1.4: Connectives

Logical connectives are used to build complex assertions from simpler pieces. There are five logical connectives in Propositional Logic. This table summarizes them, and they are explained below.

symbol	nickname	what it means
\(\lnot\)	not	“It is not the case that ______”
\(\&\)	and	“Both ______ and ______”
\(\lor\)	or	“Either ______ or ______”
\(\Rightarrow\)	implies	“If ______ then ______”
\(\Leftrightarrow\)	iff	“______ if and only if ______”

Remark \(1.4.1\).

As we learn to write proofs, it will be important to be able to produce a deduction in Propositional Logic from a sequence of assertions in English. It will also be important to be able to retrieve the English meaning from a sequence of assertions in Propositional Logic, given a symbolization key. The above table should prove useful in both of these tasks.

1.4A. Not \((\lnot)\).

As an example, consider how we might symbolize these assertions:

1. Mary is in Barcelona.
2. Mary is not in Barcelona.
3. Mary is somewhere other than Barcelona.

In order to symbolize Assertion 1, we will need one 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 assertions, we can write a new symbolization key and use \(B\) to mean something else.

Now, Assertion 1 is simply \(B\).

Since Assertion 2 is obviously related to Assertion 1, we do not want to introduce a different letter to represent it. To put it partly in English, the assertion means “It is not true that \(B\).” For short, logicians say “Not \(B\).” This is called the logical negation of \(B\). In order to convert it entirely to symbols, we will use “\(\lnot\)” to denote logical negation. Then we can symbolize “Not \(B\)” as \(\lnot B\).

Assertion 3 is about whether or not Mary is in Barcelona, but it does not contain the word “not.” Nevertheless, they both mean “It is not the case that Mary is in Barcelona.” As such, we can translate both Assertion 2 and Assertion 3 as \(\lnot B\). \[\text{An assertion can be symbolized as \(\lnot A\) 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 Assertion 4 can be translated as \(R\).

What about Assertion 5? Saying the widget is irreplaceable means that it is not the case that the widget is replaceable. So even though Assertion 5 is not negative in English, we symbolize it using negation as \(\lnot{R}\).

Assertion 6 can be paraphrased as “It is not the case that the widget is irreplaceable.” Now, as we have already discussed, “The widget is irreplaceable” can be symbolized as “\(\lnot R\).” Therefore Assertion 6, can be formulated as “it is not the case that \(\lnot R\).” Hence, it is the negation of \(\lnot R\), so it can be symbolized as \(\lnot \lnot R\). This is a double negation. (However, if you think about the assertion in English, it is another way of saying the same thing as Assertion 4. In general, we will see that if \(A\) is any assertion, then \(A\) and \(\lnot\lnot A\) are “logically equivalent.”)

More examples:

7. Elliott is short.
8. Elliott is tall.

If we let \(S\) mean “Elliot is short,” then we can symbolize Assertion 7 as \(S\).

However, it would be a mistake to symbolize Assertion 8 as \(\lnot{S}\). If Elliott is tall, then he is not short—but Assertion 8 does not mean the same thing as “It is not the case that Elliott is short.” It could be that he is not tall but that he is not short either: perhaps he is somewhere between the two (average height). In order to symbolize Assertion 8, we would need a new assertion letter.

For any assertion \(A\):

• If \(A\) is true, then \(\lnot{A}\) is false.
• If \(\lnot{A}\) is true, then \(A\) is false.

Using “\(\mathsf{T}\)” for true and “\(\mathsf{F}\)” for false, we can summarize this in a truth table for negation: \[\begin{array}{c||c}
\mathcal{A} & \neg \mathcal{A} \\
\hline \mathrm{T} & \mathrm{F} \\
\mathrm{F} & \mathrm{T}
\end{array}\]

Exercise \(1.4.2\).

Using the given symbolization key, translate each English-language assertion into .

\(M\): Those creatures are men in suits.
\(C\): Those creatures are chimpanzees.
\(G\): Those creatures are gorillas.

1. Those creatures are not men in suits.
2. It is not the case that those creatures are not gorillas.
3. Of course those creatures are not chimpanzees!

Exercise \(1.4.3\).

Using the same symbolization key, translate each symbolic assertion into English:

1. \(G\)
2. \(\lnot M\)
3. \(\lnot\lnot C\)

1.4B. And \((\&)\).

Consider these assertions:

9. Adam is athletic.
10. Barbara is athletic.
11. Adam is athletic, and Barbara is also athletic.

We will need separate assertion letters for Assertions 9 and 10, so we define this symbolization key:

\(A\): Adam is athletic.
\(B\): Barbara is athletic.

Assertion 9 can be symbolized as \(A\).

Assertion 10 can be symbolized as \(B\).

Assertion 11 can be paraphrased as “\(A\) and \(B\). ” In order to fully symbolize this assertion, we need another symbol. We will use “\(\&\).” We translate “\(A\) and \(B\) ” as \(A\& B\). We will call this connective “and” (but many logicians call it conjunction).

Notice that we make no attempt to symbolize “also” in Assertion 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 Propositional Logic.

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.

Assertion 12 is obviously a conjunction. The assertion 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 deduction: Since \(B\) means “Barbara is athletic,” one might paraphrase the assertions as “\(B\) and energetic.” This would be a mistake. Once we translate part of an assertion as \(B\), any further structure is lost. \(B\) is an atomic assertion; it is nothing more than true or false. Conversely, “energetic” is not an assertion; on its own it is neither true nor false. We should instead paraphrase the assertion as “\(B\) and Barbara is energetic.” Now we need to add an assertion letter to the symbolization key. Let \(E\) mean “Barbara is energetic.” Now the assertion can be translated as \(B\& E\). \[\text{An assertion can be symbolized as \(A \& B\) if it can be paraphrased in English as "both A, and B"}\]

Assertion 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 Propositional Logic, it is important to realize that the assertion can be paraphrased as, “Barbara is athletic, and Adam is athletic.” Thus, this translates as \(B\& A\).

Assertion 14 is a bit more complicated. The word “although” sets up a contrast between the first part of the assertion and the second part. Nevertheless, the assertion says both that Barbara is energetic and that she is not athletic. In order to make the second part into an atomic assertion, we need to replace “she” with “Barbara.”

So we can paraphrase Assertion 14 as, “Both Barbara is energetic, and Barbara is not athletic.” The second part contains a negation, so we paraphrase further: “Both Barbara is energetic and it is not the case that Barbara is athletic.” This translates as \(E\&\lnot B\).

Assertion 15 contains a similar contrastive structure. It is irrelevant for the purpose of translating to Propositional Logic, so we can paraphrase the assertion 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 part? We already have the assertion 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 assertion letter. Let \(M\) mean “Adam is more athletic than Barbara.” Now the assertion translates as \(B\& M\). \[\text{Assertions that can be paraphrased "\(A\), but \(B\)" or "Although \(A\), \(B\)" are best symbolized using "and": \(A \& B\).}\]

It is important to keep in mind that the assertion letters \(A\), \(B\), and \(M\) are atomic assertions. Considered as symbols of Propositional Logic, they have no meaning beyond being true or false. We have used them to symbolize different English language assertions that are all about people being athletic, but this similarity is completely lost when we translate to . No formal language can capture all the structure of the English language, but as long as this structure is not important to the deduction there is nothing lost by leaving it out.

For any assertions \(A\) and \(B\),

\[\text{\(A \& B\) is true if and only if both \(A\) and \(B\) are true}\]

We can summarize this in the truth table for “and”: \[\begin{array}{c|c||c}
\mathcal{A} & \mathcal{B} & \mathcal{A} \& \mathcal{B} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\mathrm{T} & \mathrm{F} & \mathrm{F} \\
\mathrm{F} & \mathrm{T} & \mathrm{F} \\
\mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}\]

Exercise \(1.4.4\).

Using the given symbolization key, translate each English-language assertion into Propositional Logic.

E₁: Ava is an electrician.
E₂: Harrison is an electrician.
F₁: Ava is a firefighter.
F₂: Harrison is a firefighter.
S₁: Ava is satisfied with her career.
S₂: Harrison is satisfied with his career.

1. Ava and Harrison are both electricians.
2. Harrison is an unsatisfied electrician.
3. Neither Ava nor Harrison is an electrician.
4. Both Ava and Harrison are electricians, but neither of them find it satisfying.
5. It cannot be that Harrison is both an electrician and a firefighter.
6. Ava is neither an electrician, nor a firefighter.

Exercise \(1.4.5\).

Using the given symbolization key, translate each symbolic assertion into English.

\(J\): Romeo likes Juliet.
\(M\): Mercutio likes Juliet.
\(T\): Romeo likes Tybalt.

1. \(M \& J\)
2. \(J \& \lnot T\)
3. \(\lnot M \& J\)

1.4C. Or \((\lor)\).

Consider these assertions:

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 assertions 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.

Assertion 16 is “Either \(D\) or \(M\).” To fully symbolize this, we introduce a new symbol. The assertion becomes \(D \lor M\). We will call this connective “or” (but many logicians call it disjunction).

Assertion 17 is only slightly more complicated. There are two subjects, but the English assertion 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 \(D \lor E\). \[\text{An assertion can be symbolized as \(A \lor B\) if it can be paraphrased in English as "Either \(A\), or \(B\)"}\]

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 Assertion 17, above. I might play with Denison, with Ellery, or with both Denison and Ellery. merely says that I will play with at least one of them. This is called an inclusive or.

The symbol “\(\lor'\)’ represents an inclusive or. So \(D \lor E\) 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 truth table for “or”: \[\begin{array}{c|c||c}
\mathcal{A} & \mathcal{B} & \mathcal{A} \vee \mathcal{B} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\mathrm{T} & \mathrm{F} & \mathrm{T} \\
\mathrm{F} & \mathrm{T} & \mathrm{T} \\
\mathrm{F} & \mathrm{F} & \mathrm{F}
\end{array}\]

Like “and,” the connective “or” is commutative: \({A}\lor{B}\) is logically equivalent to \({B}\lor{A}\) (see Exercise \(1.7.5(2)\)). \[\text{In mathematical writing, "or" always means inclusive or}\]

These assertions 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 \(S_1\) mean that you get soup and \(S_2\) mean that you get salad.

Assertion 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 “or” and “not.” It becomes \(\lnot S_1 \lor \lnot S_2\).

Assertion 19 also requires negation. It can be paraphrased as, “It is not the case that either you get soup or you get salad.” We use parentheses to indicate that “not” negates the entire assertion \(S_1 \lor S_2\), not just \(S_1\) or \(S_2\): “It is not the case that \((S_1 \lor S_2)\).” This becomes simply \(\lnot (S_1 \lor S_2)\).

Notice that the parentheses are doing important work here. The assertion \(\lnot S_1 \lor S_2\) would mean “Either you will not have soup, or you will have salad.”

Assertion 20 is an exclusive or. We can break the assertion into two parts. The first part says that you get one or the other. We translate this as \((S_1 \lor S_2)\). The second part says that you do not get both. We can paraphrase this as, “It is not the case that both you get soup and you get salad.” Using both “not” and “and,” we translate this as \(\lnot(S_1 \& S_2)\). Now we just need to put the two parts together. As we saw above, “but” can usually be translated as “and.” Assertion 20 can thus be translated as \((S_1 \lor S_2) \& \lnot(S_1 \& S_2)\).

Although “\(\lor'\)’ is an inclusive or, the preceding paragraph illustrates that we can symbolize an exclusive or in Propositional Logic. We just need more than one connective to do it.

Exercise \(1.4.6\).

Using the given symbolization key, translate each English-language assertion into Propositional Logic.

\(M\): Those creatures are men in suits.
\(C\): Those creatures are chimpanzees.
\(G\): Those creatures are gorillas.

1. Those creatures are men in suits, or they are not.
2. Those creatures are either gorillas or chimpanzees.
3. Either those creatures are chimpanzees, or they are not gorillas.

Exercise \(1.4.7\).

Give a symbolization key and symbolize the following assertions in Propositional Logic.

1. Either Alice or Bob is a spy, but not both.
2. Either Bob is a spy, or it is the case both that the code has been broken and the German embassy is in an uproar.
3. Either the code has been broken or it has not, but the German embassy is in an uproar regardless.
4. Alice may or may not be a spy, but the code has been broken in any case.

Exercise \(1.4.8\).

Using the given symbolization key, translate each assertion into English.

J: Romeo likes Juliet.
M: Mercutio likes Juliet.
T: Romeo likes Tybalt.

1. \(M \lor T\)
2. \(T \lor (\lnot J \& M)\)
3. \(\lnot (M \lor J) \& \lnot T\)

1.4D. Implies \((\Rightarrow)\).

For the following assertions, let \(R\) mean “You will cut the red wire” and \(B\) mean “The bomb will explode.”

21. If you cut the red wire, then the bomb will explode.
22. The bomb will explode if you cut the red wire.
23. The bomb will explode only if you cut the red wire.

Assertion 21 can be translated partially as “If \(R\), then \(B\).” We can rephrase this as “\(R\) implies \(B\).” We will use the symbol “\(\Rightarrow\)” to represent “implies”: the assertion becomes \(R\Rightarrow B\). We call this connective “implies” or “if-then” (but many logicians call it a conditional). The assertion on the left-hand side (\(R\) in this example) is called the hypothesis, and the assertion on the right-hand side (\(B\)) is called the conclusion.

Assertion 22 tells us that if you cut the red wire, then the bomb will explode. Thus, it is logically equivalent to Assertion 21, so it can be symbolized as \(R \Rightarrow B\).

Assertion 23 is also a conditional assertion that tells us something must be true if some other thing is true. Since the word “if” appears in the second half of the assertion, it might be tempting to symbolize this in the same way as Assertion 21 and 22. That would be a mistake.

The implication \(R\Rightarrow B\) says that if \(R\) were true, then \(B\) would also be true. It does not say that your cutting the red wire is the only way that the bomb could explode. Someone else might cut the wire, or the bomb might be on a timer. The assertion \(R\Rightarrow B\) does not say anything about what to expect if \(R\) is false. Assertion 3 is different. It says that the only conditions under which the bomb will explode involve your having cut the red wire; i.e., if the bomb explodes, then you must have cut the wire. As such, Assertion 3 should be symbolized as \(B \Rightarrow R\).

Remark \(1.4.9\).

The paraphrased assertion “ \(A\) only if \(B\)” is logically equivalent to “If \(A\), then \(B\).”

“If \(A\), then \(B\)” means that if \(A\) is true, then so is \(B\). So we know that if the hypothesis is true, but the conclusion is false, then the implication “If \(A\), then \(B\)” is false. (For example, if you cut the red wire, but the bomb does not explode, then Assertion 21 is obviously false.) We now consider the other possible situations, and determine whether the assertion “If \(A\), then \(B\)” is true or not.

• Suppose, for instance, that you do not cut the red wire. Then Assertion 21 is not a lie, whether the bomb explodes or not, because the assertion does not promise anything in this case. Thus, we consider Assertion 21 to be true in this case. In general, if \({A}\) is false, then the implication “\({A} \Rightarrow {B}\)” is true. (It does not matter whether \({B}\) is true or not.)
• The only remaining case to consider is when you cut the red wire and the bomb does explode. In this case, Assertion 21 has told the truth. In general, if \({A}\) and \({B}\) are true, then the implication “\({A} \Rightarrow {B}\)” is true. \[\text{A \(\Rightarrow\) B is true unless A is true and B is false. In that case, the implication is false.}\]

We can summarize this with a truth table for “implies.” \[\begin{array}{c|c||c}
\mathcal{A} & \mathcal{B} & \mathcal{A} \Rightarrow \mathcal{B} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\mathrm{T} & \mathrm{F} & \mathrm{F} \\
\mathrm{F} & \mathrm{T} & \mathrm{T} \\
\mathrm{F} & \mathrm{F} & \mathrm{T}
\end{array}\]

Remark \(1.4.10\).

Logic students are sometimes confused by the fact that \({A}\Rightarrow {B}\) is true whenever \({A}\) is false, but it is actually quite natural. For example, suppose a teacher promises, “If you do all of the homework, then you will pass the course.” A student who fails to do all of the homework cannot accuse the teacher of a falsehood, whether he passes the course or not.

Also, people often use this principle when speaking sarcastically. An example is the assertion, “If Rudy is the best player on the team, then pigs can fly.” We all know that pigs cannot fly, but, logically, the assertion is true as long as Rudy is not the best player on the team.

Warning.

The connective “implies” is not commutative: you cannot swap the hypothesis and the conclusion without changing the meaning of the assertion, because it is easy to find a situation in which \({A}\Rightarrow{B}\) is true, but \({B}\Rightarrow{A}\) is false. (Namely, suppose is false and is true.)

Let us go back to the example with which we started our discussion of “\(\Rightarrow\),” in which \(R\) is the assertion “You will cut the red wire,” and \(B\) means “The bomb will explode.” There are many different ways of saying \(R \Rightarrow B\) in English. Here are some of the ways; all of these mean the same thing!

• If you cut the red wire, then the bomb will explode.
• You cutting the red wire implies that the bomb will explode.
• In any circumstances in which you cut the red wire, the bomb will explode.
• Whenever you cut the red wire, the bomb will explode.
• The bomb will explode whenever you cut the red wire.
• The bomb exploding is a necessary consequence of you cutting the red wire.
• You cutting the red wire is sufficient to ensure that the bomb will explode.
• You cutting the red wire guarantees that the bomb will explode.
• You cut the red wire only if the bomb will explode.
• If the bomb does not explode, you must not have cut the red wire.
• Either you will not cut the red wire, or the bomb will explode.

Exercise \(1.4.11\).

Using the given symbolization key, translate each English-language assertion into Propositional Logic.

\(A\): Mister Ace was murdered.
\(B\): The butler committed the murder.
\(C\): The cook committed the murder.
\(D\): The Duchess is lying.
\(E\): Mister Edge was murdered.
\(F\): The murder weapon was a frying pan.

1. If Mister Ace was murdered, then the cook did it.
2. If Mister Edge was murdered, then the cook did not do it.
3. If the murder weapon was a frying pan, then the culprit must have been the cook.
4. If the murder weapon was not a frying pan, then the culprit was either the cook or the butler.
5. Either the Duchess is lying, or it was Mister Edge who was murdered.
6. If Mister Ace was murdered, he was done in with a frying pan.
7. The cook murdered Mister Edge, but she did not use the frying pan.

Exercise \(1.4.12\).

Give a symbolization key and symbolize the following assertions in Propositional Logic.

1. If Gregor plays first base, then the team will lose.
2. If either Gregor or Evan plays first base, then there will not be a miracle.
3. If neither Gregor nor Evan plays first base, then there will be a miracle.
4. The team will lose if there is no miracle.
5. If there is a miracle, then Gregor’s mom will not bake cookies.

Exercise \(1.4.13\).

For each deduction, write a symbolization key and translate the deduction as well as possible into .

1. If Dorothy plays the piano in the morning, then Roger wakes up cranky. Dorothy plays piano in the morning if she is not distracted. So if Roger does not wake up cranky, then Dorothy must be distracted.
2. It will either rain or snow on Tuesday. If it rains, Neville will be sad. If it snows, Neville will be cold. Therefore, Neville will either be sad or cold on Tuesday.
3. If Zoog remembered to do his chores, then things are clean but not neat. If he forgot, then things are neat but not clean. Therefore, things are either neat or clean—but not both.

Exercise \(1.4.14\).

Using the given symbolization key, translate each assertion into English.

\(J\): Remeo likes Juliet.
\(M\): Mercutio likes Juleit.
\(T\): Romeo likes Tybalt.

1. \(M \Rightarrow J\)
2. \(J \vee(M \Rightarrow \neg T)\)
3. \((T \Rightarrow J) \&(M \Rightarrow J)\)

1.4E. Iff \((\Leftrightarrow)\).

Consider these assertions:

24. The figure on the board is a triangle if it has exactly three sides.
25. The figure on the board is a triangle only if it has exactly three sides.
26. The figure on the board is a triangle if and only if it has exactly three sides.

Let \(T\) mean “The figure is a triangle” and \(S\) mean “The figure has exactly three sides.”

Assertion 24 can be rephrased as, “If the figure has exactly three sides, then it is a triangle.” So it can be translated as \(S \Rightarrow T\).

Assertion 25 is importantly different. Remark \(1.4.9\) tells us that it can be translated as \(T\Rightarrow S\).

Assertion 26 says two things: that “\(T\) is true if \(S\) is true” and that “\(T\) is true only if \(S\) is true.” The first half is Assertion 1, and the second half is Assertion 25; thus, it can be translated as \[(S\Rightarrow T) \& (T \Rightarrow S) .\]
However, this “if and only if” comes up so often that it has its own name. We call this connective “iff,” which is short for “if and only if” (but many logicians call it a biconditional).

Because we could always write \(({A}\Rightarrow{B})\&({B}\Rightarrow{A})\) instead of \({A}\Leftrightarrow{B}\), we do not strictly speaking need to introduce a new symbol for “iff.” Nevertheless, it is useful so often that it is commonly accepted as one of the basic logical connectives. \[\text{A \(\Leftrightarrow\) B is true if and only if A and B have the same truth value (either both are true or both are false).}\]

This is the truth table for “iff”: \[\begin{array}{c|c||c}
\mathcal{A} & \mathcal{B} & \mathcal{A} \Leftrightarrow \mathcal{B} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\mathrm{T} & \mathrm{F} & \mathrm{F} \\
\mathrm{F} & \mathrm{T} & \mathrm{F} \\
\mathrm{F} & \mathrm{F} & \mathrm{T}
\end{array}\]

Exercise \(1.4.15\).

Using the given symbolization key, translate each English-language assertion into Propositional Logic.

E₁: Ava is an electrician.
E₂: Harrison is an electrician.
F₁: Ava is a firefighter.
F₂: Harrison is a firefighter.
S₁: Ava is satisfied with her career.
S₂: Harrison is satisfied with his career.

1. If Ava is not an electrician, then neither is Harrison, but if she is, then he is too.
2. Ava is satisfied with her career if and only if Harrison is not satisfied with his.
3. Harrison and Ava are both firefighters if and only if neither of them is an electrician.

Exercise \(1.4.16\).

Using the given symbolization key, translate each assertion into English.

\(J\): Romeo likes Juliet.
\(M\): Mercutio likes Juliet.
\(T\): Romeo likes Tybalt.
\(Y\): Romeo likes Yorick.

1. \(T \Leftrightarrow Y\)
2. \(M \Leftrightarrow(J \vee Y)\)
3. \((J \Leftrightarrow M) \&(T \Rightarrow Y)\)