · · · If NO RESULT --> NO TRIGGER.

Forming the contrapositive:
1) Reverse the order of statements around the arrow);
2) Negate every single piece--that is, "multiply" every single piece by "-1"
· · · · · change each + to a - (implied yes becomes no, and vice versa)
· · · · · switch out "or" (which becomes "and") and "and" (which becomes "or").

example:
STATEMENT: If you are HUMAN, then you are a MAMMAL. [If human --> mammal.]
CONTRAPOSITIVE: If you are NOT a MAMMAL, then you are NOT HUMAN. [If not mammal --> not human.]

• Common Formal Logic translations
. . . . . “all”, “any”, “whenever”, etc. = If
. . . . . “some” = at least one
. . . . . “most” = more than half
. . . . . "none is" = all are not
. . . . . “only if” = then ( --> )
. . . . . “unless” = If NOT (start your statement there, with the "if" on the trigger side)
. . . . . “Not X unless Y.” (necessary condition) = If not Y --> not X // If X --> Y.
. . . . . . . . . . . ex: “Not a mother unless female.”
. . . . . . . . . . . . . . . . If not female --> not MOM
. . . . . . . . . . . . . . . . If MOM --> female.
. . . . . "only" (necessary condition) = If NOT __ --> NOT __.
. . . . . . . . . . . ex: “Only X are Y.”
. . . . . . . . . . . . . If not X --> not Y
. . . . . . . . . . . . . If Y --> X.
. . . . . . . . . . . ex: “Only females are mothers.”
. . . . . . . . . . . . . If not female --> not MOM
. . . . . . . . . . . . . If MOM --> female.
. . . . . “No As are B.” (mutual exclusion) = If A --> not B // If B --> not A.
. . . . . . . . . . . . . ex: “No ballerinas are clumsy.”
. . . . . . . . . . . . . . . Q: What two groups do we know about in this statement?
. . . . . . . . . . . . . . . A: Ballerinas and Clumsy people:
. . . . . . . . . . . . . . . . . . . . If B --> not C
. . . . . . . . . . . . . . . . . . . . If C --> not B.

• How do we know when to use Formal Logic?
Two big hints that we can use formal logic in an LR stimulus:
. . . an abundance of obnoxiously dense wording
. . . logic-specific language like “must”, "always", "each", "every", "any", “unless”, “needed”, “no __ without __”, “only”, etc.

• "Could be true" in formal logic statements
Can something be a mammal and NOT be human? YES!
You can have the result of this statement without having the trigger. This is true of every formal logic statement, no matter what the trigger and result are: The triggers of the two versions can never go together (because they cause opposing results), but the results of any statement and its contrapositive are not inconsistent (that is, they "could be true" together, but don't have to be).


There are three major "types" of Formal Logic statements on the LSAT:

• Basic (Sufficient) Formal Logic Statements:
. . . . . “If X, then Y.” If X --> Y
. . . . . “All X are Y.” If X --> Y
. . . . . “Whenever X, Y also happens.”
. . . . . . . . . . If X --> Y.
. . . . . . . . . . Contrapositive: If NOT Y --> NOT X

Basic If/Then statements (or SUFFICIENT statements) deal with sufficient causes. A sufficient cause is something that is sufficient--on its own--to bring about a result.
. . . . . If SUFFICIENT CAUSE --> result.
. . . . . If NO result --> NO SUFFICIENT CAUSE.

BUT… A sufficient cause is not necessarily the only possible cause. So we cannot assume that having the result necessarily means we must have had the sufficient cause. Here’s an example:

Let’s say when I get a sunburn, my skin peels:
. . . . . If sunburn --> skin peels.
But when the weather is dry, my skin also peels.
. . . . . If dry weather --> skin peels.
And a certain lotion also makes my skin peel.
. . . . . If lotion --> skin peels.

So sunburn, dry weather, and lotion are all sufficient causes, but if all we know is that my skin is peeling, we cannot guess what the particular cause actually was. The only other thing I know FOR SURE is this:
. . . . . If my skin is NOT peeling --> NO sunburn, NO dry weather, and NO lotion.

• Necessary Statements: “only if”, “unless”, etc.
NECESSARY Formal Logic Statements deal with necessary conditions. A necessary condition is something without which a RESULT is NOT POSSIBLE.
. . . . . If NO NECESSARY CONDITION --> NO result.
. . . . . If result --> NECESSARY CONDITION.

BUT… A necessary condition is NOT necessarily sufficient –on its own- to bring about a result. A necessary condition merely allows a result, and that isn’t a CERTAIN relationship, so we can’t translate any formal logic with the necessary condition as the trigger. But… having the result appear proves that we must have met the necessary condition, and that is a CERTAIN relationship, so we can write a formal logic statement based upon that concept. For example:
. . . . . “You can FLY only if you have WINGS.”
. . . . . “You can’t fly unless you have wings.”

The main reason necessary statement can seem confusing is the concept of time. We tend to think that we need the wings before we can fly. But the wings don’t make us fly; they only allow us to do so. Logic doesn’t care about time. So what do we know FOR SURE? If you’re FLYING, then we know you have WINGS.

If the RESULT is there, then we know we must have met the NECESSARY CONDITION. The necessary condition is only your “permission slip”-–it allows you to do a thing, but doesn't force you to do it-–so the presence of the necessary condition will not be on the trigger side in a necessary statement; it will only occur on the result side of the statement or contrapositive … lacking the necessary condition, however, is a definite cause--it prevents you from having the result--so the lack of the necessary condition will occur on the trigger side of either the statement or its contrapositive.
. . . . . “X only if Y.” If X --> Y.
. . . . . . Contrapositive: If NOT Y --> NOT X.
. . . . . No X unless Y. If NOT Y --> NOT X.
. . . . . . Contrapositive: If X --> Y.
. . . . . . . . . . [Notice these two are the contrapositives of one another.]

To reverse (order of terms) without negating or to negate without reversing would be to mistake a sufficient cause for a necessary condition, or vice versa. For example, in the "sunburn" example above, to assume any single possible (sufficient) cause was the definite (necessary) cause--or in the "wings" example, to assume that having wings (meeting the necessary condition) actually makes me fly (sufficient)--would be a necessity vs. sufficiency flaw! So… N vs. S happens whenever you reverse without negating or negate without reversing--or, in other words, whenever the arrow is backwards.

Necessary and sufficient statements are treated the same way; the only difference between them is an issue of semantics: it’s all about the verbal clues we use to translate the initial Formal Logic statement. Once the initial statement is written, all Formal Logic statements are equal.

• Complex/Compound Statements: “and” & “or”:
Here’s why we need that third step in writing the contrapositive (the “and” becomes “or” and “or” becomes “and” part). Once you know why that happens, it makes it much easier to use. Take the following example:
. . . . . If A and B --> C

Q: What causes us (FORCES us) to have C?
A: The combination of A and B.

The combination of A and B is a very specific situation, and if we remember that the contrapositive is basically the "equal-but-opposite truth," then with the original version of the rule in mind, we can ask ourselves: “How can we NOT have C?”
The answer: A could be missing, or B could be missing, or both A and B could be missing. Any of those situations will allow us to NOT have C (they’re all valid exceptions).

Back to that last statement:
. . . . . If A and B --> C

Now we know the contrapositive is: If NOT C --> NOT A or NOT B

Important note: The word “or” in Formal Logic is not mutually exclusive – “A or B” includes as possibilities A only, B only, and both A and B… so, in other words, you don’t need the word “both” unless the option of "both" is explicitly being excluded--i.e., "A or B, but not both."

So “and” in the first half of a Complex/Compound FL statement becomes “or” in the second half of its contrapositive, and the opposite is also true: “or” in the second half becomes “and” in the first half of its contrapositive.

And now the other variation on this theme:
. . . . . If X or Y --> Z


What do we know? There are really two things that cause Z (two separate, individually sufficient causes for Z): X and Y. Either one alone is sufficient, so we can think of them as two separate statements:
. . . . . If X --> Z
. ...and...
. . . . . If Y --> Z

Let’s handle them separately at first. Their contrapositives are:
. . . . . If X --> Z
. . . . . . Contra: If NOT Z --> NOT X
. ...and...
. . . . . If Y -->Z
. . . . . . Contra: If NOT Z --> NOT Y

So we see that NOT Z actually causes both NOT X and NOT Y, so let’s put it all together: If NOT Z --> NOT X and NOT Y.

Now we see why we turn “or” in the first half of a complex/compound If/Then statement into “and” in the second half of its contrapositive, and why the opposite is true as well (“and” in the second half becomes “or” in the first half of its contrapositive… i.e., if a single trigger has two CERTAIN results, and we know that either result is missing, then we must also know that we are missing the trigger).

Other useful things to know about Formal Logic:

• Combining Statements:
If we have multiple statements working together (which is often the case in Logic Games), we want to combine statements to deduce further rules and simplify the game. Here’s how we combine rules:

Always write the contrapositive of every rule you write immediately after translating the initial rule. When a result in one statement appears in another rule as a trigger, combine those statements. For instance, say we had the following rules:
. . . . . If A --> B
. . . . . . Contra: If NOT B --> NOT A
. . . . . If B --> C
. . . . . . Contra: If NOT C --> NOT B
. . . . . If D --> NOT C
. . . . . . Contra: If C --> NOT D

As we find common elements--elements that appear in more than one rule--we combine statements:
. . . . . If A --> B --> C --> NOT D.

If we have combined everything correctly, when we combine the remaining statements, we will get this:
. . . . . If D --> NOT C --> NOT B --> NOT A.

And if we’ve done everything correctly again, you’ll notice this: the statement we just made is the contrapositive of the first (highlighted) one. [Note that some very complex, compound logic chains--like those you might see in a logic game of selection--might not allow a perfect 1:1 contrapositive matchup like this.]

• Funky Translations:
If you find yourself stuck, keep a few very familiar examples in mind, examples you can plug into the exact structure of the statement you’re trying to translate, and use them as decoder hints. Here are a few examples:

Mutual exclusion (you can't have both)
. . . . . ex: “No ballerinas are clumsy.”
Ask yourself, “What two groups do I know about in this statement?”
If I am a BALLERINA, then I’m not clumsy; and if I am CLUMSY, then I’m not a ballerina:
. . . . . If B --> not C.
. . . . . If C --> not B.

Necessary condition (if you have one, you must have had the other; if you're missing one, you're missing the other)
. . . . . ex: “Only females are mothers.”
Does being female automatically make someone a mother? No… but if you aren’t female, then you cannot be a mother; we also know that being a mother proves she is female.
. . . . . If not female --> not MOM.
. . . . . If MOM --> female.

Perhaps the best way to get comfortable with formal logic is to just take your time at first, stopping to ask, “What do I know for sure? What are the exceptions to that statement?” Don't worry; you will get faster with proper practice.
Last edited by gilchristh on Sat Aug 01, 2009 10:07 pm, edited 3 times in total.

cpaige
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Re: Formal logic basics

Postby cpaige » Fri Jul 31, 2009 11:47 am

haven't read it yet, but it looks great. not to doubt you (because it seems like you know what you're talking about), but what training do you have in formal logic? are you a philosophy major or something?

coldjesusbeer
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Re: Formal logic basics

Postby coldjesusbeer » Fri Jul 31, 2009 12:07 pm

Great write-up, I'm bookmarking this thread to keep an eye on completed versions.

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gilchristh
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Re: Formal logic basics

Postby gilchristh » Fri Jul 31, 2009 12:45 pm

cpaige wrote:haven't read it yet, but it looks great. not to doubt you (because it seems like you know what you're talking about), but what training do you have in formal logic? are you a philosophy major or something?

Was not a philosophy major, but I had more than enough classes for a minor (didn't apply for the minor). I've been teaching the LSAT for 7 years, though, so I'm extremely familiar with the formal logic on the LSAT (which is very limited in scope) and these notes are designed specifically for LSAT formal logic--conversational language, words and phrases you'll often see on the LSAT, etc.

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gilchristh
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Re: Formal logic basics

Postby gilchristh » Fri Jul 31, 2009 1:52 pm

K, I've read through these a bit and cleaned 'em up, but I could use a few pairs of fresh eyes before I declare them ready. Anyone want to help me proof 'em?

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MURPH
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Re: Formal logic basics

Postby MURPH » Sat Aug 01, 2009 5:46 am

Wow you deserve a sticky and all the glory and fame that goes with it. Awesome

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TheLuckyOne
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Re: Formal logic basics

Postby TheLuckyOne » Sat Aug 01, 2009 7:33 am

gilchristh wrote:K, I've read through these a bit and cleaned 'em up, but I could use a few pairs of fresh eyes before I declare them ready. Anyone want to help me proof 'em?


Under "necessary statements" you wrote: You can only fly if you have wings...Well, I am uncertain if it's the same as "You can fly only if you have wings". Seems to reverse the relationship. Basically saying that "if you have wings the only thing you could do is fly".

Under Complex/Compound Statements: “and” & “or” you wrote: Important note: The word “or” in Formal Logic is not mutually exclusive – “A or B” includes as possibilities A only, B only, and both A and B… so, in other words, you don’t need the word “both” unless the option of "both" is explicitly being excluded--i.e., "A or B, but not both."
I think the text in those square brackets was supposed to be bold or something like that..

I skimed certain parts, but these are the only things I've noticed.

BTW, great job!!!!

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gilchristh
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Re: Formal logic basics

Postby gilchristh » Sat Aug 01, 2009 12:33 pm

TheLuckyOne wrote:
gilchristh wrote:K, I've read through these a bit and cleaned 'em up, but I could use a few pairs of fresh eyes before I declare them ready. Anyone want to help me proof 'em?


Under "necessary statements" you wrote: You can only fly if you have wings...Well, I am uncertain if it's the same as "You can fly only if you have wings". Seems to reverse the relationship. Basically saying that "if you have wings the only thing you could do is fly".

Under Complex/Compound Statements: “and” & “or” you wrote: Important note: The word “or” in Formal Logic is not mutually exclusive – “A or B” includes as possibilities A only, B only, and both A and B… so, in other words, you don’t need the word “both” unless the option of "both" is explicitly being excluded--i.e., "A or B, but not both."
I think the text in those square brackets was supposed to be bold or something like that..

I skimed certain parts, but these are the only things I've noticed.

BTW, great job!!!!

Awesome, thank you! Yep, right on both accounts, and fixed.

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gilchristh
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Re: Formal logic basics

Postby gilchristh » Sat Aug 01, 2009 12:33 pm

MURPH wrote:Wow you deserve a sticky and all the glory and fame that goes with it. Awesome

Hahahaha... thanks :wink:

CroCop
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Re: Formal logic basics

Postby CroCop » Sat Aug 01, 2009 9:32 pm

Someone PM a moderator and get this thing a sticky!

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gilchristh
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Re: Formal logic basics

Postby gilchristh » Sun Aug 02, 2009 3:43 pm

bump for any possible edits?

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wrigley
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Re: Formal logic basics

Postby wrigley » Sun Aug 02, 2009 3:55 pm

good job, thanks for posting!




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