Notes
In recitations you will be expected to work in groups to solve challenging problems illustrating the course content covered by Sean in the preceding Wednesday, Thursday, Monday classes where applicable. I will provide different ways to approach the same content that will challenge you to think creatively about it, and so improve your understanding. You should expect to think hard during recitation. I typically walk around and provide guidance, a joke to lighten the mood, some formal strategy, some strategy for solving problems under pressure, etc.
Disjunctions should be rather familiar by now. Let’s remind ourselves of some examples. Consider the following:
John is bald or John is a man
Now, we know that only one side has to be true for this to be evaluated as true. Both sides are true. What about the following:
John has hair or John is a man
Again, this should evaluate to true since at least one side of the disjunction is true. We would translate the first sentence as:
BvM
And we would translate the second sentence as:
HvM
What if we translated the first sentence in the following manner:
MvH
Do you think this changes the meaning of the sentence? No, this does not. Disjunctions, like conjunctions, are commutative. We can also build them up as we have done with the conjunctions. That is, we can build longer disjunctions in the following manner:
HvFvN
This might capture the meaning of some sentence like the following:
John either has hair or is funny or is neat
We can also combine the disjunction and conjunction in various ways. Consider the following;
John is either going to the beach and swimming or laying around at home
We might translate this sentence in the following way:
(B&S)vL
This may lead to more complex sentences such as the following:
John is a bald instructor or a bald student
Which could be captured as:
(B&I)v(B&S)
Not both
Now that we have covered conjunctions and negations, and introduced a disjunction into the truth table, now let’s look at disjunction in more depth. More specifically, let’s look at the negations of disjunctions. That is, let’s look at sentences like the following:
It is not the case that either John has hair or John is a woman.
Now, this is true. This should be straightforward for our symbolization from English into the connectives. We will symbolize this in the following manner:
~(HvW)
Is this previous statement equivalent to:
~Hv~W
What do you think? This sentence says that it is not the case that John has hair or it is not the case that John is a woman.
I think they are not equivalent. The first translation above is much stronger than the second. I do not want to leave open the possibility that one of these is true. I am making a claim that both of the disjuncts is false. That is,
~H&~W
Now, in many circumstances you won’t hear the sentence as we have prepared it here. Our English is a little stilted. I have used the locution ‘it is not the case that John has hair or is a woman’. Instead, you will hear something like the following:
John is neither bald nor a woman
We will translate sentences using neither nor as the negation of disjunctions. Since this locution is the logical equivalent of saying either of the following two sentences:
(1) John does not have hair and John is not a woman
(2) It is not the case that John has hair and it is not the case that John is a woman
Then we will translate these sentences all as the negation of disjunction. Thus, all of these sentences are logically equivalent for our purposes.
Conditional
With our conditional we have an antecedent and a consequent. The antecedent recall comes first and the consequent second. There are many ways you will see this written in English. Think about translations of the following conditional sentences:
(1) If John goes to the store John will buy candy
(2) John buys candy if John goes to the store
(3) In case John goes to the store, John buys candy
(4) Provided that John goes to the store, John buys candy
We would translate all of these in the following manner:
(1) S -> C
This is because they all have the same form. The ‘if’ you will note, when it is apparent, will always introduce the antecedent while the ‘then’ will always introduce the consequent. Provided, on the condition that, in case, etc., are all equivalent for our purposes to ‘if’. So when you see them, be sure to translate them as the antecedent. Let’s see how well you have mastered this concept:
Translate the following sentence into our symbolic language using the material conditional:
If John went to the store, then either John bought candy or John bought mints and John did not buy both candy and mints
The following would be an acceptable translation:
S -> [(CvM)&~(C&M)]
Do you think this captures the meaning of the sentence. I hope you recall the meaning of the second portion of the sentence. We have an exclusive sense of or going on in the consequent of the material conditional. Now, what is the main connective of the sentence? Exactly, this is the conditional. Now, let’s look at a few more examples of English translations with the material conditional.
Only If
Consider the following sentences:
(1) John will buy candy only if John goes to the store
(2) John will buy candy if John goes to the store
Should they be translated as the following:
(1) C -> S
That is, should we translate both of these sentences in the same way? What about the other direction? Should we translate these sentences in the following direction?
(2) S -> C
What do you think?
I should point out that it is not necessary that we have to translate them both into the same form. Indeed, one will be translated in one direction and the other in the other direction. Recall that if sentences always introduce the antecedent. That suggests that the second sentence should be translated as above. What about the other sentence? It has only if rather than simply the if. Don’t let this fool you though! When you see only if, this is different from simply the if. When you see ‘only if’ you will need to note that this introduces the consequent. That is, the first translation will be translated in the first manner.
Now, consider the following:
(1) If John goes to the store then John buys candy
(2) If it is not the case that John buys candy then it is not the case that John goes to the store
How might we symbolize these? Get together and provide a solution:
(1) S -> C
(2) ~C -> ~S
These are logically equivalent. That suggests that a comparable way to translate sentences such as ‘only if’ would be as (2) above. That is, sentences like the following:
(1) If John doesn’t go to the store then John doesn’t buy candy
Can be translated in the following:
(2) If John buys candy then John goes to the store
(3) John buys candy only if John goes to the store
(4) ~S -> ~C
(5) C -> S
What about the following:
John’s tree will grow only if it receives enough light
T -> L
This is misleading though. This says that if the tree grows then it will have enough light. Yet, this is incorrect. This direction suggests that there is a causal relation between the tree growing because it has enough light in the sense that growing comes first or precedes the acquisition of light. The problem is the tense involved. The material conditional when combined with tense may lead us astray. We will avoid many of these constructions. We will not try to capture tense with this logic. There are more advanced logics that can handle them though. We have to stop somewhere! Note, you probably remember a similar argument provided when we discussed certain conjunction sentences. I repeat here, we won’t be able to translate such sentences in our logic.
If and Only If
We can introduce another derived connective today. This will be what we call the biconditional. The name should be suggestive since it is simply two conditionals. One will go one direction and the other in the other. Consider the following:
John goes to the store if and only if John buys candy
We will learn to translate this in the following manner:
S <- -> C
You can surely see that the conditionals go both ways. Now consider the following:
If John goes to the store then John buys candy and if John buys candy then John goes to the store
This reveals the structure of the bi-conditional. Try to formalize this without the two arrows. Just use the other connectives that we have.
Hopefully, your construction looks like the following:
(C -> S) & (S -> C)
Our bi-conditional is the abbreviation of this. Anytime you have a bi-conditional you will have the opportunity to replace it with this as they are logically equivalent..
Unless
One final translation will be considered today. Consider the following:
John goes to the store unless John buys candy
This will be tricky. I want you to think about unless in the following manner. When you see unless between two sentences, I want you to translate what follows as ‘if not’. So, the above sentence will be translated as:
~C -> S
Now, I want you to think with me for a moment about this sentence as compared to the following:
John goes to the store or John buys candy
How do you think we would translate this sentence?
CvS
Note, that they are logically equivalent! That suggests that when you translate unless sentences, you may translate them as disjunctions. For instance,
John goes to the store unless John buys candy
Is readily translated as:
SvC
Or even as:
~C -> S
That ends our discussion for today.