The problem with sentences is that it’s often unclear what they mean. Does “St. Trinian’s is a pretty little girls’ school” refer to a pretty school for little girls, or a fairly small school, or one for girls who are pretty and little?
Are these “little” girls young, or of modest stature? Is the “school” a building, a co-moving grouping (as of fish), or an intellectual lineage?1
We want our beliefs to be true, but if we don’t even know what they mean, we’re in trouble. A single sentence might be true in some sense, false in some other sense, and meaningless in a third. If you believe “St. Trinian’s is a pretty little girls’ school,” what do you believe?
The meaning of a sentence depends on the meaning of its parts. The logical tradition attempted to find a fixed scheme for extracting sentence meaning from word meanings. Unfortunately, this is impossible.
In “the eggplant is a fruit,” probably what is meant is that all eggplants are fruits. In “the dog is a Samoyed,” probably what is meant is that some dog is a Samoyed. We can reasonably assume these meanings from our background understanding of their topics. This knowledge is nowhere in the sentence. The meaning depends on its parts—but not only on them.
This problem is pervasive. Linguists catalog many distinct ways a sentence can be ambiguous. On analysis, almost any sentence can be read with multiple meanings.
Rationalism’s diagnosis is that natural languages—English, Chinese, Tamil—are hopelessly broken. They are incapable of adequately expressing true beliefs. To be rational, we have to at least know what we’re talking about. Modern rationalism’s first major improvement on traditional logic replaced natural language sentences with mathematical formulae.
Let’s start with “the eggplant is a fruit” versus “the dog is a Samoyed.” The immediate problem is that nobody can explain what words like “the” and “is” mean.2 They caused trouble for logicians for millennia, until Gottlob Frege banned them.
Frege’s 1879 invention of modern formal logic fixed several outstanding defects in traditional, Aristotelian logic:
- He ditched natural language for a formal system that has compositional semantics: the meaning of a formula can be derived unambiguously from the meanings of its parts. It can’t be meaningless, and it can’t have more than one interpretation.3 That eliminated all problems of syntactic ambiguity, such as whether a little girls’ school is for little girls, or is little itself.
- Frege was able to rigorously separate deduction from intuition, and to reject the latter. The distinction between the two had been somewhat nebulous in previous rationalist systems.
- He solved several long-standing technical problems, in which Aristotelian logic gave outright wrong answers, by introducing a new device, “nested quantifiers.”
In Fregean epistemology, we can eliminate the ambiguity of “is”:
|∀x eggplant(x) ⇒ fruit(x)||False|
|∃x dog(x) ∧ Samoyed(x)||True|
The symbol ∀ means “for all” and ⇒ means “implies.” So the first formula reads “For all things—let’s take an arbitrary example, and call it ‘x’—if that thing (named ‘x’) is an eggplant, it implies that it (‘x’) is also a fruit.” That’s what “the eggplant is a fruit” was supposed to mean.
∀ is called the universal quantifier, because it states a universal truth.
The symbol ∃ means “there is some” and ∧ means “and.” So the second formula reads “There is some thing—call it ‘x’—which is a dog, and is a Samoyed.” This captures the fact that, in this case, we are talking about a dog, not about dogs in general.
∃ is the existential quantifier, because it says something exists. The formula is not quite right as a translation of “the dog is a Samoyed,” however. That sentence is meant to be about the dog—this particular dog, whose identity should be clear in context. The logical formula only asserts that there exists some dog, somewhere, that is a Samoyed.
The meaning of “the dog,” taken out of context, is necessarily indeterminate: which dog it refers to depends on the situation.
If your dog is in the National Dog Registry, it has an ID number, like maybe 1514670. Then we can write, unambiguously,
|dog(dog1514670) ∧ Samoyed(dog1514670)||True|
In English: “dog #1514670 is a dog, and dog #1514670 is a Samoyed.”
But what if you see a dog, and it’s obviously a Samoyed, but you don’t know its registry number? What do you believe then?
This seemingly trivial question holds a key to meta-rationality! We will return to it repeatedly in The Eggplant.
On the meta-rational account, beliefs and reasoning are almost always context-dependent. There is no way to “fix” the belief “the dog is a Samoyed” to eliminate the context-dependence of “the.”
In Part Two, we’ll look in detail at how context-dependence works in reasonableness. In Part Three, we’ll see how rationality partially eliminates context-dependence, in order to generalize understanding. However, that move separates rationality from concrete reality. Rationality has to depend on reasonableness to bridge the gap.
In Parts Four and Five, we’ll see how meta-rationality selectively integrates reasonableness and rationality to make both work better.
- 1.This example originates in the grammar guide for the artificial human language Loglan, which made predicate calculus (formal logic) speakable in order to eliminate ambiguity.
- 2.During his impeachment proceedings, President Clinton explained that his earlier testimony, to the effect that he had not had sex with Monica Lewinsky, was not perjury, because “It depends upon what the meaning of the word ‘is’ is. If the—if he—if ‘is’ means is and never has been, that is not—that is one thing. If it means there is none, that was a completely true statement.” The Articles of Impeachment declared that “President Clinton is guilty of what C.S. Lewis called ‘verbicide,’ murder of the plain spoken word. His attempt to invoke the literal truth defense fails under the reasonableness test.” We’ll discuss both “literal truth” and “reasonableness” a little later in this book.
- 3.I’m skimming over exceptions here. Compositionality tries to guarantee that a logical formula can’t be more ambiguous, or less meaningful, than its parts. As we’ll see, its parts may themselves be ambiguous or meaningless. Also, it turned out that self-reference can make a formula meaningless even though all its parts are well-behaved.