Conditional Sentences
We learned how to identify these sentences.
For example, the sentence "Anna studied and she got an A on the exam" is a conjunction.
Why?
It joins two smaller statements in this form:
______________ and ______________
First: Anna studied.
Second: Anna got an A on the midterm.
We will call these small statements atomic propositions. So, the word "and" should be called "a propositional connective" because it connects propositions.
But consider connecting these propositions with the word "if":
If Anna studied, then she got an A on the exam.
The phrase "If ......, then........" signifies a conditional sentence.
So, conditionals are phrases of the form ‘If...., then'.
Review:
Identify the propositional connectives:
1. ________ and __________
2. _________ or _____________
3. It is not the case that ______________
4. If ___________ then ________________
There are different kinds of conditionals:
Indicative mood: If litmus paper is put in acid, then it will turn red.
Subjunctive mood: If you were to the concert, then you would enjoy it.
The conditionals we will work with in propositional logic are indicative conditionals.
If __________________, then__________________.
In a conditional, the phrase that immediately follows ‘if' is called the antecedent, the phrase that immediately follows ‘then' is called the consequent.
The word ‘then' does not always appear in a conditional:
If you lived here, you would be home by now.
If I do not eat, I get hungry.
We will use an arrow (®) to symbolize conditional sentences:
A ® B
This reads: If A, then B
Or,
If A is true, then B is true.
As I said last week, the letters are just symbols. You may use different letters.
For example, the sentence:
If you study, you will pass the exam.
Can be translated as:
S ® P
Again, the symbol used to abbreviate if-then is the arrow (® ).
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".
In a conditional, the first constituent is called the antecedent, and the second constituent is called the consequent. When a conditional is stated in standard form in English, it is easy to identify the antecedent and the consequent:
‘if' introduces the antecedent
‘then' introduces the consequent
Solve:
Now translates the following conditional sentences into symbolic logic:
5. If I study too hard, I will not enjoy life.
6. If I do not study, I will not pass.
7. If he catches you, he will hurt you.
8. If you look outside, you will see a bird.
9. If it is raining, I will stay home.
10. If Oswald didn't shoot Kennedy, someone else did.
11. I do math well, if I drink a lot of coffee.
12. You will have money, if you work.
13. If you do not pay, I will pay.
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Now read carefully the notes below.
Conjunctions
Not all sentences containing the word ‘and' are correctly analyzed as conjunctions.
John and Susan are roommates.
This is clearly not equivalent to ‘Jane is a roommate' and ‘Susan is a roommate'!
Lyn and Linda are married.
This is not equivalent to ‘Lyn is married' and ‘Linda is married'.
Some sentences have the logical form of conjunction but do not contain the word ‘and'.
The ice-covered river is beautiful but treacherous.
For the purposes of translation, ‘but' is a stylistic variant of ‘and'. Translating ‘but' as ‘and' will correctly represent the truth conditions of the component sentences.
Stylistic variants of ‘and' include ‘however', ‘yet', ‘still', ‘although'.
Solve:
Which of the following sentences are conjunctions, namely, compound sentences that can correctly be symbolized by using ‘&'? For cases that can be symbolized as conjunctions, give a symbolization using a capital letter for each proposition. (Note: In deciding whether a given sentence is a conjunction, consider what the sentence asserts, not whether you can turn the sentence into a conjunction. In addition to ‘but', the following words may function as stylistic variants of ‘and': ‘however', ‘yet', ‘still', ‘although'.)
Example:
Although the waves were breaking, the surf was low.
This sentence is a conjunction. (‘Although' is a stylistic variant of ‘and'.) Let ‘W' = ‘The waves were breaking' and ‘S' = ‘The surf was low'. The symbolization is: W & B
Now your turn:
1. Paul McCartney went one way, and John Lennon went another.
2. A large star is large, but a large flea is small.
3. Although it is raining, I am happy.
4. It is raining, but I am in a good mood.
5. Bonnie and Clyde were classmates.
6. Al and Ann are lovers.
7. Ken and Kathy are siblings.
8. Beth is a top student, not to mention a great athlete.
9. We'll always have death and taxes.
10. Beethoven and Mozart were great composers.
11. Two and two are four.
12. Although she loved him, she left him.
13. All sound arguments are valid, and they have true premises as well.
14. Both the right front fender and the passenger side door were damaged as a result of the accident.
15. Justice and tolerance are valuable.
Read:
Some sentences are more challenging to translate into symbolic logic.
Consider:
Samantha doesn't exercise or count calories.
Let E stand for Samantha exercises.
Let C stand for Samantha counts calories.
What does this sentence really tell us? Samantha neither exercises nor counts calories.
Thus, the sentence should be translated as:
- E & - C
Notice that this is logically equivalent to - (E v C)
- E & - C is equivalent to - (E v C)
In other words, "neither" is equivalent to "not either."
*Notice this is different from negating the whole conjunction.
Again, the sentence A & B means both. Negation of this statement is "Not both," so it would be incorrect to say that the negation of A & B means
Not A and Not B.
"Not both" means - ( A & B),which is equivalent to "either not one or not the other."
"either not one or not the other" means - A v - B
So, - ( A & B) is equivalent to - A v - B
Once again compare these:
- E & - C is equivalent to - (E v C)
and
- ( A & B) is equivalent to - A v - B
Solve:
Translate the two sentences into symbols:
1. Samantha does not both exercise and count calories.
2. Neither stocks nor bonds provide an absolutely risk-free investment. (S = Stocks provide an absolutely risk-free investment opportunity. B = Bonds provide an absolutely risk-free investment opportunity.)
Negations
Consider this sentence:
Greg is not in town.
The negation of this sentence is:
Greg is in town.
Now, your turn. Write down (un full sentences) how you would negate the following sentences:
Solve:
1. Fred is not very smart.
2. Minh isn't a bad student.
3. Not every car is fuel-efficient.
Now Consider this sentence:
Keeping your promises is not always good.
Which of the following is a correct negation of this statement:
a) Keeping your promises is always good.
b) Keeping your promises is always not good.
Now consider how should we negate statements containing the word "some," as in:
Some people can't get enough.
If we say "Some people can get enough," are we really providing a negation of the original claim? Think of an appropriate word that would contradict this claim, in particular the word "some."
Write down your proposal here:
Conditionals
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' translates as ‘if not'.
To translate a sentence with ‘unless', write ‘If not ________, then ________', and fill in the blanks. Letting ‘D' = ‘The grass will die' and ‘R' = ‘It rains', the sentence above is translated as
How would you translate this sentence?
Be careful! Some sentences are disguised conditionals.
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. Rather, me hurting you depends on whether you keep on bother me. So the sentence is really a conditional of the form:
If you ......., then I will.......
How about this sentence:
Keep trying, and you will succeed.
We see here the word "and" but clearly the meaning of this sentence is that you will succeed if you keep trying.
How about:
One false move and I shoot.
Logically Equivalent Statements
Consider whether these two statements are the same:
1. Either you pay or I pay.
2. If you do not pay, I will pay.
Now, translate the sentences into symbolic logic.
We can conclude that:
P v Q is logically the same as - P ® Q
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Next, determine the main connective and write down accordingly whether the statement is a conditional, conjunction, negation, or a disjunction. You solved similar problems last week but this week the exercise is more challenging.
Translate the following sentences using the suggested abbreviations.
1. If nothing is perfect, then I can't be blamed for my mistakes. (N = Nothing is perfect. B = I can be blamed for my mistakes.)
2. But if everything is perfect, I neither make mistakes nor can I be blamed for them. (E = Everything is perfect. M = I make mistakes.)
3. So if I either make mistakes or am blamed for my mistakes, then it is not true that everything is perfect.
4. If Art neither diets nor exercises, he will gain weight. (D = Art diets. E = Art exercises. G = Art gains weight.)
5. If Art doesn't gain weight, then he either diets or exercises.
6. Art neither diets nor exercises, but he still does not gain weight.
7. Dinosaurs cannot be cloned unless scientists can fill in the missing gaps in dinosaur DNA.
8. Dinosaurs cannot be cloned unless scientists can both obtain samples of dinosaur DNA and fill in the missing gaps in dinosaur DNA. (S = Scientists can obtain samples of dinosaur DNA.)