The version of logical language we’re using is often called Sentential Logic or SL. It is called sentential logic, because the basic units of the language will represent entire sentences.
In SL, capital letters are used to represent basic sentences. Considered only as a symbol of SL, the letter A could mean any sentence. So when translating from English into SL, it is important to provide a symbolization key. The key provides an English language sentence for each sentence letter used in the symbolization. For example, consider this argument:
There is an apple on the desk.
If there is an apple on the desk, then Jenny made it to class.
∴ Jenny made it to class.
This is obviously a valid argument in English. In symbolizing it, we want to preserve the structure of the argument that makes it valid. What happens if we replace each sentence with a letter? Our symbolization key would look like this:
A: There is an apple on the desk.
B: If there is an apple on the desk, then Jenny made it to class.
C: Jenny made it to class.
We would then symbolize the argument in this way:
There is no necessary connection between some sentence A, which could be any sentence, and some other sentences B and C, which could be any sentences. The structure of the argument has been completely lost in this translation. The important thing about the argument is that the second premise is not merely any sentence, logically divorced from the other sentences in the argument. The second premise contains the first premise and the conclusion as parts. Our symbolization key for the argument only needs to include meanings for A and C, and we can build the second premise from those pieces. So we symbolize the argument this way:
If A, then C.
This preserves the structure of the argument that makes it valid, but it still makes use of the English expression `If… then…' Although we ultimately want to replace all of the English expressions with logical notation, this is a good start.
The sentences that can be symbolized with sentence letters are called atomic sentences, because they are the basic building blocks out of which more complex sentences can be built. Whatever logical structure a sentence might have is lost when it is translated as an atomic sentence. From the point of view of SL, the sentence is just a letter. It can be used to build more complex sentences, but it cannot be taken apart.
There are only twenty-six letters of the alphabet, but there is no logical limit to the number of atomic sentences. We can use the same letter to symbolize different atomic sentences by adding a subscript, a small number written after the letter. We could have a symbolization key that looks like this:
A1: The apple is under the armoire.
A2: Arguments in SL always contain atomic sentences.
A3: Adam Ant is taking an airplane from Anchorage to Albany.
A294: Alliteration angers otherwise affable astronauts.
Keep in mind that each of these is a different sentence letter. When there are subscripts in the symbolization key, it is important to keep track of them.
Here is an interesting thing to note, and it makes sense. Whenever a variable is defined, we use capital letters and you can use any one you want, and it is often a good idea to choose one that represents the sentence well, like using B for “Barbara is awesome.” However, when we are
talking abstractly or discussing rules in general, we use lower-case letters, italicize them, and often start with p and go from there.
Logical connectives are used to build complex sentences from atomic components. There are five logical connectives in SL. They are summarized below. Today we’re looking at the first 3 and we’ll be looking at the other 2 in another lesson.
~ = negation: `It is not the case that p', ~p
& = conjunction: `Both p and q’ p & q
v = disjunction: `Either p or q' p v q (yes, that’s just a lower case v, but technically it’s a different character)
⊃ = conditional: `If p then…q' p ⊃ q (you can copy and paste “⊃” or you can use >)
↔ = biconditional: ‘p if and only if q’ p ↔ q (you can copy and paste “↔” or you can use <>)
Consider how we might symbolize these sentences:
1. Mary is in Barcelona.
2. Mary is not in Barcelona.
3. Mary is somewhere besides Barcelona.
In order to symbolize sentence 1, we will need one sentence letter. We can provide a symbolization key:
B: Mary is in Barcelona.
Note that here we are giving B a different interpretation than we did in the previous section. The symbolization key only specifies what B means in a specific context. It is vital that we continue to use this meaning of B so long as we are talking about Mary and Barcelona. Later, when we are symbolizing different sentences, we can write a new symbolization key and use B to mean something else.
Now, sentence 1 is simply B.
Since sentence 2 is obviously related to the sentence 1, we do not want to introduce a different sentence letter. To put it partly in English, the sentence means `Not B.' In order to symbolize this, we need a symbol for logical negation. We will use ~.
Now we can translate `Not B' to ~B, which is sentence 2.
Sentence 3 is about whether or not Mary is in Barcelona, but it does not contain the word `not.' Nevertheless, it is logically equivalent to sentence 2.
They both mean: It is not the case that Mary is in Barcelona.
As such, we can translate both sentence 2 and sentence 3 as ~B. A sentence can be symbolized as ~A if it can be paraphrased in English as `It is not the case that A.' Consider these further examples:
4. The widget can be replaced if it breaks.
5. The widget is irreplaceable.
6. The widget is not irreplaceable.
If we let R mean `The widget is replaceable', then sentence 4 can be translated as R.
What about sentence 5? Saying the widget is irreplaceable means that it is not the case that the widget is replaceable. So even though sentence 5 is not negative in English, we symbolize it using negation as ~R.
Sentence 6 can be paraphrased as `It is not the case that the widget is irreplaceable.' Using negation twice, we translate this as ~R. The two negations in a row each work as negations, so the sentence means `It is not the case that it is not the case that R.' If you think about the sentence in English, it is logically equivalent to sentence 4. So when we define logical equivalence is SL, we will make sure that R and ~~R are logically equivalent.
7. Elliott is happy.
8. Elliott is unhappy.
If we let H mean `Elliot is happy', then we can symbolize sentence 7 as H. However, it would be a mistake to symbolize sentence 8 as ~H. If Elliott is unhappy, then he is not happy, but sentence 8 does not mean the same thing as `It is not the case that Elliott is happy.' It could be that he is not happy but that he is not unhappy either. Perhaps he is somewhere between the two. In order to symbolize sentence 8, we would need a new sentence letter.
For any sentence A: If A is true, then ~A is false. If ~A is true, then A is false.
Using `T' for true and `F' for false, we can summarize this in a characteristic truth table for negation:
Consider these sentences:
9. Adam is athletic.
10. Barbara is athletic.
11. Adam is athletic, and Barbara is also athletic.
We will need separate sentence letters for 9 and 10, so we define this symbolization key:
A: Adam is athletic.
B: Barbara is athletic.
Sentence 9 can be symbolized as A.
Sentence 10 can be symbolized as B.
Sentence 11 can be paraphrased as `A and B.' In order to fully symbolize this sentence, we need another symbol. We will use `&.' We translate `A and B' as A&B. The logical connective `&' is called conjunction, and A and B are each called conjuncts.
Notice that we make no attempt to symbolize `also' in sentence 11. Words like `both' and `also' function to draw our attention to the fact that two things are being conjoined. They are not doing any further logical work, so we do not need to represent them in SL.
Some more examples:
12. Barbara is athletic and energetic.
13. Barbara and Adam are both athletic.
14. Although Barbara is energetic, she is not athletic.
15. Barbara is athletic, but Adam is more athletic than she is.
Sentence 12 is obviously a conjunction. The sentence says two things about Barbara, so in English it is permissible to refer to Barbara only once. It might be tempting to try this when translating the argument: Since B means `Barbara is athletic', one might paraphrase the sentences as `B and energetic.' This would be a mistake. Once we translate part of a sentence as B, any further structure is lost. B is an atomic sentence; it is nothing more than true or false. Conversely, `energetic' is not a sentence; on its own it is neither true nor false. We should instead paraphrase the sentence as `B and Barbara is energetic.' Now we need to add a sentence letter to the symbolization key. Let E mean `Barbara is energetic.' Now the sentence can be translated as B &E. A sentence can be symbolized as A &B if it can be paraphrased
in English as `Both A, and B.' Each of the conjuncts must be a sentence.
Sentence 13 says one thing about two different subjects. It says of both Barbara and Adam that they are athletic, and in English we use the word `athletic' only once. In translating to SL, it is important to realize that the sentence can be paraphrased as, `Barbara is athletic, and Adam is athletic.' This translates as B &A.
Sentence 14 is a bit more complicated. The word `although' sets up a contrast between the first part of the sentence and the second part. Nevertheless, the sentence says both that Barbara is energetic and that she is not athletic. In order to make each of the conjuncts an atomic sentence, we need to replace `she' with `Barbara.'
So we can paraphrase sentence 14 as, `Both Barbara is energetic, and Barbara is not athletic.' The second conjunct contains a negation, so we paraphrase further: `Both Barbara is energetic and it is not the case that Barbara is athletic.' This translates as E &:B.
Sentence 15 contains a similar contrastive structure. It is irrelevant for the purpose of translating to SL, so we can paraphrase the sentence as `Both Barbara is athletic, and Adam is more athletic than Barbara.' (Notice that we once again replace the pronoun `she' with her name.) How should we translate the second conjunct? We already have the sentence letter A which is about Adam's being athletic and B which is about Barbara's being athletic, but neither is about one of them being more athletic than the other. We need a new sentence letter. Let
R mean `Adam is more athletic than Barbara.' Now the sentence translates as B &R.
Sentences that can be paraphrased `A, but B' or `Although A, B' are best symbolized using conjunction: A &B. It is important to keep in mind that the sentence letters A, B, and R are atomic sentences. Considered as symbols of SL, they have no meaning beyond being true or false. We have used them to symbolize different English language sentences that are all about people being athletic, but this similarity is completely lost when we translate to SL. No formal language can capture all the structure of the English language, but as long as this structure is not important to the argument there is nothing lost by leaving it out.
Which of these three statements are saying the same thing?
1) Mike and George are boxers
2) Mike is a boxer and George is a boxer
3) Mike and George are boxing each other
The first 2 are the same – but the third says something different. Although it involves the word “and” it is not being used as a conjunction. It is telling us that they are partaking in an action with the other. If we were to treat it as a conjunction, it would be: Mike is boxing each other and George is boxing each other. However, we could say that this sentence is saying, “Mike is boxing George and George is boxing Mike”, but this is changing things up a bit. However, the
two sentences would be expressing nearly the same ideas. Remember that a conjunction just joins two propositions.
For any sentences A and B, A &B is true if and only if both A and B are true. We can summarize this in the characteristic truth table for conjunction:
A B A & B
T T T
T F F
F T F
F F F
Conjunction is symmetrical because we can swap the conjuncts without changing the truth-value of the sentence. Regardless of what A and B are, A &B is logically equivalent to B &A.
Are these valid arguments?
“Harry is short and John is tall, therefore Harry is short.”
“Harry is short. John is tall. Therefore, Harry is short and John is tall.”
They are both valid. Think about validity means and then about what is being said. Remember that in a valid argument, if the premises are true, the conclusion must also be true. In other words, if the premises are true, there is no way that the conclusion can ever be false. So, with both of these arguments, if the premises are true, the conclusions have to be true as well. Thus, they are valid. They are sound if Harry is actually short and John is actually tall, but when doing symbolic logic, we just care about validity. Soundness is much too practical and important.
Looking into the future, this is what truth tables will be telling us: we’ll be making complex ones that will let us know when all of the premises are true, and if the conclusion is true every time that the premises are all true, then the argument is valid.
Consider these sentences:
16. Either Denison will play golf with me, or he will watch movies.
17. Either Denison or Ellery will play golf with me.
For these sentences we can use this symbolization key:
D: Denison will play golf with me.
E: Ellery will play golf with me.
M: Denison will watch movies.
Sentence 16 is `Either D or M.' To fully symbolize this, we introduce a new symbol. The sentence becomes D vM. The `v' connective is called disjunction, and D and M are called disjuncts.
Sentence 17 is only slightly more complicated. There are two subjects, but the English sentence only gives the verb once. In translating, we can paraphrase it as. `Either Denison will play golf with me, or Ellery will play golf with me.' Now it obviously translates as D v E.
A sentence can be symbolized as A v B if it can be paraphrased in English as `Either A, or B.' Each of the disjuncts must be a sentence.
Sometimes in English, the word `or' excludes the possibility that both disjuncts are true. This is called an exclusive or. An exclusive or is clearly intended when it says, on a restaurant menu, `Entrees come with either soup or salad.' You may have soup; you may have salad; but, if you want both soup and salad, then you have to pay extra.
At other times, the word `or' allows for the possibility that both disjuncts might be true. This is probably the case with sentence 17, above. I might play with Denison, with Ellery, or with both Denison and Ellery. Sentence 17 merely says that I will play with at least one of them. This is called an inclusive or.
The symbol `v' represents an inclusive or. So D v E is true if D is true, if E is true, or if both D and E are true. It is false only if both D and E are false. We can summarize this with the characteristic truth table for disjunction:
A B A v B
T T T
T F T
F T T
F F F
Like conjunction, disjunction is symmetrical. AvB is logically equivalent to BvA.
These sentences are somewhat more complicated:
18. Either you will not have soup, or you will not have salad.
19. You will have neither soup nor salad.
20. You get either soup or salad, but not both.
We let S1 mean that you get soup and S2 mean that you get salad.
Sentence 18 can be paraphrased in this way: `Either it is not the case that you get soup, or it is not the case that you get salad.' Translating this requires both disjunction and negation. It becomes S1 v S2.
Sentence 19 also requires negation. It can be paraphrased as, `It is not the case that either that you get soup or that you get salad.' We need some way of indicating that the negation does not just negate the right or left disjunct, but rather negates the entire disjunction. In order to do this, we put parentheses around the disjunction: `It is not the case that (S1 vS2).' This becomes simply ~(S1 v S2). Notice that the parentheses are doing important work here. The sentence ~S1vS2 would mean `Either you will not have soup, or you will have salad.'
Sentence 20 is an exclusive or. We can break the sentence into two parts. The first part says that you get one or the other. We translate this as (S1 v S2). The second part says that you do not get both. We can paraphrase this as, `It is not the case both that you get soup and that you get salad.' Using both negation and conjunction, we translate this as ~(S1 &S2). Now we just need to put the two parts together. As we saw above, `but' can usually be translated as a conjunction. Sentence 20 can thus be translated as (S1 v S2)& ~(S1 &S2).
Although `v' is an inclusive or, we can symbolize an exclusive or in SL. We just need more than one connective to do it.
You must use parenthesis in logic like you might anywhere else. AS the previous example showed you, leaving them out can completely change the meaning of a sentence. The rules are simple: use parentheses to connect only 2 statements at a time and when 2 things are in a parentheses, they basically become one thing. So, if I wanted to symbolize
“Adam went to the store, Mark went to the play, John did not go to the store, and Belle did not go to the play” we could write
(A & M) & (~J & ~B) or
A & (M & (~J & ~B)) or a whole lot of other ways by shifting around parentheses.
(I’ll leave it to you to figure out what each letter represents)