Soon we will learn how to evaluate arguments in Sentential Logic —arguments whose premises and conclusions are SL sentences. In real life, though, we’re not interested in evaluating arguments in some artificial language; we’re interested in evaluating arguments presented in natural languages like English. So in order for our evaluative procedure of SL argument to have any real-world significance, we need to show how SL arguments can be fair representations of natural-language counterparts. We need to show how to translate sentences in English into SL.
We already have some hints about how this is done. We know that simple English sentences are represented as capital letters in SL. We know that our operators—tilde, dot, wedge, horseshoe, and triple-bar—are the SL counterparts of the English locutions ‘not’, ‘and’, ‘or’, ‘if/then’, and ‘if and only if’, respectively. But there is significantly more to say on the topic of the relationship between English and SL. Our operators—alone or in combination—can capture a much larger portion of English than that short list of words and phrases.
Tilde, Dot, and Wedge
Consider the word ‘but’. In English, it has a different meaning from the word ‘and’. When I say “Donald Trump is rich and generous,” I communicate one thing; when I say “Donald Trump is rich, but generous,” I communicate something slightly different. Both utterances convey the assertions that Trump is rich, on the one hand, and generous on the other. The ‘but’-sentence, though, conveys something more—namely, that there’s something surprising about the generosity in light of the richness, that there’s some tension between the two. But notice that each of those utterances is true under the same circumstances: when Trump is both rich and generous; the difference between ‘but’ and ‘and’ doesn’t affect the truth-conditions. Since the meanings of our SL operators are specified entirely in terms of their effects on truth-values, SL is blind to the difference in meaning between ‘and’ and ‘but’. Since the truth-conditions for compounds featuring the two words are the same—true just in case both components are true, and false otherwise—we can use the dot to represent both. ‘Donald Trump is rich and generous’ and ‘Donald Trump is rich, but generous’ would both be rendered in SL as something like ‘R • G’ (where ‘R’ stands for the simple sentence ‘Trump is rich’ and ‘G’ stands for ‘Trump is generous’). Again, switching from English into SL is a strategy for dealing with the messiness of natural language: to conduct the kind of rigorous logical analyses involved in evaluating deductive arguments, we need a simpler, tamer language; the slight difference in meaning between ‘and’ and ‘but’ is one of the wrinkles we need to iron out before we can proceed.
There are other words and phrases that have the same effect on truth-value as ‘and’, and which can therefore be represented with the dot: ‘although’, ‘however’, ‘moreover’, ‘in addition’, and so on. These can all be used to form conjunctions.
There are fewer ways of forming disjunctions in English. Almost always, these feature the word ‘or’, sometimes accompanied by ‘either’. Whenever we see ‘or’, we will translate it into SL as the wedge. As we discussed, the wedge captures the inclusive sense of ‘or’—one or the other, or both. The exclusive sense—one or the other, but not both—can also be rendered in SL, using a combination of symbols. ‘Hillary Clinton or Donald Trump will win the election, but not both’. How would we translate that into SL? Let ‘H’ stand for ‘Hillary Clinton will win’ and ‘D’ stand for ‘Donald Trump will win’. We know how to deal with the ‘or’ part: ‘Hillary Clinton will win or Donald Trump will win’ is just ‘H ∨ D’. How about the ‘not both’ part? That’s the claim, paraphrasing slightly, that it’s not the case that both Hillary and Trump will win; that is, it’s the negation of the conjunction: ‘~ (H • D)’. So we have the ‘or’ part, and we have the ‘not both’ part; the only thing left is the word ‘but’ in between. We just learned how to deal with that! ‘But’ gets translated as a dot. So the proper SL translation of ‘Hillary Clinton or Donald Trump will win the election, but not both’ is this:
(H ∨ D) • ~ (H • D)
Notice we had to enclose the disjunction, ‘H ∨ D’, in parentheses. This is to remove ambiguity: without the parentheses, we wouldn’t know whether the wedge or the (middle) dot was the main operator, and so the construction would not have been well-formed. In SL, the exclusive sense of ‘or’ is expressed with a conjunction: it conjoins the (inclusive) ‘or’ claim to the ‘not both’ claim— one or the other, but not both.
It is worth pausing to reflect on the symbolization of ‘not both’, and comparing it to a complementary locution—‘neither/nor’. We symbolize ‘not both’ in SL as a negated conjunction; ‘neither/nor’ is a negated disjunction. The sentence ‘Neither Donald Trump nor Beyoncé will win the election’ would be rendered as ‘~ (D ∨ B)’; that is, it’s not the case that either Donald or Beyoncé will win.
When we discussed the syntax of SL, it was useful to use and analogy to arithmetic to understand the interactions between tildes and parentheses. Taking that analogy too far in the case of negated conjunctions and disjunctions can lead us into error. The following is true in arithmetic:
- (2 + 5) = -2 + -5
We can distribute the minus-sign inside the parentheses (it’s just multiplying by -1). The following, however, are not true in logic (The triple-bar is a logical equals-sign; it indicates that the components have the same truth-conditions (meaning)):
~ (p • q) ≡ ~ p • ~ q [WRONG]
~ (p ∨ q) ≡ ~ p ∨ ~ q [WRONG]
The tilde cannot be distributed inside the parentheses in these cases. For each, the left- and right- hand components have different meanings. To see why, we should think about some concrete examples. Let ‘R’ stand for ‘Donald Trump is rich’ and ‘G’ stand for ‘Donald Trump is generous’. ‘~ (R • G)’ symbolizes the claim that Trump is not both rich and generous. Notice that this claim is compatible with his actually being rich, but not generous, and also with his being generous, but not rich. The claim is just that he’s not both. Now consider the claim that ‘~ R • ~ G’ symbolizes. The main operator in that sentence is the dot; it’s a conjunction. Conjunctions make a commitment to the truth of each of their conjuncts. The conjuncts in this case symbolize the sentences ‘Trump is not rich’ and ‘Trump is not generous’. That is, this conjunction is committed to Trump’s lacking both richness and generosity. That is a stronger claim than saying he’s not both: if you say he’s not both, that’s compatible with him being one or the other; ‘~ R • ~ G’, on the other hand, insists that both are ruled out. So, generally speaking, a negated conjunction makes a different (weaker) claim than the conjunction of two negations.
There is also a difference between a negated disjunction and the disjunction of two negations. Consider ‘~ (R ∨ G)’. That symbolizes the sentence ‘Trump is neither rich nor generous’. In other words, he lacks both richness and generosity. That’s a much stronger claim that the one symbolized by ‘~ R ∨ ~ G’—the disjunction ‘Either Trump isn’t rich or he isn’t generous’. He lacks one or the other quality (or both; the disjunction is inclusive). That’s compatible with his actually being rich, but not generous; it’s also compatible with his being generous, but not rich.
Did you notice what happened there? I used the same language to describe the claim symbolized by ‘~ (R • G)’ and ‘~ R ∨ ~ G’. Both merely assert that he isn’t both rich and generous; he may be one or the other. I also described the claims made by ‘~ (R ∨ G)’ and ‘~ R • ~ G’ the same way. Both make the stronger claim that he lacks both characteristics. This is true in general: negated conjunctions are equivalent to the disjunction of two negations; and negated disjunctions are equivalent to the conjunction of two negations. The following logical equivalences are true (They’re often referred to as “DeMorgan’s Laws,” after the nineteenth century English logician Augustus DeMorgan, who was apparently the first to formulate in the terms of the modern formal system developed by his fellow countryman and contemporary, George Boole. DeMorgan didn’t discover these equivalences, however. They have been known to logicians since the ancient Greeks):
~ (p • q) ≡ ~ p ∨ ~ q
~ (p ∨ q) ≡ ~ p • ~ q
If you want to distribute that tilde inside the parentheses (or, alternatively, moving from right to left, pull the tilde outside), you have to change the wedge to a dot (and vice versa).
Horseshoe and Triple-Bar
There are many English locutions that we can symbolize using the horseshoe and the triple-bar—especially the horseshoe. In fact, as we shall see, it’s possible to render claims translated with the triple-bar using the horseshoe instead (along with a dot). We will look at a representative sample of the many ways in which conditionals and biconditionals can be expressed in English, and talk about how to translate them into SL using the horseshoe and triple-bar.
The canonical presentation of a conditional uses the words ‘if’ and ‘then’, as in ‘If the Democrats win back Congress, then a lot of new legislation will be passed’. But the word ‘then’ isn’t really necessary: ‘If the Democrats win back Congress, a lot of new legislation will be passed’ makes the same assertion. It would also be symbolized as ‘D ⊃ L’ (with ‘D’ and ‘L’ standing for the obvious simple components). The word ‘if’ can also be replaced. ‘Provided the Democrats win back Congress, a lot of new legislation will be passed’ also makes the same claim.
Things get tricky if we vary the placement of the ‘if’. Putting it in the middle of sentence, we get ‘Your pain will go away if you drink this herbal tea every day for a week’, for example. Compare that sentence to the one we considered earlier: ‘If you drink this herbal tea every day for a week, then your pain will go away’. Read one, then the other. They make the same claim, don’t they? Rule of thumb: whatever follows the word ‘if’, when ‘if’ occurs on its own (without the word ‘only’; see below), is the antecedent of the conditional. We would translate both of these sentences as something like ‘D ⊃ P’ (where ‘D’ is for drinking the tea, and ‘P’ is for the pain going away).
The word ‘only’ changes things. Consider: ‘I will win the lottery only if I have a ticket’. A sensible claim, obviously true. I’m suggesting this is a conditional. Let ‘W’ stand for ‘I win the lottery’ and ‘T’ stand for ‘I have a ticket’. Which is the antecedent and which is the consequent? Which of these two symbolizations is correct:
T ⊃ W
W ⊃ T
To figure it out, let’s read them back into English as canonical ‘if/then’ claims. The first says, “If I have a ticket, then I’ll win the lottery.” Well, that’s optimistic! But clearly false—something only a fool would believe. That can’t be the correct way to symbolize our original, completely sensible claim that I will win only if I have a ticket. So it must be the second symbolization, which says that if I did win the lottery, then I had a ticket. That’s better. Generally speaking, the component occurring before ‘only if’ is the antecedent of a conditional, and the component occurring after is the consequent.
The claim in the last example can be put differently: having a ticket is a necessary condition for winning the lottery. We use the language of “necessary and sufficient conditions” all the time. We symbolize these locutions with the horseshoe. For example, being at least 16 years old is a necessary condition for having a driver’s license (in most states). Let ‘O’ stand for ‘I am at least 16 years old’ and ‘D’ stand for ‘I have a driver’s license. ‘D ⊃ O’ symbolizes the sentence claiming that O is necessary for D. The opposite won’t work: ‘O ⊃ D’, if we read it back, says “If I’m at least 16 years old, then I have a driver’s license.” But that’s not true. Plenty of 16-year-olds don’t get a license. There are additional conditions besides age: passing the test, being physically able to drive, etc.
Another way of putting that point: being at least 16 years old is not a sufficient condition for having a driver’s license; it’s not enough on its own. An example of a sufficient condition: getting 100% on every test is a sufficient condition for getting an A in a class (supposing tests are the only evaluations). That is, if you get 100% on every test, then you’ll get an A. If ‘H’ stands for ‘I got 100% on all the tests’ and ‘A’ stands for ‘I got an A in the class’, then we would indicate that H is a sufficient condition for A in SL by writing ‘H ⊃ A’. Notice that it’s not a necessary condition: you don’t have to be perfect to get an A. ‘A ⊃ H’ would symbolize a falsehood.
To define a concept is to provide necessary and sufficient conditions for falling under it. For example, a bachelor is, by definition, an unmarried male. That is, being an unmarried male is necessary and sufficient for being a bachelor: you don’t qualify as a bachelor is you’re not an unmarried male, and being an unmarried male is enough, on its own, to qualify for bachelorhood. It’s for circumstances like this that we have the triple-bar. Recall, the phrase that triple-bar is meant to capture the meaning of is ‘if and only if’. We’re now in a position to understand that locution. Consider the claim that I am a bachelor if and only if I am an unmarried male. This is really a conjunction of two claims: I am a bachelor if I’m an unmarried male, and I’m a bachelor only if I’m an unmarried male. Let ‘B’ stand for ‘I’m a bachelor’ and ‘U’ stand for ‘I’m an unmarried male. Our claim is then B if U, and B only if U. We know how to deal with ‘if’ on its own between two sentences: the one after the ‘if’ is the antecedent of the conditional. And we know how to deal with ‘only if’: the sentence before it is the antecedent, and the sentence after it is the consequent. To symbolize ‘I am a bachelor if and only if I am an unmarried male’ using horseshoes and a dot, we get this:
(U ⊃ B) • (B ⊃ U)
The left-hand conjunct is the ‘if’ part; the right-hand conjunct is the ‘only if’ part. The purpose of the triple-bar is to give us a way of symbolizing such claims more easily, with a single symbol. ‘I am a bachelor if and only if I am an unmarried male’ can be translated into SL as ‘B ≡ L’, which is just shorthand for the longer conjunction of conditionals above. And given that ‘necessary and sufficient’ is also just a conjunction of two conditionals, we use triple-bar for that locution as well. (Also, the phrase ‘just in case’ can be used to express a biconditional claim.)
At this point, you may have an objection: why include triple-bar in SL at all, if it’s dispensable in favor of a dot and a couple of horseshoes? Isn’t it superfluous? Well, yes and no. We could do without it, but having it makes certain translations easier. As a matter of fact, this is the case for all of our symbols. It’s always possible to replace them with combinations of others. Consider the horseshoe. It’s false when the antecedent is true and the consequent false, true otherwise. So really, it’s just a claim that it’s not the case that the antecedent is true and the conclusion false—a negated conjunction. We could replace any p ⊃ q with ~ (p • ~ q). And the equivalences we saw earlier— DeMorgan’s Laws—show us how we can replace dots with wedges and vice versa. It’s a fact (I won’t prove it; take my word for it) that we could get by with only two symbols in our language: tilde and any one of wedge, dot, or horseshoe. (In fact, it’s possible to get by with only one symbol: if we defined a new two-place operator that’s true when both components are false, and false otherwise, that would do the trick. The symbol typically used for this truth-function is ‘|’, called the “Sheffer stroke” after the logician (Henry Sheffer) who first published this result) So yeah, we have more symbols than we need, strictly speaking. But it’s convenient to have the number of symbols that we do, since they line up neatly with English locutions, making translation between English and SL much easier than it would be otherwise.
Translate the following into SL, using the bolded capital letters to stand for simple sentences.
1. Harry Lime is a Criminal, but he’s not a Monster.
2. If Thorwald didn’t kill his wife, then Jeffries will look foolish.
3. Rosemary doesn’t love both Max and Herman.
4. Michael will not Kill Fredo if his Mother is still alive.
5. Neither Woody nor Buzz could defeat Zurg, but Rex could.
6. If either Fredo or Sonny takes over the family, it will be a Disaster.
7. Eli will get rich only if Daniel doesn’t drink his milkshake.
8. Writing a hit Play is necessary for Rosemary to fall in Love with Max.
9. Kane didn’t Win the election, but if the opening of the Opera goes well he’ll regain his Dignity.
10. If Dave flies into the Monolith, then he’ll have a Transformative experience; but if he doesn’t fly into the Monolith, he will be stuck on a Ghost ship.
11. Kane wants Love if and only if he gets it on his own Terms.
12. Either Henry keeps his Mouth shut and goes to Jail for a long time or he Rats on his friends and lives the rest of his life like a Schnook.
13. Only if Herman builds an Aquarium will Rosemary Love him. 14. Killing Morrie is sufficient for keeping him Quiet.
15. Jeffries will be Vindicated, provided Thorwald Killed his wife and Doyle Admits he was right all along.
16. Collaborating with Cecil B. DeMille is necessary to Revive Norma’s career, and if she does not Collaborate with DeMille, Norma may go Insane.
17. Either Daniel or Eli will get the oil, but not both.
18. To have a Fulfilling life as a toy, it is necessary, but not sufficient, to be Played with by children.
19. The Dude will get Rich if Walter’s Plan works, and if the Dude gets Rich, he’ll buy a new Bowling ball and a new Carpet.
20. Either the AE-35 Unit is really malfunctioning or HAL has gone Crazy; and if HAL has gone Crazy, then the Mission will be a failure and neither Dave nor Frank will ever get home.