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Philosophy, Coursework, Phi 170

Symbolic Notations

Symbolic Notations

Week 3

Worksheet Three

*Please read everything below carefully and fill in whenever you are asked to fill in your answers.

Topic: Symbolic Notations

We already learned that how to symbolize declarative sentences.

For example, this declarative sentence:  “You studied and you got an A on the exam” can be put into symbolic notations likes this:

S & A

This notation stands for “You studied (S) and (&) you got an A on the exam (A).”

But now we will learn how to symbolize conditional sentences, meaning sentences that do not have the word “and” but instead have the phrase “If….then…” as in: “”If you study, then you will get an A.”

Well, to convey in symbols the idea of something conditional or hypothetical, we will use the symbol of an arrow ->  as in:     A -> B

So when you see the arrow between two letters, you should read the notation like this: If A, then B.

Practice:

X -> Y This stands for IF X, then Y.

P -> Q    This stands for If P, then Q.

M -> T  This stands for If M then T   (If today is Monday, then tomorrow is Tuesday.)

Meaning:

If M is true, then T is true, meaning: IF it’s true that today is Monday, then it must be true that tomorrow is Tuesday.

Now your turn. Put these 3 sentences into symbolic notations. If you do not know how to type an arrow let me know asap.

  1. If today is Wednesday, then tomorrow is Thursday. W -> T
  2. If today is Friday, then tomorrow is Saturday. F -> S
  3. If today is Sunday, then I do not have to work. S -> -W

Remember, the letters are just symbols. You may use different letters. For instance:

“If you study hard, you will pass the exam.”

This sentence can be translated as:

S -> P    or  H -> E

Remember, every conditional statement divides into two constituents, which do not play equivalent roles (in contrast to conjunction and disjunction). Once again, the constituents of a conditional A-> B are respectively called the antecedent and the consequent.

The word ‘antecedent’ means “that which leads”, and the word ‘consequent’ means “that which follows”.

Now your turn; symbolize these conditional sentences:

Remember: ‘If’ introduces the antecedent, ‘then’ introduces the consequent.

  1. If I have the money, I will buy a new car. M ->B
  2. If it rains, I will go to the movies. R -> M
  3. I will go to the movies, if it does not rain. –R -> M
  4. If you tell me the truth, I will trust you. T -> Y
  5. I will trust you, if you tell me the truth. T ->Y
  6. If it rains, I will cancel the trip. R -> T
  7. I will work late, if I have the energy. E -> W
  8. If I have the time, I will call them. T -> C
  9. I will pay the bill, if I get the money. M -> B

Consider again the conditional sentence: If the weather is nice, I will go to the beach.

Can I say the same thing but in a different way? Yes:  I will go to the beach, if the weather is nice.

Am I saying the same thing in the second sentence? Do I mean exactly the same?

Yes. The meaning of these two sentences is identical.

This is why the symbolic notation of the two sentences below will be exactly the same:

If the weather is nice, I will go to the beach.        N->B

I will go to the beach, if the weather is nice.        N->B

Please do NOT translate the second one as:  B -> N.    You would be saying here that if you go to the beach then the weather will suddenly become nice. But this is absurd. The weather does not care if you go to the beach.   You cannot make the weather nice by going to the beach.

So go back to sentences: 3, 5, and 7 and make corrections if necessary.

Now create two conditional sentences, first in English, and then translate them into symbols.

  1. 1. If I feel hungry, I will eat F -> E
  2. I will play, if it does not rain -R -> P

Translate the sentences into symbols. Then say which part of the sentence is the antecedent and which part is the consequent.

  1. I will work late, CONSEQUENT if I have the energy. ANTECEDENT E ->W
  2. If I have the time, ANTECEDENT I will call them. CONSEQUENT T -> C
  3. If you are born in America, ANTECEDENT you are automatically an American citizen. CONSEQUENT A -> C
  4. I would be right now in Russia, CONSEQUENT if New York were a Russian territory. ANTECEDENT T -> R
  5. If the universe is expanding, ANTECEDENT then space has to be finite. CONSEQUENT  U -> F

Consider the sentence: I will not help you, if you do not tell me the truth.

Let, H will stand for “I will help you.” T will stand for “You tell me the truth.”

Only one is a proper translation of the sentence above. Which one? What does the other (incorrect) translation say?

  1. – H ® – T If – H, then – T it means ‘if I will not help you, then you do not tell me the truth’ which is incorrect
  2. – T ® – H               If – T, then – H CORRECT

Translates the following conditional sentences into symbols (the first is already translated):

  1. If I study too hard, I will not enjoy life. S-> – E
  2. If I do not go to the meeting, I will not be able to register my company. –M -> -R
  3. If he catches you, he will hurt you. C -> H
  4. If you look outside, you will see a mountain. L -> M
  5. If Oswald didn’t shoot Kennedy, someone else did. –O -> E
  6. I do math well, if I drink a lot of coffee. C -> M
  7. You will have money, if you work. W -> M
  8. If you do not pay, I will pay. –P -> W
  9. You will not get the job, if you do not apply. –A -> -J

Now create 5 conditional sentences by yourself; then translate the sentences into symbols.

  1. 1. If I wash utensils, then I will not cook U -> -C
  2. I will steal, if I am provoked P -> S
  3. If I do not read, I will not understand –R -> -U
  4. If you told me, I would have done it T -> D
  5. I will not tell, if I do not know –T -> -K

Sentences with the word “unless”

Just like conjunctions do not have to contain the word “and”, conditionals do not always need the phrase “if…, then….”.

Consider: The grass will die unless it rains.

‘Unless’ means ‘if not’.

To translate into symbols a sentence with ‘unless’, think of: ‘If not ________, then ________’.

So the sentence: The grass will die unless it rains can be translated as – R -> D.

Now translate this sentence: I will go to the beach, unless I am sick.

Write your translation here:  -S -> B

There are other conditional sentences that do not need the word “If” or the word “unless.”

Consider this sentence:

Keep on bothering me and I will hit you.

The meaning of this sentence is clearly not that I am telling you to bother me and then I will hit you! Therefore, you should translate this sentence not as B & H   but rather as B-> H .

The sentence is really a conditional of the form:

If you ……., then I will…….

How about this sentence: “Keep trying, and you will succeed.”

Please translation here the sentence above: K -> S

How about:  “One false move and I shoot.”   Please translate this sentence.

Your translations: F -> S

Consider this sentence:

“Unless you tell me the truth, I will not speak with you.”  Is the translation below a correct one?

– T  ->  – S CORRECT

(Negation of T, followed by an arrow, the symbol for a conditional sentence, followed by negation of S)

Again, all conditional sentences are of the kind:  “If it rains, then I will stay home.”  Sentences with the word “unless” also have the logic of a condition.

Translate into symbolic logic this sentence: “Unless I study, I will fail the exam.”

Translate here: -S -> F

Now create your own sentence with the word “unless”, and then translate that sentence into symbols.

Unless he says he is sorry, I am not talking to him –S -> -T 

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Question 

Worksheet Three

Philosophy 170: Introduction to Logic

Symbolic Notations

Symbolic Notations

Week 3

Day 5. Wednesday, February 17

Worksheet Three

*Please read everything below carefully and fill in whenever you are asked to fill in your answers.

Topic: Symbolic Notations

We already learned that how to symbolize declarative sentences.

For example, this declarative sentence:  “You studied and you got an A on the exam” can be put into symbolic notations likes this:

S & A

This notation stands for “You studied (S) and (&) you got an A on the exam (A).”

But now we will learn how to symbolize conditional sentences, meaning sentences that do not have the word “and” but instead have the phrase “If….then…” as in: “”If you study, then you will get an A.”

 

Well, to convey in symbols the idea of something conditional or hypothetical, we will use the symbol of an arrow ->  as in:     A -> B

So when you see the arrow between two letters, you should read the notation like this: If A, then B.

Practice:

X -> Y   this stands for: IF X, then Y

P -> Q    this stands for: If P, then Q

M -> T  this stands for: If M then T   (as is: If today is Monday, then tomorrow is Tuesday.)

Meaning:

If M is true, then T is true; meaning: IF it’s true that today is Monday, then it must be true that tomorrow is Tuesday.

Now your turn.  Put these 3 sentences into symbolic notations.  If you do not know how to type an arrow let me know asap.

  1. If today is Wednesday, then tomorrow is Thursday.
  2. If today is Friday, then tomorrow is Saturday.
  3. If today is Sunday, then I do not have to work.

Remember, the letters are just symbols.  You may use different letters.  For instance:

“If you study hard, you will pass the exam.”

This sentences can be translated as:

S -> P    or  H -> E

Remember, every conditional statement divides into two constituents, which do not play equivalent roles (in contrast to conjunction and disjunction). Once again, the constituents of a conditional A-> B are respectively called the antecedent and the consequent.

The word ‘antecedent’ means “that which leads”, and the word ‘consequent’ means “that which follows”.

Now your turn; symbolize these conditional sentences:

Remember: ‘If’ introduces the antecedent, ‘then’ introduces the consequent.

  1. If I have the money, I will buy a new car.
  2. If it rains, I will go to the movies.
  3. I will go to the movies, if it does not rain.
  4. If you tell me the truth, I will trust you.
  5. I will trust you, if you tell me the truth.
  6. If it rains, I will cancel the trip.
  7. I will work late, if I have the energy.
  8. If I have the time, I will call them.
  9. I will pay the bill, if I get the money.

Consider again the conditional sentence: If the weather is nice, I will go to the beach.

Can I say the same thing but in a different way?  Yes:  I will go to the beach, if the weather is nice.

Am I saying the same thing in the second sentence?  Do I mean exactly the same?

Yes. The meaning of these two sentences is identical.

This is why the symbolic notation of the two sentences below will be exactly the same:

If the weather is nice, I will go to the beach.        N->B

I will go to the beach, if the weather is nice.        N->B

Please do NOT translate the second one as:  B -> N.    You would be saying here that if you go to the beach then the weather will suddenly become nice.  But this is absurd.  The weather does not care if you go to the beach.   You cannot make the weather nice by going to the beach.

So go back to sentences: 3, 5, and 7 and make corrections if necessary.

Now create two conditional sentences, first in English, and then translate them into symbols.

1.

2.

Translate the sentences into symbols. Then say which part of the sentence is the antecedent and which part is the consequent.

  1. I will work late, if I have the energy.
  2. If I have the time, I will call them.
  3. If you are born in America, you are automatically an American citizen.
  4. I would be right now in Russia, if New York were a Russian territory.
  5. If the universe is expanding, then space has to be finite.

Consider the sentence: I will not help you, if you do not tell me the truth.

Let, H will stand for “I will help you.” T will stand for “You tell me the truth.”

Only one is a proper translation of the sentence above. Which one?  What does the other (incorrect) translation say?

  1. – H ® – T If – H, then – T
  2. – T ® – H               If – T, then – H

Translates the following conditional sentences into symbols (the first is already translated):

  1. If I study too hard, I will not enjoy life. S-> – E
  2. If I do not go to the meeting, I will not be able to register my company.
  3. If he catches you, he will hurt you.
  4. If you look outside, you will see a mountain.
  5. If Oswald didn’t shoot Kennedy, someone else did.
  6. I do math well, if I drink a lot of coffee.
  7. You will have money, if you work.
  8. If you do not pay, I will pay.
  9. You will not get the job, if you do not apply.

Now create 5 conditional sentences by yourself; then translate the sentences into symbols.

1.

2.

3.

4.

5.

Sentences with the word “unless”

Just like conjunctions do not have to contain the word “and”, conditionals do not always need the phrase “if…, then….”.

Consider: The grass will die unless it rains.

‘Unless’ means ‘if not’.

To translate into symbols a sentence with ‘unless’, think of: ‘If not ________, then ________’.

So the sentence: The grass will die unless it rains can be translated as – R -> D.

Now translate this sentence: I will go to the beach, unless I am sick.

Write your translation here:

There are other conditional sentences that do not need the word “If” or the word “unless.”

Consider this sentence:

Keep on bothering me and I will hit you.

The meaning of this sentence is clearly not that I am telling you to bother me and then I will hit you! Therefore, you should translate this sentence not as B & H   but rather as B-> H .

The sentence is really a conditional of the form:

If you ……., then I will…….

How about this sentence: “Keep trying, and you will succeed.”

Please translation here the sentence above:

How about:  “One false move and I shoot.”   Please translate this sentence.

Your translations:

Consider this sentence:

“Unless you tell me the truth, I will not speak with you.”  Is the translation below a correct one?

– T  ->  – S

(Negation of T, followed by an arrow, the symbol for a conditional sentence, followed by negation of S)

Again, all conditional sentences are of the kind:  “If it rains, then I will stay home.”  Sentences with the word “unless” also have the logic of a condition.

Translate into symbolic logic this sentence: “Unless I study, I will fail the exam.”

Translate here:

Now create your own sentence with the word “unless”, and then translate that sentence into symbols.