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Introduction to Logic

Logic

A big idea on the LSAT is logic. The word logic gets thrown around in everyday speech. But on the LSAT, as is the case with the other ideas and big words here, there is a very particular meaning to this idea of logic. The logic on the LSAT breaks down into two different types of logic.

Causation: If you are an Economics major or any social science major you will learn about statistical inferences. On the LSAT it doesn’t get into the details of statistical inferences, but it does rely on a basic intuitive understanding of what causation is.

Lawgic: This other branch is the conditional logic. Which is intimately tied to this notion of validity which is covered in depth in the arguments curriculum. But if you take a formal logic class, symbolic logic and predicate logic you will learn about Universal Quantifiers and Conditional Logic. On the flip side of that is Existential Quantifiers, which is also an idea that would have been covered in any college level logic curriculum. This plays a huge part on the LSAT, your ability in an abstract way to manipulate symbols according to the rules of Conditional Logic or Existential Quantifiers.

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Universal Quantifiers Overview

Warning: This "lesson" will be too much. The good news is that there's no substantive content in this "lesson" at all. It's simply a list of lessons coming up.

So don't freak out okay? Everything's going to be fine.

Before we start, let me give you a brief overview on what this curriculum entails. First, we need to understand why we even need to learn the stuff in this curriculum, and why it’s important. It's got a lot to do with what Conditional Logic is. What is a Universal Quantifier?

At the heart of it, they’re relationships. They’re a very unique type of relationship between Sufficient and Necessary conditions. This relationship is better expressed in the new language which is called "lawgic," and of course the language that the LSAT expresses this relationship in is called English.

So we are going to learn about the deficiencies of English, the shortcomings of it, and why we even need to learn this new language.

Before learning about the mechanics of the language, we need to learn about the sufficient and necessary conditions, because this is the idea behind what Conditional Logic is and what is a Universal Quantifier? It’s Sufficient and Necessary conditions. So when learning about Sufficient and Necessary Conditions we need to learn about Contrapositives.

Once we complete those two legs of the curriculum we'll know what it is that we are doing, why we are doing it, and what this thing is that we are learning. Then we need to get to the nitty-gritty of the mechanics of actually using this new language, this language called lawgic. We say this language is better than English to express logical ideas, well we then need to be able to use it. To begin using it we have to first develop a nose for which sentences in English actually express the idea of Conditional Logic. To do this we use words which we call indicators, they’re known as logical operators, they play a particular role in the English language, which the role is being a logical operator, it dictates a logical relationship of the ideas in a sentence.

We then need to get familiar with these indicators. The LSAT has fallen in love with four sets with four groups of indicators that they repeatedly go back to. They use them to indicate to you that we are talking about Conditional Logic and Universal Quantifiers. That is why they are called logical indicators.

In Group 1 there is the translation rule and the list, and once we learn this we go to Group 2 and learn about the translation rule and list. Then we move on to Group 3 and within Group 3 we need to take a detour to talk about this very special word called "or." "Or" is notoriously ambiguous and confusing for students. But within the diagram is a clear cut definition of the three definitions of "or" laid out for you, and the lesson on how you distinguish between each of them, and what you do with it once you realize you’re in definition 1 as opposed to definition 2. Then moving on from Group 3, we go to Group 4 which is the final group of words in which they use to indicate to you this relationship of Conditional logic.

Once you finish up Group 4, we do a summary of the four groups and move on to Advanced Logical indicators. Under this curriculum we will learn about truth tables and how to tell apart "Or v. Not Both." We will then learn about "And/Or" being embedded in a Conditional statements and the rules in dealing with that. Plus what De Morgan’s law is, which is really a law about how to take the contrapositive of "And/Or" embedded in Conditionals.

Then after that the second lesson in Advanced Logical Indicators is Bi-Conditionals which also plays a huge part on the Logic Games portion of the LSAT. Under Bi-Conditionals you’ll see the English language indicators of the Bi-Conditionals typically come in four shapes, and these four words typically try to indicate Bi-Conditional relationships.

Once you’ve completed the advanced curriculum of the Bi-Conditional, we move on to the Mastery level curriculum, which has to do with Embedded Conditionals. Which is probably one of the hardest types of conditionals and also very rare (thank-goodness). Once you complete this curriculum you will have successfully completed the curriculum in Conditional Logic/Universal Quantifiers.

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Universal Quantifiers

Conditional Logic/Universal Quantifiers are very useful for many Logical Reasoning questions and more than half of the Logic Games questions.

Conditional Logic is a language, just like how English, French, and Chinese are languages.

Except where as those are natural languages, Conditional Logic is an artificial language used to aid our understanding and manipulation of abstract ideas.

The major relationship in Conditional Logic is that of the sufficient condition and the necessary condition.

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English is not Good Enough

Under language, the thing you need to understand is that the LSAT is written in English. So naturally, the question becomes why are we learning a different language? Which I am telling you that logic is a different language, I know you haven’t seen it yet, but have faith and take my word for it, logic is a different language. Later on you will see that it actually is.

The entire LSAT, you flip it open and not a single bit of it is written in logic, all of it is written in English. So why are we learning this different language? Well it’s because the language that you have, the language that you use to think about ideas, write about ideas, and speak about ideas, those all determine the way you understand those ideas.

So English, I submit, is not very powerful. As a tool to understand and manipulate abstract symbols, and manipulate logical ideas - a lot of which we already talked about in the other curriculum under Arguments where you learn about validity - English is just a blunt tool. It’s enough for a lot of people - there are people taking the LSAT that just simply don’t need to defer to this. But, its not for most people, at least most of the students I’ve had the pleasure of teaching, this just wasn’t enough.

Certainly logicians thought it wasn't enough. Logicians when they first invented this logic/language they thought, 'Hey, to better understand the world of logic we need a better tool to straighten our minds out. So this is why they created the Logic language. The limitations of the Logic language is that it’s narrower than the English language, which means that English is broader than Logic.

English captures things that this Logic language has no business, desire, or intention of capturing. Ideas like sentiments, an example would be the smell of lilacs in the spring breeze brings a peace and calm to my mind. That’s not something this Logic language has anything to say about. But this notion of all Jedi use the force, yes that sentence is something Logic has something to say about. Which is why under Conditional Logic and Universal Quantifiers the Indicators section is where you will be able to tell those two apart.

The other thing about English is that there are so many ways to express the same idea. And the LSAT writers are really good at figuring out all the different ways in English to express the same idea. To an LSAT student that is kind of annoying, because really you just want to get to the underlying relationships. Especially when it’s something Logic can speak to, like all Jedi use the force. You just want to be able to see, past all the superficial language differences contained in English, what the core relationships really are.

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Why Learn Logic?

At its very core English is just a language, it’s a language that expresses ideas. As the previous lesson alluded to, some of these ideas are broader than what the capabilities of logic can express. Like the fact that you enjoyed a steak dinner two weeks ago. That’s not something logic can capture, it’s just beyond the scope of what logic can do.

But some ideas are completely within logic. It’s for those ideas that we’re going to learn this new language called Logic, to express. It's also called Conditional Logic or you can think of it as a Universal Quantifiers.

So this is one of these ideas: “If you're a Jedi, then you use the force.” So in this lesson lets just explore what really this idea is. What does it mean to say, “If you're a Jedi, then you use the force?” Well an idea is just an idea. In order to talk about an idea, in order to give an expression to an idea, we have to impose some kind of language on it. This is already a language.

Let’s try a different language, a picture language. So say you go around the world and collect all the Jedi. You’re an omniscient-being, and as you look down into the universe, (I am more omniscient so I give you orders), your job is to find me all of the Jedis in the universe and collect them. Each of the Jedis will represent a little red dot. So you go around and find Yoda, and you pick him up and put him in your pocket. And you walk around some more and find Luke Skywalker, so you collect him. You keep walking and you find Qui-Gon Jinn and his pupil, Obi-Wan Kenobi and grab them too. So after you have grabbed all the Jedi and put them together, I tell you to put a pen around them and label this pen "J" for all the Jedi in the universe and no other Jedi exist out there.

Then the next thing you do, is you go around and look for all the force users. But you saved yourself a lot of trouble because you already collected all the Jedi, and all the Jedi are force users. So there's at least these people. But then there are more, because there is Darth Vader. And then there is all of Darth Vader's force using minions out there, the Sith-Lords. They’re all force users, but they’re not Jedi.

So you see this is a different way of expressing that idea. This is a way of expressing the idea in English. This is a way of expressing the idea in venn diagrams. Or in terms of sets, you can think of it like here is a set, here's a members of the set. Here's a larger set which contains the smaller set and here are the members of the larger set. Larger set is called force users, smaller set is called Jedi. Really these two are very much overlapping so let's get rid of this and just use this.

So at bottom, that's the idea you're trying to represent. Every time we talk about this notion of Conditional Logic, every time we talk about this notion of Universal Quantifiers, in short, this entire curriculum, we’re going to be talking about this idea right here. Do you see the sense in which this is a Universal Quantifier? It's quantifying over the entire universe of Jedi, and it’s saying of the Jedi that they’re all force users, not just some, not just most, but every single one.

Alternate transcript with all reference to Jedi and Force replaced with cat and mammal:

At its very core English is just a language, it’s a language that expresses ideas. As the previous lesson alluded to, some of these ideas are broader than what the capabilities of logic can express. Like the fact that you enjoyed a steak dinner two weeks ago. That’s not something logic can capture, it’s just beyond the scope of what logic can do.

But some ideas are completely within logic. It’s for those ideas that we’re going to learn this new language called Logic, to express. It's also called Conditional Logic or you can think of it as a Universal Quantifiers.

So this is one of these ideas: “If you're a cat, then you are a mammal.” So in this lesson lets just explore what really this idea is. What does it mean to say, “If you're a cat, then you are a mammal?” Well an idea is just an idea. In order to talk about an idea, in order to give an expression to an idea, we have to impose some kind of language on it. This is already a language.

Let’s try a different language, a picture language. So say you go around the world and collect all the cats. You’re an omniscient-being, and as you look down into the universe, (I am more omniscient so I give you orders), your job is to find me all of the cats in the universe and collect them. Each of the cats will represent a little red dot. So you go around and find Garfield, and you pick him up and put him in your pocket. And you walk around some more oh Hello Kitty and you collect her as well. You keep walking and there's Princess Carolyn and Sylvester and you grab them too. So after you have grabbed all the cats and put them together, I tell you to put a pen around them and label this pen "C" for all the cats in the universe and no other cats exist out there.

Then the next thing you do, is you go around and look for all the mammals. But you saved yourself a lot of trouble because you already collected all the cats, and all cats are mammals. So there's at least these items. But then there are more, because there is Snoopy. And there is Lassie and Toto and Bruiser and me and you on and on and on. We are all mammals, but we're not cats.

So you see this is a different way of expressing that idea. This is a way of expressing the idea in English. This is a way of expressing the idea in venn diagrams. Or in terms of sets, you can think of it like here is a set, here's a members of the set. Here's a larger set which contains the smaller set and here are the members of the larger set. Larger set is called mammals, smaller set is called cats. Really these two are very much overlapping so let's get rid of this and just use this.

So at bottom, that's the idea you're trying to represent. Every time we talk about this notion of Conditional Logic, every time we talk about this notion of Universal Quantifiers, in short, this entire curriculum, we’re going to be talking about this idea right here. Do you see the sense in which this is a Universal Quantifier? It's quantifying over the entire universe of cats, and it’s saying of the cats that they’re all mammals, not just some, not just most, but every single one.

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Too Many Expressions in English

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Lawgic is a Language

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Sufficiency & Necessity: What Does the Arrow Mean?

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The Contrapositive

Contrapositives will save your life on the LSAT. Often, you’ll think you got an answer choice right. “Duh, they want me to infer that ‘All business school students are greedy.’ Hmm… but I don’t see it. WHAT IS GOING ON LSAT?!” Well, that’s because the right answer choice says “If you’re not greedy, you’re not a business school student.” See, same thing! I mean that too. Contrapositives are logically equivalent statements. You can think of them as being genetic twins. They’re the same.

So which one is the contrapositive? Actually, they each are the contrapositive of the other.

Consider this example.

English: All dogs are unselfish.

Lawgic: D → /S

English: If something is selfish, then it’s not a dog.

Lawgic: S → /D

First, let’s agree that these sentences say the exact same thing. Agreed? Good. D → /S is the contrapositive of S → /D just as much as S → /D is the contrapositive of D → /S. They’re contrapositives of each other. Just like twins. So what does “contrapositive” mean? It just means that you’re referring to the only other way in the Lawgic language of expressing that particular conditional relationship. See how that’s different from English? In English, there are myriad ways to express that idea. In Lawgic, there are two. They are called contrapositives of each other.

To get from one statement in Lawgic to its contrapositive, you apply a two step transformation process.

Step 1. Switch the two symbols around the arrow.

Step 2. Slap a negation sign on each symbol.

Step 3. There is no Step 3. It’s a two step process.

You have to remember that when you slap a negation onto a symbol that’s already negated, the negation goes away. Negating “not selfish” becomes “selfish.”

LET’S REVIEW

Contrapositives are very useful for the LSAT. To get the contrapositive from a Lawgic statement, you “flip and negate.”

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Contrapositive Mistakes

Contrapositive Mistakes: when you do only one of the two steps for getting a proper contrapositive. You have to do both.

  1. flip

  2. negate

Doing just one and not the other will result in a mistake.

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The Negation

IT’S NOT WHAT YOU THINK.

When we say “negation” – the “/” in Lawgic – we mean something very specific. What we mean is the contradiction. The logical opposite. What we mean is “it’s not the case that…”

For example, what is the negation (or contradiction or binary opposition) of the idea “rich?” Did you say “poor?” That’s wrong. “Poor” is our natural language conception of opposition (not binary opposition). Black is to white as rich is to poor. On the LSAT, however, we’re speaking of contradiction, of binary opposition, of logical negation. The opposite of “rich” is “not rich.” What’s the difference? “Not rich” captures middle class too where as “poor” does not. “Not rich” also captures well-to-do, getting-by, leaves, love, friendship, etc. Weird right? It’s owing to the difference between your ordinary conception of opposition from that of the logical conception of contradiction. The latter neatly, in a ruthlessly binary manner, separates the entire universe of things into two categories. Here, we have “rich” and there we have everything else, i.e., “not rich.” What happens, then, when you negate “not rich?” Well, you’re back to “rich.”

On the LSAT, the rule of thumb is to assume that they’re talking about contradiction. For example, if you say that the rate of violent crimes has not risen in the last 5 years, you must not assume that the rate of violent crimes has fallen (which would be the natural opposition). Why? Because the rate could have just stayed the same. It could have just not risen.

LET’S REVIEW

“/” means “negation.” “Negation” means logical negation. It means to take the entire universe of things and split it into two halves. Over here, we have “cold,” and over there, we have everything else: “not cold.” Do not confuse this notion of contradiction with your ordinary understanding of opposition.

Featured image: No parking sign against a bright sky

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Logical Indicators

CERTAIN WORDS IN ENGLISH INDICATE LOGICAL STRUCTURE.

The words in the 4 translation groups, which we will soon learn about, and the words “some” “most” and “few” (which we will learn about in later lessons) are special. They are logical indicators. They are also called logical operators.

As logical indicators, these words perform a very specific function in the English language. They lay down the structure of the sentence as opposed to the content.

For example, let’s consider “if.”

I can say, “ifyou are a man,you are mortal.”

Or I can say “ifyou eat cheeseburgers, you will be able to fly."

Of course, these sentences are distinct. They express very different ideas. The first sentence is about the inevitable mortality of all men and the second is about cheeseburgers. But, these different ideas are not informed by the word “if.” Rather, they are informed by the other words in the sentences, like “mortal” and “cheeseburgers.”

So, what does “if” do? It lays down the structure of the logical relationship. In other words, what “if” does is to establish that whatever these ideas may be in this particular sentence (in the first example, being a man and being mortal, in the second, eating cheeseburgers and being able to fly) they exist in a conditional relationship with each other where the idea following the “if” is the sufficient condition and the other idea is the necessary condition.

Once you realize this, you see that it doesn’t matter what ideas go into this structure. "If X, Y" says that X is sufficient for Y and Y is necessary for X. You can define X and Y to be whatever you want. X is being a man and Y is being mortal. So we’ve recreated the first sentence.

X is eating a cheeseburger and Y is being able to fly. So, eating a cheeseburger is sufficient for being able to fly. That sentence would, of course, be false. Awesome, but false. Yet, that doesn’t change the fact that it expresses a conditional relationship between the two ideas. Of course, the reason that it’s false is because that conditional relationship does not hold in our world. Who knows, maybe there is a world where eating cheeseburgers guarantees your ability to take flight. That world would be almost as crazy as one in which being a Jedi allows you to use the Force, whatever that is.

LET’S REVIEW

Some words in English express logical relationships. We call them logical indicators or logical operators. These words are special and you should train to become sensitive to them.

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Logical Indicators and Translations

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Basic Translation: Group 1

GROUP 1: IF, WHEN, WHERE, ALL, THE ONLY, EVERY, ANY

All the words in this group follow this translation rule:

The ideas introduced by (i.e., immediately following) these words are the sufficient conditions.

Let’s try it

If zombies attack New York City, the real estate market will slip.

Step 1) Identify the logical operator

Here, it’s “If.”

Step 2) Identify the two main concepts (or groups, categories, events or ideas)

Here, it’s the two events “zombies attack NYC” and “real estate market will slip.”

Step 3) Assign symbols to the two main concepts

Here, “Z” for “zombies attack NYC” and “RES” for “real estate market will slip.”

Step 4) Apply the translation rule

Here, the ideas introduced by these words are the sufficient conditions.

Z → RES

Step 5) Find the contrapositive

/RES → /Z

Step 6) Translate back to English

If the real estate market doesn’t slip, then zombies are not attacking NYC.

All done!

You see why this is called a “mechanism?” Because it’s mechanistic! You don’t really have to think very much about these sentences. It’s almost like a computer algorithm. You put in the data and it puts out the results. The computer doesn’t know what it’s doing. It’s simply following instructions. That’s what you should do for now. You have to be fast at this, really fast.

There’s an advantage to using a mechanism. It helps to elucidate the meaning of certain opaque sentences. A sentence using the “If” construct, or really any of the constructions in Group 1, probably doesn’t need elucidation no matter how complicated the ideas may be. But a less clear construct, like “unless” (that’s Group 3) when coupled with more complicated ideas may spell trouble. But as long as you coldly follow the mechanism you’ll be in the clear.

LET’S REVIEW

Group 1 logical indicators: if, when, where, all, the only, every, any

Group 1 rule: the idea immediately following the logical indicator is the sufficient condition.

Featured image: OldBook-attribution-Thalita Carvalho

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Clarification for "the only"

In our discussion forums, a student brought up confusion regarding the logical indicator "the only." You can see the discussion here.

In response, this is a clarification lesson.

The short answer is: just follow the rule. "The only" introduces sufficient conditions.

There are only two usages of "the only" to form grammatical sentences.

The first grammatically correct usage:

The only X's (insert modifiers) are the Y's (insert modifiers).

In this form, the Y's must call back, point back, refer back to the X's.

The translation is X → Y.

For example:

The only kids with messy hair are the kids with brown eyes.

The only kids (X) with messy hair (modifier) are the kids (Y) with brown eyes (modifier).

Notice that Y refers back to X. You could have just as easily said:

The only kids with messy hair are the ones with brown eyes.

Translated: If you're a kid with messy hair, then you have brown eyes. If you don't have brown eyes, you don't have messy hair.

Another example:

The only students who got an A, are the ones who studied.

The only students (X) who got an A (modifier), are the ones (Y) who studied (modifier).

Translated: If you got an A, then you studied. If you didn't study, then you didn't get an A.

One more example:

The only philosophical works that continue to be studied with diligence and high scrutiny by generations upon generations of scholars and sages are those that offer deep insight into the nihilistic nature of man's existence.

The only philosophical works (X) that continue to be studied with diligence and high scrutiny by generations upon generations of scholars and sages (modifier) are those (Y) that offer deep insight into the nihilistic nature of man's existence (modifier).

Translated: If you're a philosophical work that blah blah blah, then you offer deep insight blah blah blah. If you don't offer deep insight, then you're not a philosophical work.

The second grammatically correct usage:

Y's (insert modifier) are the only X's (insert modifier).

In this form, the X's must call back, point back, refer back to the Y's.

The translation is X → Y.

For example:

Kids with brown eyes are the only kids with messy hair.

Kids (Y) with brown eyes (modifier) are the only kids (X) with messy hair (modifier).

Notice that X refers back to Y. You could have just as easily said:

Kids with brown eyes are the only ones with messy hair.

Translated: If you have messy hair, then you have brown eyes. If you don't have brown eyes, then you don't have messy hair.

Another example:

Horses are the only animals that are strong.

Horses (Y) are the only animals (X) that are strong (modifier).

Notice "animals" (X) points back to "horses" (Y). You can't say "Horses are the only dogs that are strong" because wtf.

Translated: If you're a strong animal, then you are a horse. If you are not a horse, then you are not a strong animal.

One more example:

Maniacal and over the top performances are the only effective ones.

Maniacal and over the top (modifier) performances (Y) are the only effective (modifier) ones (X).

Notice the modifiers appear first.

Translated: If it's an effective performance, then it was maniacal and over the top. If it was not maniacal or not over the top, then it was not effective.

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Basic Translation: Group 2

GROUP 2: ONLY, ONLY IF, ONLY WHEN, ONLY WHERE, ALWAYS, MUST

All the words in this group follow this translation rule:

The ideas introduced by (i.e., immediately following) these words are the necessary conditions.

Let’s try it

There will be an accomplishment of lasting value only if there is hard work.

Step 1) Identify the logical operator

It’s “only if.”

Step 2) Identify the two main concepts (or groups, categories, events or ideas)

It’s the two events “accomplishment of lasting value” and “there is hard work.”

Step 3) Assign symbols to the two main concepts

“ALV” for “accomplishment of lasting value” and “HW” for “there is hard work.”

Step 4) Apply the translation rule

The ideas introduced by these words are the necessary conditions.

ALV → HW

Step 5) Find the contrapositive

/HW → /ALV

Step 6) Translate back to English

If there is no hard work, then there is not an accomplishment of lasting value.

All done!

LET’S REVIEW

Group 2 logical indicators: only, only if, only when, only where, always, must, requires

Group 2 rule: the idea immediately following the logical indicator is the necessary condition.

Featured image: NotebookPen -attribution- Jain Basil Aliyas

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Quiz - Group 1 and 2 Translations 1 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original I would not be able to see Arun if he were in the next room.

Question 2

Tap to restore original Amar would not want to learn to ski if he were over 40.

Question 3

Tap to restore original When things cost more, people buy and use less of them.

Question 4

Tap to restore original Where the judges are independent there will be a good legal system.

Question 5

Tap to restore original All trees are perennial plants with an elongated stem.

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Quiz - Group 1 and 2 Translations 2 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original If Yishan disliked the cold, she would not spend her vacation in Alaska.

Question 2

Tap to restore original Earth is the only home we’ve ever known.

Question 3

Tap to restore original Any person who knows what is truly right will automatically do it.

Question 4

Tap to restore original Max would ask the police to investigate only if he were not guilty.

Question 5

Tap to restore original Empires begin to decline only when their ruling class is corrupt.

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Quiz - Group 1 and 2 Translations 3 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original It’s always sunny in philadelphia.

Question 2

Tap to restore original Every time a NASA program pushes the frontiers of humanity, our collective confidence swells.

Question 3

Tap to restore original The only oral myths that have survived are the ones that were eventually written down.

Question 4

Tap to restore original When abundant rain falls in sub-Saharan Africa, hurricanes afterward hit the United States mainland with particular frequency.

Question 5

Tap to restore original Where there is no communication, family ties become frayed and eventually snap.

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Quiz - Group 1 and 2 Translations 4 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Cyrenaic hedonism believes pleasure is the only good.

Question 2

Tap to restore original Every mating ritual can be conceptualized as a game to illuminate the underlying strategies.

Question 3

Tap to restore original All Martian gullies are poleward of 30° latitude.

Question 4

Tap to restore original When a mix of fresh and dried grass clippings is mixed into garden soil, plant growth decreases.

Question 5

Tap to restore original Big business would be perceived more favorably by the public if they were more efficient.

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Quiz - Group 1 and 2 Translations 5 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original When the sky is clear, the atmospheric pressure is high.

Question 2

Tap to restore original The body of anyone infected by virus X will, after a week, produce antibodies to fight the virus.

Question 3

Tap to restore original Economic conditions improved only when the policy of free trade had been implemented.

Question 4

Tap to restore original Where a moral duty exists, it supersedes any legal duty and any other kind of duty.

Question 5

Tap to restore original All self-driving cars must be programmed to kill people.

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Quiz - Group 1 and 2 Translations 6 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original If this business introduced minor variations into its operating system software, unauthorized access to all the computers at the same time could be virtually eliminated.

Question 2

Tap to restore original When an acidic liquid such as vinegar is added to baking soda the resulting mixture fizzes.

Question 3

Tap to restore original All self-driving cars drive safer than non-self-driving cars.

Question 4

Tap to restore original In tall palms, the terminal bud on the main stem is the only one to develop.

Question 5

Tap to restore original Damage becomes abundant only where a continuing supply of breeding material is available.

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Quiz - Group 1 and 2 Translations 7 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original All microscopic organisms must be capable of feeling pain.

Question 2

Tap to restore original Businesses do the environmentally “right“ thing only if doing so makes good business sense.

Question 3

Tap to restore original Only the highest flying birds of prey nest in the summits of the Andes.

Question 4

Tap to restore original Some people claim that any internally consistent scientific theory is plausible.

Question 5

Tap to restore original The number of beneficial soil bacteria increases whenever plant material is mixed into garden soil.

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Quiz - Group 1 and 2 Translations 8 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original In surrounding counties, where hunting is permitted, the size of the deer population has not increased in the last eight years.

Question 2

Tap to restore original Every planetary society will be endangered by impacts from space.

Question 3

Tap to restore original Any valid moral judgment about a particular action must be formed on the basis of its consequences.

Question 4

Tap to restore original Only Italian plumbers can fly while wearing raccoon suits.

Question 5

Tap to restore original Cruise ships make unscheduled stops only when the necessities of safety demand that such course of action is prudent.

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Quiz - Group 1 and 2 Translations 9 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Chlamydiae can grow only where their host cells grow.

Question 2

Tap to restore original In Aristotle's view, universals exist only where they are instantiated.

Question 3

Tap to restore original Roses always provide a stunning display of color.

Question 4

Tap to restore original Someone who has more than one overdue book out on loan from the library at the same time must be fined.

Question 5

Tap to restore original Where the rights of businesses and the duty of government conflict, the main issue is finding a successful compromise.

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Quiz - Group 1 and 2 Translations 10 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Where health insurance is provided by private insurance companies, people who are wealthy generally receive better health care than do people who are unable to afford health insurance.

Question 2

Tap to restore original Every human being who ever was lived out their lives on planet Earth.

Question 3

Tap to restore original Good deeds must be based on unselfish motives.

Question 4

Tap to restore original It has been claimed that an action is morally good only if it benefits another person and was performed with that intention.

Question 5

Tap to restore original Only the first stanza of the poem Defence of Fort M'Henry is commonly sung today.

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Basic Translation: Group 3

GROUP 3: UNLESS, UNTIL, OR, WITHOUT

These are my favorites because they’re notoriously confusing for students. I’m not sadistic or anything. That’s not why I like them. I like them because of how easily this apparent difficulty can be overcome by sticking to your translation mechanism.

All the words in this group follow this translation rule:

You pick either idea, then negate that idea, then make that idea the sufficient condition.

Let’s try it

Don’t go to Hawaii, unless you want to have a good time.

Step 1) Identify the logical operator

Here, it’s “unless.”

Step 2) Identify the two main concepts (or groups, categories, events or ideas)

Here, it’s the two events “not going to Hawaii” and “having a good time.”

Step 3) Assign symbols to the two main concepts

Here, “/H” for “not going to Hawaii” and “GT” for “having a good time.”

Step 4) Apply the translation rule

Here, pick an idea - “**/**H” - and negateit - “H” - and make it the sufficientcondition.

H → GT

Step 5) Find the contrapositive

/GT → /H

Step 6) Translate back to English

If you don’t want to have a good time, then don’t go to Hawaii.

All done!

The thing about this group is that it indicates necessity, doesn’t it? Unless is emphasizing the necessity of what comes after it. But then it’s simultaneously warning you about the consequences of not following through with that necessity. So, with “Don’t go to Hawaii, unless you want to have a good time,” the “unless” is indicating that “wanting to have a good time” is the necessary condition. Okay, but it’s the necessary condition for what? Well, the sentence doesn’t explicitly say. It’s implied that it’s the necessary condition of “going to Hawaii.” Is that too confusing? Then just straightforwardly follow the translation mechanism and you’ll be golden.

LET’S REVIEW

Group 3 logical indicators: unless, until, or, without

Group 3 rule: You pick either idea, then negate that idea, then make that idea the sufficient condition.

Featured image: Creative Commons License paul bica

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Quiz - Group 3 Translations 1 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Without physical exercise, health deteriorates.

Question 2

Tap to restore original Unless a railroad serves its customers well, it will not be a successful business.

Question 3

Tap to restore original Without a policy of freedom of speech, governments respond to dangerous ideas irrationally.

Question 4

Tap to restore original Farmers do not know their income for a given calendar year until tax returns are calculated and submitted the following April.

Question 5

Tap to restore original Unless medical research findings are brought to peer review by a medical journal, peer review will not occur.

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Quiz - Group 3 Translations 2 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Heegun or Wong will attend the dinner.

Question 2

Tap to restore original Without the support of large corporate sponsors, opera companies could not afford to produce any but the most famous of operas.

Question 3

Tap to restore original One should not intentionally misrepresent another person’s beliefs unless one’s purpose in doing so is to act in the interest of that other person.

Question 4

Tap to restore original Until there is no shred of doubt that nuclear dumps are safe, it makes sense to situate them where they pose the least threat to the public.

Question 5

Tap to restore original Unless something is done about the alcohol problem at this university, I'll have to transfer to a university where there are no fraternities.

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Quiz - Group 3 Translations 3 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Opera is scheduled for Tuesday or Thursday.

Question 2

Tap to restore original In their interpretation of language, consumers should not presume that vagueness indicates an attempt to deceive on the part of manufacturers unless those manufacturers would reap large benefits from successful deception.

Question 3

Tap to restore original Jihane or Ortega must be appointed to the appeals court.

Question 4

Tap to restore original Without Brown's being present, no new large store can be attracted to the downtown area.

Question 5

Tap to restore original Unless Charles is an expert in some branch of psychology, he should not offer a solution to Patrick's behavioral problem.

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Why is “Or” So Confusing?

The word “or” and the often interchangeably used phrase “either or” are very confusing.

They are confusing because they are ambiguous. That means they have more than one meaning. They have three meanings.

  1. The "inclusive or" meaning “and/or”
  2. The "exclusive or" meaning "or, but not both"
  3. Simply "and"

Context will tell you which meaning is intended and let me save you a lot of trouble and tell you right now that the meaning that the LSAT summons most often is the first one, the inclusive "or" meaning “and/or.” (Those of you who are very clever will note that's why we included “or” as a Group 3 logical indicator.)

Now, let’s take a look at the other two meanings first.

(2) You may sit eitherat one end of the table orthe other end.

(3) Jane is a faster eater than eitherMary orJon.

The "or"/“either or” functions differently in those sentences.

For sentence (2) the "or"/“either or” says that you get to have one possibility, at the exclusion of the other possibility. You may sit at one end of the table, or you may sit at the other end, but certainly you cannot sit at both ends of the table. We call this “exclusive or.” We’ll revisit this interpretation of “or” when we cover bi-conditionals in a later advanced logic lesson since this meaning translates into a bi-conditional.

For (3) the "or"/“either or” is simply communicating "and." We interpret that sentence to mean that Jane lays waste to her hamburgers and fries faster than Mary and faster than Jon. It’s odd, I know. The word “or” means “and” when used in this context. English sucks doesn't it?

Notice that this sentence is comparative. Can you think of other comparative usages of "or"/"either or?" I'll bet that they all are interpreted as "and."

Now, let's return to our first meaning.

(1) Jon enrolls in either Economics 101 orPolitical Science 101 this semester.

Some of you think that Jon must take one class at the exclusion of the other. Some of you disagree, i.e. you think that Jon could take both classes.

Those of you in the latter group have an interpretation of "or"/“either or” that is more in line with how the LSAT wants you to think about the word.

According to this sentence, Jon must take at least one of the two classes. That much is certain. But, he could take both. We call this “inclusive or.” This is the “or” that we included in our Group 3 logical indicators.

"Or" and "either or" is confusing. Remember the rule of thumb is to read them like in sentence (1), the inclusive usage of the word.

If the LSAT wants to communicate the "and" (incredibly rare) meaning or the "exclusive or" meaning (this happens often enough), it will be explicit and the context will disambiguate.

I'll talk more about “exclusive or” in the later bi-conditional logic lessons, but for now, you can think of it as “one or the other, but not both.” Understand that of the two choices, you have to choose at least one and at most one. In other words, you have to choose exactly one. For example, if we say “you must eat either the steak or the cod, but not both.” You have to eat at least one of steak or cod and not both steak and cod. That means you eat exactly one of the two, no more, no less.

Now, on the other hand, the “inclusive or” means you have to choose at least one, but you could have both. Think about it this way: the minimum I need to choose is 1, but the maximum is 2. For example, reference sentence (1).

LET’S REVIEW

“Or” and “either or” are confusing. They have three different meanings.

  1. The "inclusive or" meaning “and/or”
  2. The "exclusive or" meaning "or, but not both"
  3. Simply "and"

The "inclusive or" is the version of "or" we include in the Group 3 translations because it is the one used most commonly on the LSAT.

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Clarification for "Either Or"

I want to stress "either or" for a moment because I know that colloquially, we tend to use that phrase to mean only "exclusive or."

But, that's not always true on the LSAT.

So, if you're still confused about how "either or" could possibly be interpreted inclusively, consider the following spectrum of sentences using that phrase:

  1. I am now either in Shanghai or Chicago.
  2. I will marry either Alice or Beth.
  3. From a police officer: "Either you stop or I'll shoot."
  4. I will either watch Lord of the Rings or the Matrix.
  5. From the admissions team at Yale Law School: "If you either score a 180 on your LSAT or get a Nobel Peace Prize, you will be admitted."

Even though every sentence uses "either or," you must see that there's a sliding scale of interpretation from (1) mostly likely exclusive to (5) mostly likely inclusive.

I'm listing below the presumptions about our world that modulate the strength of exclusivity in each of the sentences and you should see that these presumptions of exclusivity reduce in strength as we move down the list.

  1. presuming the laws of physics hold true.
  2. presuming that bigamy is not legal.
  3. presuming the cop isn't trying to fuck you over - it's starting to get weak right? He might be trying to trick you into stopping to get an easier shot.
  4. presuming that I won't watch both - see how very weak this one is? Why can't I watch both?
  5. presuming that the admissions team at Yale Law School speaks and understands English, this just has to be inclusive.

You should also see that these presumptions have nothing to do with the phrase "either or." Instead, it's all coming from the other words in the sentence. It's all coming from context.

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Basic Translation: Group 4

GROUP 4: NO, NONE, NOT BOTH, NEVER, CANNOT

All the words in this group follow this translation rule:

You pick either idea, then negate that idea, then make that idea the necessary condition.

Let’s try it

None of the Americans attended the dictator's party.

Step 1: Identify the logical operator

Here, it’s “none.”

Step 2: Identify the two main concepts (or groups, categories, events or ideas)

Here, it’s the two events “being an American” and “attending the dictator’s party.”

Step 3: Assign symbols to the two main concepts

Here, “A” for “being an American” and “ADP” for “attending the dictator’s party.”

Step 4: Apply the translation rule

Here, pick an idea - “A” - and negate it - “/A” - and make it the necessary condition.

ADP → /A

Step 5: Find the contrapositive

A → /ADP

Step 6: Translate back to English

If you’re an American, then you did not attend the dictator’s party.

All done!

LET’S REVIEW

Group 4 logical indicators: no, none, not both, never, cannot

Group 4 rule: You pick either idea, then negate that idea, then make that idea the necessary condition.

Featured image: Oldkeys-attribution-Steven De Polo

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Neither Nor

Translations:

pandas → not cute enough to enter

koalas → not cute enough to enter

blue dress → jane not select

black dress → jane not select

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Quiz - Group 4 Translations 1 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Lazy cats never develop heart disease.

Question 2

Tap to restore original No holiday falls within the month of August.

Question 3

Tap to restore original No birds are trees.

Question 4

Tap to restore original Mesopotamian cities never had marketplaces.

Question 5

Tap to restore original None of the recent technological advances in producing electric power at photovoltaic plants can be applied to producing power at traditional plants.

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Quiz - Group 4 Translations 2 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original No flowers with scent provide a stunning display of color.

Question 2

Tap to restore original Heegun and Wong cannot both attend the dinner.

Question 3

Tap to restore original None of the volleyball players at yesterday's office beach party came to work today.

Question 4

Tap to restore original Superior conductors are never satisfied with their own performance.

Question 5

Tap to restore original A layer of topsoil six to eight inches thick cannot support the continued cultivation of corn.

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Quiz - Group 4 Translations 3 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Opera cannot be scheduled for both Tuesday and Thursday.

Question 2

Tap to restore original Jihane and Ortega cannot both be appointed to the appeals court.

Question 3

Tap to restore original Countries cannot be both strong and moral.

Question 4

Tap to restore original In countries with nationalized health insurance, no one who needs a familiar medical treatment in order to stay alive is denied that treatment.

Question 5

Tap to restore original None of the foundation's money was used for the weapons research.

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Group 3 or Group 4?

Sometimes you'll come across sentences with both Group 3 and Group 4 indicators. Naturally, you wonder what to do. Do I follow Group 3 translation rules or do I follow Group 4 translation rules?

I guess you kind of lucked out. As it happens, you can just blindly apply Group 3 translation or Group 4 translation and get to the same results. You just have to remember to use the other "logical indicator" as a negation.

For example: Never go to Hawaii unless you want to have a good time.

You can pick either one to use as logical indicator. The other “logical indicator” becomes a negation.

Group 3: Unless

Two ideas: [never go to hawaii] and [have good time]

Symbols: /H and GT

Apply rule (negate, sufficient): H → GT

Contrapositive: /GT → /H

Group 4: Never

Two ideas: [go to hawaii] and [unless you want to have a good time]

Symbols: H and /GT

Apply rule (negate, necessary): H → GT

Contrapositive: /GT → /H

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Quiz - Group 3 or 4 Translations 1 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Plant life cannot survive without atmospheric carbon.

Question 2

Tap to restore original A well-paid police force cannot be effective without a good legal system.

Question 3

Tap to restore original No nation can have a successful economy unless at some point scientists have communicated well with the public.

Question 4

Tap to restore original No restrictions should be placed on the sale of merchandise unless sale of that merchandise could endanger innocent people.

Question 5

Tap to restore original None of the pandas that relocated to this part of the forest will prosper unless we drive out the poachers.

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Miscellaneous Logical Indicators

GROUP MISCELLANY: IS ESSENTIAL, IS REQUIRED, IS/ARE

These logical indicators show up often enough on the LSAT that we should be familiar with how to handle their translations.

"is essential" and "is required" and "is necessary"

The phrases “is essential” and “is required” and “is necessary” are predicates.  They point to their subjects and say that those subjects are essential, are required, are necessary.

For example:

Happiness is essential to living a good life.

What's essential?  Happiness.  So, Happiness is the necessary condition.

gl → H

/H → /gl

Consider another example:

Practice is required to be a skilled artist.

What's required?  Practice.  So, Practice is the necessary condition.

sa → P

/P → /sa

Note: Contrast "is required" v. the Group 2 logical indicator "requires."

"is/are"

The verbs "is" and "are" have so many meanings and usages that you can write dissertations on them.  Of course, not every usage of "is/are" means that we're in conditional logic territory.  But, some usages do signal exactly that.  For example:

Cows are big.

Translated:

C → b

/b → /C

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4 Translation Groups Summary

THE 4 GROUPS OF COMMONLY USED CONDITIONAL INDICATORS

Group 1: if, when, where, all, the only, every, any

The idea immediately following the logical indicator is the sufficientcondition.

Group 2: only, only if, only when, only where, always, requires, must

The idea immediately following the logical indicator is the necessary condition.

Group 3: unless, until, or, without

You pick either idea, then negate that idea, then make that idea the sufficient condition.

Group 4: no, none, not both, never, cannot

You pick either idea, then negate that idea, then make that idea the necessary condition

Take a break and reinforce this idea!

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Quiz - Mixed Translations 1 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original No plan for eliminating a neighborhood problem that requires demolishing basically sound houses should be carried out until all other possible alternatives have been thoroughly investigated.

Question 2

Tap to restore original Anyone who is twenty-one years old or older is legally allowed to purchase alcohol in the United States.

Question 3

Tap to restore original None of the chemicals used for cleaning the Sistine Chapel will affect the original dyes.

Question 4

Tap to restore original Desert fan palm oases occur only where permanent water sources are available.

Question 5

Tap to restore original Uranium can be economically extracted only where it is present in high concentrations.

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Quiz - Mixed Translations 2 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Without analyses of eggs from museums, the studies linking pesticides with the decline of birds of prey would have been impossible.

Question 2

Tap to restore original The rule-changing procedure outlined in the building's regulations states that only if a group of tenants can obtain the signatures of 10 percent of the tenants on a petition to change a rule will the proposed change be put to a majority vote of all the tenants in the building.

Question 3

Tap to restore original A legally impermissible action is never morally excusable.

Question 4

Tap to restore original The only times that the hospital's emergency room staff attends to relatively less serious emergencies are times when there is no critical emergency to attend to.

Question 5

Tap to restore original Every surviving civilization is obliged to become spacefaring.

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Quiz - Mixed Translations 3 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Unless forced to do so by the government, automakers do not make changes in automotive technology that is not related to profitability.

Question 2

Tap to restore original All prehistoric tools that are 13,000 years or older are made of wood.

Question 3

Tap to restore original The progression from latent to acute W can occur only when the agent that causes acute W absorbs large quantities of fat from the patient's blood.

Question 4

Tap to restore original A flight is not late unless it arrives more than fifteen minutes past its scheduled arrival time.

Question 5

Tap to restore original Expert opinion based on a scientific technique is admissible only where the technique is generally accepted as reliable in the relevant scientific community.

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Quiz - Mixed Translations 4 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original Unless bookstores generate a high sales volume, however, they cannot get discounts from publishers.

Question 2

Tap to restore original Senator Beton concludes that government funding of the arts cannot lead to the creation of works of true artistic excellence.

Question 3

Tap to restore original All currently-active land volcanoes are located on the southern half of Hawaii Island.

Question 4

Tap to restore original Any woman who really has faith in herself will be dubbed arrogant by her fellows.

Question 5

Tap to restore original Large inequalities in wealth always threaten the viability of true democracy.

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Quiz - Mixed Translations 5 w/ Answers

Quiz Instructions: Translate the sentences from English into Lawgic

Question 1

Tap to restore original There are no naturally occurring strains of wheat that have Rhizobium bacteria living in their roots.

Question 2

Tap to restore original None of the motors manufactured by EM Industries are quiet enough to use in home appliances.

Question 3

Tap to restore original Words can be explained satisfactorily only when they refer to things that can be seen or touched.

Question 4

Tap to restore original All planets in our solar system were subject to the Late Heavy Bombardment.

Question 5

Tap to restore original Every visiting official had his own private mansion in the capital of the Tang dynasty.

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Chaining Conditional Statements Together

Featured image: Creative Commons License Ella's Dad

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Chaining Conditional Statements Together - Exercise

Instructions: First, translate each sentence into Lawgic according to the translation rules you just learned. Then, chain them up.

Exercise #1

Tom must select cat food or Jerry.

If Tom selects Jerry, he must select Milk.

Exercise #2

If Mr. White makes crack, then he also cooks meth.

If he makes LSD, then he cannot cook meth.

Mr. White can make heroin only if he makes LSD.

Exercise #3

If Joffrey kills Sansa, he cannot kill Arya.

If Joffrey kills Sansa, he cannot kill Robb.

He cannot kill Robb unless he kills Bran.

If he kills Bran he must kill Rickon.

See video below for answers, but not before you give it a try yourself.

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Conditional Indicators Drill

Conditional Indicators Drill Flashcards

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