Section 1: From sentences to predicates
Consider the following argument, which is obviously valid in English:
If everyone knows logic, then either no one will be confused or everyone will. Everyone will be confused only if we try to believe a contradiction. This is a logic class, so everyone knows logic. .˙. If we don’t try to believe a contradiction, then no one will be confused.
In order to symbolize this in SL, we will need a symbolization key.
L
: Everyone knows logic.
N
: No one will be confused.
E
: Everyone will be confused.
B
: We try to believe a contradiction.
Notice that \(N\) and \(E\) are both about people being confused, but they are two separate sentence letters. We could not replace \(E\) with ¬\(N\). Why not? ¬\(N\) means ‘It is not the case that no one will be confused.’ This would be the case if even one person were confused, so it is a long way from saying that everyone will be confused.
Once we have separate sentence letters for \(N\) and \(E\), however, we erase any connection between the two. They are just two atomic sentences which might be true or false independently. In English, it could never be the case that both no one and everyone was confused. As sentences of SL, however, there is a truth-value assignment for which N and E are both true.
Expressions like ‘no one’, ‘everyone’, and ‘anyone’ are called quantifiers . By translating \(N\) and \(E\) as separate atomic sentences, we leave out the quantifier structure of the sentences. Fortunately, the quantifier structure is not what makes this argument valid. As such, we can safely ignore it. To see this, we translate the argument to SL:
\(L\) → (\(N\)∨\(E\))
\(E\) → \(B\)
\(L\)
.˙. ¬\(B\) → \(N\)
This is a valid argument in SL. (You can do a truth table to check this.)
Now consider another argument. This one is also valid in English.
Willard is a logician. All logicians wear funny hats.
.˙. Willard wears a funny hat.
To symbolize it in SL, we define a symbolization key:
L
: Willard is a logician.
A
: All logicians wear funny hats.
F
: Willard wears a funny hat.
Now we symbolize the argument:
\(L\)
\(A\)
.˙. \(F\)
This is invalid in SL. (Again, you can confirm this with a truth table.) There is something very wrong here, because this is clearly a valid argument in English. The symbolization in SL leaves out all the important structure. Once again, the translation to SL overlooks quantifier structure: The sentence ‘All logicians wear funny hats’ is about both logicians and hat-wearing. By not translating this structure, we lose the connection between Willard’s being a logician and Willard’s wearing a hat.
Some arguments with quantifier structure can be captured in SL, like the first example, even though SL ignores the quantifier structure. Other arguments are completely botched in SL, like the second example. Notice that the problem is not that we have made a mistake while symbolizing the second argument. These are the best symbolizations we can give for these arguments in SL.
Generally, if an argument containing quantifiers comes out valid in SL, then the English language argument is valid. If it comes out invalid in SL , then we cannot say the English language argument is invalid. The argument might be valid because of quantifier structure which the natural language argument has and which the argument in SL lacks.
Similarly, if a sentence with quantifiers comes out as a tautology in SL, then the English sentence is logically true. If it comes out as contingent in SL , then this might be because of the structure of the quantifiers that gets removed when we translate into the formal language.
In order to symbolize arguments that rely on quantifier structure, we need to develop a different logical language. We will call this language quantified logic, QL.