Section 2: Connectives
Logical connectives are used to build complex sentences from atomic components. There are five logical connectives in SL. This table summarizes them, and they are explained below.
| symbol | what it is called | what it means |
|
¬
& ∨ → ↔ |
negation
conjunction disjunction conditional biconditional |
‘It is not the case that. . .’
‘Both. . . and . . .’ ‘Either. . . or . . .’ ‘If . . . then . . .’ ‘. . . if and only if . . .’ |
Negation
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 ¬\(B\).
Sentence 3 is about whether or not Mary is in Barcelona, but it does not contain the word ‘not.’ Nevertheless, it is obviously 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 ¬\(\mathcal{A}\) if it can be paraphrased in English as ‘It is not the case that \(\mathcal{A}\).’ |
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 in SL, we will make sure that \(R\) and ¬¬\(R\) are logically equivalent.
More examples:
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 allow for the possibility that he is indifferent, we would need a new sentence letter to symbolize sentence 8.
For any sentence \(\mathcal{A}\): If \(\mathcal{A}\) is true, then¬\(\mathcal{A}\) is false. If¬\(\mathcal{A}\) is true, then \(\mathcal{A}\) is false. Using ‘T’ for true and ‘F’ for false, we can summarize this in a characteristic truth table for negation:
| \(\mathcal{A}\) | ¬\(\mathcal{A}\) |
|
T F |
F T |
We will discuss truth tables at greater length in the next chapter.
Conjunction
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 \(\mathcal{A}\) & \(\mathcal{B}\) if it can be paraphrased in English as ‘Both \(\mathcal{A}\), and \(\mathcal{B}\).’ 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 ‘\(\mathcal{A}\), but \(\mathcal{B}\)’ or ‘Although \(\mathcal{A}\), \(\mathcal{B}\)’ are best symbolized using conjunction: \(\mathcal{A}\) & \(\mathcal{B}\) |
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.
For any sentences \(\mathcal{A}\) and \(\mathcal{B}\), \(\mathcal{A}\) & \(\mathcal{B}\) is true if and only if both \(\mathcal{A}\) and \(\mathcal{B}\) are true. We can summarize this in the characteristic truth table for conjunction:
| \(\mathcal{A}\) | \(\mathcal{B}\) | \(\mathcal{A}\) & \(\mathcal{B}\) |
| 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 \(\mathcal{B}\) and \(\mathcal{A}\) are, \(\mathcal{A}\) & \(\mathcal{B}\) is logically equivalent to \(\mathcal{B}\) & \(\mathcal{A}\).
Disjunction
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 \(D\)∨\(M\). The ‘∨’ 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 \(D\)∨\(E\).
| A sentence can be symbolized as \(\mathcal{A}\) ∨ \(\mathcal{B}\) if it can be paraphrased in English as ‘Either \(\mathcal{A}\), or \(\mathcal{B}\).’ 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 ‘∨’ represents an inclusive or. So \(D\)∨\(E\) 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:
| \(\mathcal{A}\) | \(\mathcal{B}\) | \(\mathcal{A}\)∨\(\mathcal{B}\) |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Like conjunction, disjunction is symmetrical. \(\mathcal{A}\)∨\(\mathcal{B}\) is logically equivalent to \(\mathcal{B}\)∨\(\mathcal{A}\). 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 \(S\) 1 mean that you get soup and \(S\) 2 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 ¬\(S\) 1 ∨¬\(S\) 2 .
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 (\(S1\)∨\(S2\)).’ This becomes simply ¬(\(S1\)∨\(S2\)). Notice that the parentheses are doing important work here.
The sentence¬\(S\) 1 ∨\(S\) 2 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 (\(S\) 1 ∨\(S\) 2 ). 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 ¬(\(S\) 1 & \(S\) 2 ). 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 (\(S\) 1 ∨\(S\) 2 )&¬(\(S\) 1 & \(S\) 2 ). Although ‘∨’ is an inclusive or, we can symbolize an exclusive or in SL. We just need more than one connective to do it.
Conditional
For the following sentences, 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 only if you cut the red wire.
Sentence 21 can be translated partially as ‘If \(R\), then \(B\).’ We will use the symbol ‘→’ to represent logical entailment. The sentence becomes \(R\) → \(B\). The connective is called a conditional . The sentence on the left-hand side of the conditional (\(R\) in this example) is called the antecedent . The sentence on the right-hand side (\(B\)) is called the consequent .
Sentence 22 is also a conditional. Since the word ‘if’ appears in the second half of the sentence, it might be tempting to symbolize this in the same way as sentence 21. That would be a mistake.
The conditional \(R\) → \(B\) 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 sentence \(R\) → \(B\) does not say anything about what to expect if \(R\) is false. Sentence 22 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, sentence 22 should be symbolized as \(B\) → \(R\).
It is important to remember that the connective ‘→’ says only that, if the antecedent is true, then the consequent is true. It says nothing about the causal connection between the two events. Translating sentence 22 as \(B\) → \(R\) does not mean that the bomb exploding would somehow have caused your cutting the wire. Both sentence 21 and 22 suggest that, if you cut the red wire, your cutting the red wire would be the cause of the bomb exploding. They differ on the logical connection. If sentence 22 were true, then an explosion would tell us— those of us safely away from the bomb— that you had cut the red wire. Without an explosion, sentence 22 tells us nothing.
| The paraphrased sentence ‘\(\mathcal{A}\) only if \(\mathcal{B}\)’ is logically equivalent to ‘If \(\mathcal{A}\), then \(\mathcal{B}\).’ |
‘If \(\mathcal{A}\) then \(\mathcal{B}\)’ means that if \(\mathcal{A}\) is true then so is \(\mathcal{B}\). So we know that if the antecedent \(\mathcal{A}\) is true but the consequent \(\mathcal{B}\) is false, then the conditional ‘If \(\mathcal{A}\) then \(\mathcal{B}\)’ is false. What is the truth value of ‘If \(\mathcal{A}\) then \(\mathcal{B}\)’ under other circumstances? Suppose, for instance, that the antecedent \(\mathcal{A}\) happened to be false. ‘If \(\mathcal{A}\) then \(\mathcal{B}\)’ would then not tell us anything about the actual truth value of the consequent \(\mathcal{B}\), and it is unclear what the truth value of ‘If \(\mathcal{A}\) then \(\mathcal{B}\)’ would be.
In English, the truth of conditionals often depends on what would be the case if the antecedent were true — even if, as a matter of fact, the antecedent is false. This poses a problem for translating conditionals into SL. Considered as sentences of SL, \(R\) and \(B\) in the above examples have nothing intrinsic to do with each other. In order to consider what the world would be like if \(R\) were true, we would need to analyze what \(R\) says about the world. Since \(R\) is an atomic symbol of SL, however, there is no further structure to be analyzed. When we replace a sentence with a sentence letter, we consider it merely as some atomic sentence that might be true or false.
In order to translate conditionals into SL, we will not try to capture all the subtleties of the English language ‘If... then....’ Instead, the symbol ‘→’ will be a material conditional . This means that when \(\mathcal{A}\) is false, the conditional \(\mathcal{A}\)→\(\mathcal{B}\) is automatically true, regardless of the truth value of \(\mathcal{B}\). If both \(\mathcal{A}\) and \(\mathcal{B}\) are true, then the conditional \(\mathcal{A}\)→\(\mathcal{B}\) is true.
In short, \(\mathcal{A}\)→\(\mathcal{B}\) is false if and only if \(\mathcal{A}\) is true and \(\mathcal{B}\) is false. We can summarize this with a characteristic truth table for the conditional.
| \(\mathcal{A}\) | \(\mathcal{B}\) | \(\mathcal{A}\)→\(\mathcal{B}\) |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The conditional is asymmetrical . You cannot swap the antecedent and consequent without changing the meaning of the sentence, because \(\mathcal{A}\)→\(\mathcal{B}\) and \(\mathcal{B}\)→\(\mathcal{A}\) are not logically equivalent.
Not all sentences of the form ‘If... then...’ are conditionals. Consider this sentence:
23. If anyone wants to see me, then I will be on the porch.
If I say this, it means that I will be on the porch, regardless of whether anyone wants to see me or not— but if someone did want to see me, then they should look for me there. If we let \(P\) mean ‘I will be on the porch,’ then sentence 23 can be translated simply as \(P\).
Biconditional
Consider these sentences:
24. The figure on the board is a triangle only if it has exactly three sides.
25. The figure on the board is a triangle 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 three sides.’
Sentence 24, for reasons discussed above, can be translated as \(T\) → \(S\).
Sentence 25 is importantly different. It can be paraphrased as, ‘If the figure has three sides, then it is a triangle.’ So it can be translated as \(S\) → \(T\).
Sentence 26 says that \(T\) is true if and only if \(S\) is true; we can infer \(S\) from \(T\), and we can infer \(T\) from \(S\). This is called a biconditional , because it entails the two conditionals \(S\) → \(T\) and \(T\) → \(S\). We will use ‘↔’ to represent the biconditional; sentence 26 can be translated as \(S\) ↔ \(T\).
We could abide without a new symbol for the biconditional. Since sentence 26 means ‘\(T\) → \(S\) and \(S\) → \(T\),’ we could translate it as (\(T\) → \(S\))&(\(S\) → \(T\)). We would need parentheses to indicate that (\(T\) → \(S\)) and (\(S\) → \(T\)) are separate conjuncts; the expression \(T\) → \(S\) & \(S\) → \(T\) would be ambiguous.
Because we could always write (\(\mathcal{A}\) → \(\mathcal{B}\)) & (\(\mathcal{B}\) → \(\mathcal{A}\)) instead of \(\mathcal{A}\) ↔ \(\mathcal{B}\), we do not strictly speaking need to introduce a new symbol for the biconditional. Nevertheless, logical languages usually have such a symbol. SL will have one, which makes it easier to translate phrases like ‘if and only if.’
\(\mathcal{A}\)↔\(\mathcal{B}\) is true if and only if \(\mathcal{A}\) and \(\mathcal{B}\) have the same truth value. This is the characteristic truth table for the biconditional:
| \(\mathcal{A}\) | \(\mathcal{B}\) | \(\mathcal{A}\)↔\(\mathcal{B}\) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |