Week 12 – Conditional and Bi-Conditional Sentences

This week we turn to conditional sentences that “go in both directions.”

What does this mean? First ask yourself whether the sentence below is still true when you reverse the condition:

If you live in NY, then you live in the US. N->U

So, is this sentence still true when we say:

If you live in the US, then you live in NY? No. Thus, this conditional sentence N -> U goes only in one direction.

Again: N-> U is true

U-> N is false

Likewise here:

If you have a dog, then you have a pet. D -> P

This claim is true. But if you reverse the condition, the sentence will be false:

If you have a pet, then you have a dog. P -> D = False

Same here:

If you are a woman, then you are a human being. W-> H True

If you are a human being, then you are a woman H-> W False

Now your turn.

Create a conditional sentence that is true but it’s not true when you reverse the condition.

If you are a vegetarian, then you eat green vegetables V->G True

If you eat green vegetables, then you are a vegetarian G->V False

In sum, is clear that the above conditions go only in one direction, meaning that it is not true that:

If you live in the US then you live in NY. (Not true)

If you are a human being, then you are a woman. (Not true)

If you have a pet then you have a dog. (Not true)

This is why:

(D v C) -> P is not the same as P -> (D v C)

N -> U is not the same as U-> N

W -> H is not the same as H -> W

Same for: “If it rains, I will stay home.” R -> S

This is different from S ->R IF S, then R says that IF I stay home (S), then it will rain (R). This is not true, since me staying home cannot magically cause the rain. The rain does not care about me. I cannot cause the rain.

So, are there any conditions that go in both directions?

Yes, consider this:

If you are Kamala Harris, then you are the current Vice President of the US.

K-> V

If you are the current Vice President of the US, then you are Kamala Harris.

V-> K

Now your turn.

Think of a true conditional sentence that will still be true when you reverse the condition.

If ice is subjected to heat, then it will melt

If ice melts then, it is subjected to heat

More examples:

“If an atom has 79 protons, then that atom is an atom of gold (Au).”

79->Au Meaning if this atom has 79 protons then it must be gold. And

since only gold has 79 protons, we can also say:

Au -> 79 Meaning, if this is gold then it has 79 protons.

(An atom that has one more proton, 80 protons, is Mercury, and if there are 78 protons then it is Platinum, and so on)

Thus, we can read this condition in both directions:

79 -> Au (If the atom has 79 protons, then it is an atom of gold.)

Au->79 (If it’s gold, then it has 79 protons.)

Again, the condition is reversible. We can say that this condition goes in both directions.

79 -> Au is true and Au -> 70 is also true.

So it is both necessary for an atom of gold to have 79 protons and it is also sufficient to have 79 protons to be an atom of gold.

Your turn:

Think of a specific atom (a chemical element). You may want to consult the periodic table. Using that chemical element as an example, create here your own conditional sentence that goes in both directions.

If an atom has 30 protons, then it must be Zinc

If it is Zinc, then it has 30 protons

30->Zn

Zn->30

To express the idea that a conditional sentence goes in both directions, meaning that the condition is not only sufficient but also necessary, we will draw the arrow head in both directions, like this: A <-> B

Logicians call these double conditionals that go in both directions:

Bi-conditionals.

Notice again, the double arrow, pointing in both directions <->

In effect we get:

A <->B which reads: If A, then B & If B, then A.

Likewise:

X <->Y reads as: If X, then Y & If Y, then X.

Notice also the “&” symbol.

Now, recall the sentence:

“If you score a 100 on the exam, you will earn an A.”

At the beginning of the semester, we most of you wrote that it is possible that getting 99 points will still get you an A, even 98 points will get you an A, and so on. So the conditional sentence above is not reversible. Just because you got an A, this does not necessarily mean you got 100 points.

But imagine a case when the teacher says: “You will receive an A+ if and only if you score 100 points.” Here the condition will be an example of a bi-conditional. It is not just sufficient but also necessary to earn 100 points to get an A+. Getting 99 is not enough to earn A+.

In the case of such a strict condition, we will use again the bi-conditional symbol. So: Only if you score 100, then you will get A+ will be symbolized as:

H <-> A+

Meaning if you get 100, then you will get an A+, but also if you got an A+, then we will know that you earned 100 points, since obtaining this perfect score was the only possibility to receive A+.

This means the bi-conditional H <-> A+ is logically equivalent to two things simultaneously:

H -> A+ and A+ -> H (at once)

Thus:

H <-> A is equivalent to a conjunction of two conditionals:

( H -> A ) & ( A -> H )

Again, this notation: (H-> A) & (A-> H) is called a conjunction of conditionals.

In effect: H <-> A = ( H -> A ) & ( A -> H )

Your turn:

Write what the bi-conditionals below are equivalent to:

P <-> Q = (P->Q) & (Q->P)
A <->B = (A-.>B) & (B->A)
A <-> (B & C) = [A->(B&C) & (B&C)->A]
(A v D) <-> (B & C) = [(AvD) -> (B&C)] & [(B&C)->(AvD)]

Now check your answers:

P <-> Q = (P -> Q) & (Q -> P)
A <-> B = (A -> B) & (B -> A)
A <-> (B & C) = [A -> (B & C)] & [(B &C) -> A)]

*Yes, in problem (3) above, you need additional square brackets that are bigger than the round brackets.

(A v D) <-> (B & C) = [(A v D) -> (B & C)] & [(B &C) -> (A v D)]

Now your turn:

Write what the bi-conditionals below are equivalent to:

K <-> Y = K->Y & Y->K
( A & B) <-> C = [(A&B) ->C] & [C->(A&B)]
P <-> ( A v B) = [P->(AvB)] & [(AvB)->P]
A <->[ ( B v C ) & D ] = {A->[(BvC) &D]} & {[(BvC) & D] & A}
K <-> ( P & R ) = [K->(P&R)] & [(P&R)->K]

Consider now these two sentences:

I’ll prepare the meal, if you wash the dishes.
I’ll prepare the meal only if you wash the dishes.

Let’s say the letter M will stand for “I prepare the meal.”

The letter W will stand for “You wash the dishes.”

We will translate the first sentence as:

W -> M

(If W, then M) Keep in mind the logic of the first sentence is this: IF you wash the dishes (W), I will prepare the meal (M).

Again, the first sentence does not begin with ‘if’ but you already know that the ‘if’ in the second part of the sentence is really the antecedent, so the W letter has to come before the arrow.

In other words, I will prepare the meal under this condition – someone has to wash the dishes.

But here the truth is that this condition is not absolutely necessary. Even if you do not wash the dishes, I may still prepare the meal.

Just like in the sentence: If you are in New York, you are in the US.

But being in NY is not absolutely necessary to be right now in the US.

In the second sentence, however, the deal is different:

I’ll prepare the meal only if you wash the dishes.

Just like in the sentence: You are the current Vice President of the US, only if you are Kamala Harris.

So the condition is strict.

Only if you agree to do the dishes, I will make the food. So if you don’t do the dishes, then forget it, I’m not making anything to eat.

So this is an instance of a bi-conditional. We will have to use the double arrow for this sentence. W <-> M

Your turn.

Create a sentence that contains the phrase “Only if….”.

Explain what you are really saying in this sentence. Is the sentence an example of a bi-conditional sentence? Explain briefly and then write the sentence in symbols.

I will go on a morning run with you if you wake up at 6 am

I will go for the morning run with you only if you wake up at 6 am

In the first sentence, I will tag the person along for a morning run if he wakes up at 6ma; if he does not wake up at that time, I may still go for the run without him

In the second sentence, I will go with the person for a morning run if only he wakes up at 6 am; if the person wakes up at 5 am or perhaps at 7 am, then I will not go with the person for the run

T<->R where R is run, and T is time (6 am)

More examples. Compare these two scenarios:

a) “If you tell me your secret, I will divulge my secret.”
b) “Only if you tell me your secret, I will tell you my secret.”

In the first sentence, I actually may divulge my secret regardless of what you decide to do. But in the second sentence, we both have to reveal our secrets. That’s the deal.

Now try to write the two sentences above in symbols. (Determine which sentence will use the double arrow head and which will be a simple right arrow head.)

Sentence I: DS->TS where TS is Tell secret, and DS is Divulge secret.

Sentence 2: DS<->TS

Your turn again.

Create two similar conditional sentences, where the second sentence contains the phrase “Only if….”.

Then try to symbolize both sentences.

I will drink the vodka if you will buy it

I will drink vodka only if you buy it.

Sentence 1: B->D where D is drink vodka while B is buy the vodka

Sentence 2: B<->D

Now let’s apply what we just learned to deducing conclusions from premises.

Let’s consider an argument where the first premise is a bi-conditional sentence, while the second premise is a single atomic statement, as in:

A <-> B Premise 1.
B Premise 2.

_________________

What should we conclude from these two premises?

Well, we can simply begin by writing in line 3 what the bi-conditional premise in line 1 is equivalent to, like this:

A <-> B Premise 1.
B Premise 2.

___________________________

(A -> B) & (B -> A)

What we just did in line 3 is exactly the same what you did on page 6 and 7.

We have to add that this step is called Bi-conditional Equivalence, which can be can abbreviated as BE. So in line 3 we have to write that we obtained the conjunction from line 1 by BE, like this:

A <-> B Premise 1.
B Premise 2.

___________________________

(A -> B) & (B -> A) 1, BE Again, here the BE stands for :Bi-

conditional Equivalence. Here I am saying

what this bi-conditional sentence in Premise 1

is equivalent to;

it is equivalent to a conjunction.

But what’s the point of what we just wrote in line 3?

As you remember, a conjunction can be simplified. We can “derive” one conjunct from a conjunction. So let’s derive by Simplification in the next line, in line 4, one of the conjuncts (the one in green, the one that says:

IF B then A, meaning: (B->A). Like this:

A <-> B Premise 1.
B Premise 2.

_______

(A -> B) & (B -> A) 1, BE (Bi-conditional Equivalence)
B -> A 3, Simp

Here we simplified line 3. In line 3, we obtained a conjunction. The first conjunct is the content of first parenthesis (in blue), the second conjunct is the second parenthesis (in green). So we have a conjunction that says “this and that”, meaning the expression was a conjunction and we simplified the conjunction to a single conjunct. The conjunct we are interested in is the one in green, the one that says:

IF B, then A. (B->A). Now we can move on to line 5. What can we derive in line 5?

Well notice that in line 2 the premise says that B is a given. So given B, what can we conclude by Modus Ponens? We can conclude A, like this:

A 2,4 MP

Here in line 5 we are saying that we obtained the letter A

from line 2 and line 4 by Modus Ponens.

Now your turn:

What can you derive from these two premises?

M <-> T Premise 1.
T Premise 2.

___________________________

(M->T) & (T->M)
T->M
M

Let’s solve one more argument with a bi-conditional premise and a declarative premise. . The first premise will be: “Only if you score 100, you will get A+”: The second premise will be “Ana got A+.”

So:

If and only if you score 100, then you will get an A+ grade.
Ana got an A+ grade.

________________

What can we conclude from this? Let’s first symbolize the sentences.

The first sentence is a bi-conditional, because the condition goes in both directions. If you get an A+, that means you got 100 points, and if you got 100 points, that means you will get an A+. So we will use the double arrow:

H <-> A+
A+

There are the two premises we have to work with. Let’s begin in line 3 by taking apart the bi-conditional, like this: H <-> A+

A+

__________

(H->A+) & (A+ ->H) 1, BE

I used here BE ( Bi-conditional Equivalence). Now I see that one of the conjuncts (the one in green) will be useful. How? The second conjunct, meaning whatever is in the second parenthesis ( A -> H) says that IF a student got A+, then she got H (Hundred points). This is useful here because in line 2, the premise says that actually we do get the A+. So given that we get the A+, I can conclude that we will get the H. Like this:

H <-> A+
A+

__________

(H->A+) & (A+ ->H) 1, BE
A+ -> H 3, Simp
H 2, 4 MP

Now your turn:

First consider the bi-conditional sentence below:

“Only if this molecule has two atoms of Hydrogen and one atom of Oxygen (M), it is a water molecule (W)”

Let’s symbolize the sentence like this:

M <-> W

So in the argument below, what can you conclude from the two premises?

(Remember to write how you obtained each step.)

M <-> W
W

__________

(M->W) & (W->M)
W->M
M

Part II

Contrapositives

Recall again the sentence:

“Only if you score 100 points, you will get an A+.”

We said that this condition is strict, meaning that if you do NOT get 100 points, then you cannot earn A+.

So this means that this sentences is also true:

– H -> – A+ (If you do NOT get 100, then you do NOT get A+)

Notice that in this sentence: N -> U (If you are NY, then you are in the US) will not be true if you say If you are not in NY, then you are not in the US.

But it will be true to say: If you are not in the US, then you are not in NY.

So, – U -> – N is the contrapositive of N->U

Your turn:

What are the contrapositives of the following conditions:

a) NJ -> US= -US->-NJ
b) A -> B = -B->-A
c) – X -> – Y = Y->X
d) – W -> V = -V->W
e) A -> – B = B->-A
f) (A v B) -> C -= -C->-(AvB)

Now check your answers:

a) NJ -> US is equivalent to: – US -> – NJ
b) A -> B is equivalent to: – B -> – A
c) – X -> – Y is: Y -> X
d) -W -> V is: – V ->W
e) A -> – B is: B -> – A
f) (A v B) -> C is: – C -> – (A v B)

Now your turn:

a) – (M v N) – > – ( O & P) is: (O&P)->(MvN)
b) [(N v C ) v ( P & V)] -> U is: -U-> -[(NvC) v(P&N)]

First, consider concepts such as “freedom,” “pornography,” “justice,” “friendship,” “art,” “disabled,” “mentally ill,” “socialism,” “free market,” “manly”, and “young.”

What do these words means? We can look up definitions of these words and yet the dictionary definitions do not settle what things qualify as pornographic, what things qualify as justice, and what kind of objects qualify as art, music, and so on.

Let’s say someone argues that the word “freedom” means: “the ability to do anything you want to do.”

Is that really freedom? How would we disagree with this person? Do we feel that we are not free just because we want to go on vacation but we can’t right now because of some financial restrictions or health concerns?

On the other hand, even if we can afford vacation, we could say that we are not really free until we are free from fear and oppression. Furthermore, let’s say you take your dog on vacation. Does this mean your dog is free?

All this means that the “the ability to do anything one wants” may not be a sufficient condition for freedom and it may not even be a necessary condition since it would be frightening if everybody could do whatever they want or whenever they want.

Next example:

Let’s say we want to define what is necessary to stay alive. So:

Necessary (but not sufficient) Conditions:

In order to stay alive, we need to eat food and drink water, or other beverages. But this is certainly not sufficient. I can have plenty to eat and drink and still not be able to stay alive, for example, due to intolerable temperature, a disease, or a lack of oxygen.

Sufficient Conditions

Let’s say we want to find a sufficient condition for “US citizen.”

Sufficient (but not necessary) Conditions:

To be a US citizen, it is sufficient to be born in the US. But being born in the US is not necessary to have an US citizenship. You can be born in another country and still become a US citizen.

Likewise:

To be a NY State resident it is sufficient or enough to reside in Brooklyn. But that’s not necessary. You can live in Queens too, for example.

Now your turn:

Give your own original example of necessary and sufficient conditions. More specifically, first think of a word (or a concept or a phrase.)

Your word:

Then, say what is necessary (what property is absolutely necessary) to be that thing, to qualify as that thing you just listed.

Now, say if that necessary thing you just mentioned would be sufficient, meaning is this condition you just described enough to constitute (to have) what you are talking about (the word you wrote)?

Can you think of anything that would indeed be sufficient to have that thing? (Recall in my example – being born in the US is sufficient to be a US citizen, it’s enough. You don’t have to be born in the US to be US citizen, but it’s enough.)

For a woman to be called a mother, she needs to give birth to a child. However, giving birth to a child is not necessary for one to be called a mother because one can adopt a child. A woman needs to only have a child that she cares for enough for the child to perceive it as its mother and for the State to allow the woman to adopt the child.

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