The Quantified Language
In the previous sections, we've built up a powerful propositional language, which we can use to study the logical forms of many different types of sentences, and to determine whether arguments involving those sentences are formally valid on the basis of those logical forms, by either providing a derivation (our positive test for validity), or by providing a counterexample by means of truth tables (our negative test for validity).
In this section, we'll introduce a new type of construction into our logical language: quantified sentences, corresponding to English constructions like "Every dog has its day". We'll then show how to apply a positive test for formal validity, by generalizing our rules for derivations to the case of quantified sentences.
Predicates and Names
Our new language will be based on a naïve Maybe an overly naïve idea. But we've got to start somewhere, and to understand more complicated theories about the structure of language, it's a good idea to try to to understand the simplest theories first. idea about language. The idea is this: simple sentences in English, and in many, perhaps all, human languages tend to have two parts. The first part is something like a name---an expression that refers to some object. The second part is something like a verb---an expression that tells you what that object is doing. Two examples of simple sentences like these are:
Alan is tired.
Grace leads the way.
More complicated sentences are then built up out of these simple sentences. For example:
- If Alan is tired, then Grace leads the way.
Because the simple sentences can be packed into more complicated sentences, and because it seems as if every logically complicated sentence needs to be built up out of simple parts, we call these sentences which have no sentential parts "Atomic sentences".
- Atomic Sentences
- Atomic Sentences are sentences that don't have other sentences as parts.
What if we want to "split the atom"? What can we say about the parts---non-sentential parts---that atomic sentences are built out of? Well, these sentences have their noun-like part, and they have their verb-like part. The traditional name for the noun-like part of a simple sentence is the sentence's "subject". The verb-like part might be a verb, but it might also be made out of several words, as in our two examples above, where the verb-like parts are "is tired" and "leads the way". So, the verb-like part of an atomic sentence is not exactly a verb. Let's call this part of the sentence a "Predicate".
These two parts of a sentence, the subject and predicate, will be the basic building blocks of our expanded formal language. We will have just one type of predicate, and we'll use capital letters "F" through "O" to stand for predicates (the letters "P, Q, R", and so on are already taken---they stand for whole atomic sentences).
We'll have two types of subjects for our atomic sentences. The first type of subject will be something like a name, in English: an expression that is supposed to refer to a single object. For these names, we'll use lowercase letters "a" through "e", which we will call constants. We'll also have a symbol for a type of expression in English that functions a lot like a name, but which doesn't have a single thing that it is intended to refer to. Some examples of such expressions are "that person", "she", "he", and "it". For these expressions, we will use lowercase letters "v, w, x, y, z", which we will call variables. If we run out of predicates, constants, or variables, we'll also allow ourselves to make new symbols by attaching numbers as subscripts to certain predicates, names and variables. So for example, R1, c2, and x22 are a predicate, a constant, and a variable respectively.
Notice that our language does not include any symbols for "compound noun phrases", like "Jules and Jim". Suppose we would like to formalize the sentence "Jules and Jim are European", and we have a scheme of abbreviation like this:
a = "Jules"
b = "Jim"
F = "is European"
We need to understand the conjunction as meaning "Jules is European and Jim is European", which would be formalized as P(a) ∧ P(b). So in this respect, even though we have a new type of symbol available for dealing with nouns, we still translate compound noun-phrases as in the previous sections.
Problem Set 15
Please translate the following set of sentences using this scheme of abbreviation:
a = "Jules"
b = "Jim"
F = "is European"
G = "is dramatic"
H = "is emotional"
(Note, Problem set 15 continues below.)
This is all very nice, but there wouldn't be very much here that's new if this was all there were to it. Luckily, there's more to English than just combining names and verbs. In particular, we have words like "Everybody" and "Somebody." And these words have a special significance for reasoning. Here are two clues that show us just how special they are.
Consider the sentence
- Everybody waits.
This sentence implies
- Tom waits.
Even though (4) is not specifically about Tom. Now, consider the sentence
- Somebody waits.
This sentence is implied by (5), even though the sentence still doesn't have any special connection to Tom.
What we're seeing here is the effect of logical structure, hidden inside the words "Anybody" and "Everybody". "Anybody" is not another sentence subject referring to some thing in the world. It is not a name or a pronoun. Neither is "Somebody." No object is connected to Tom in such a way that the inferences above would make sense if we believed that "Anybody", or "Somebody" referred to that object.
These are words with a special meaning, where that meaning does not just involve referring to some person or object.
Consider the sentences
Everything is good
Anything is good
These sentences appear to mean the same thing. But consider:
It's not the case that everything is good.
It's not the case that anything is good.
Even though both (7) and (8) seem to mean the same thing, their negations do not seem to mean the same thing. (9) seems to mean the same as "Something is bad", while (10) seems to mean the same thing as "Everything is bad".
Is this a violation of the principle of compositionality? It looks like what we're seeing is a sentence whose meaning is not determined by the meanings of its parts. Of course, this is only true if the real parts determining the meaning of the sentence are "It's not the case that", and "anything is good/everything is good", combined in exactly the same way.
In fact, what's going on here is that the phrase "Everything" conceals logical structure. Because of this structure, there is more than one way for negation to combine with a sentence having the meaning that (7) and (8) share. And the different ways that negation can combine with that meaning give you the two different meanings of (9) and (10).
Here's how we can unpack the meaning shared by (7) and (8) in order to make it clear that there are two possible ways to incorporate a negation. Consider
- For every thing, that thing is good.
This means the same as (9) and (10). But, when we want to add a negation to it, we really have two options. We could write
- It's not the case that for every thing, that thing is good.
Or we could write
- For every thing, it's not the case that that that thing is good.
It looks like (12) means the same as (9), above, while (13) means the same as (10). So the difference between (9) and (10) can be explained by saying that the logical structure shared by (7) and (8) resembles the grammatical structure of (11), with two potential "places" for a negation to be inserted.
You can notice that (11) involves the phrase "that thing", which is the subject to which the predicate "is good" is being applied. "that person" isn't exactly a name, since there's no single thing it's intended to apply to. But it is the sort of subject that we use variables to stand for. Besides the subject and the predicate, (11) seems to have one other important part: the phrase "For every thing", which the word "that thing" refers back to.
In order to capture the kind of logical structure that we're seeing in the sentences (11), (12), and (13), we'll introduce a new type of symbol, which we will call a quantifier. We'll have just two quantifiers: the "universal" quantifier "∀", which will represent universal words like "any, all, every", and the "existential" quantifier "∃", which will represent words indicating the existence of something, like "there is", or "something". In order to make it easy to see which variables are referring back to which quantifier, we'll attach to each quantifier variable, like this: "∀x". We'll call the resulting expression---built out of a quantifier and a variable---a quantificational phrase.
A quantifier phrase like ∀x corresponds to the English phrase "for every thing", and a quantifier phrase like "∃x" corresponds to the phrase "For some thing". Hence, if we have a scheme of abbreviation like
G = "is good"
our examples above can be formalized as follows: "It's not the case that for every thing that thing is good" becomes "¬∀xG(x)", while "for every thing it's not the case that that thing is good" becomes "∀x¬G(x)"
Problem Set 15
Please translate the next few sentences using the following scheme of abbreviation:
F = "is funny"
To enter the existential quantifier, you can just use the letter "E", and to enter the universal quantifier, you can just use the letter "A"--so entering "Ax" will give you ∀x, and entering "Ex" will give you ∃x
(Note, problem set 15 is continued below)
Official Statement of the language
So, to sum up, here are the different basic parts of our expanded language:
The basic building blocks of our language are:
Constants "a, b, c, d, e, c1, c2…"
Variables "v, w, x, y, z, x1, x2…"
Predicates: F, G, H, I, J, K, L, M, N, O, F1, F2…
Quantificational phrases: ∀x, ∀y…∃x, ∃y…
Where quantificational phrases are made by applying a quantifier to a variable.
Atomic formulas are built up by applying a predicate to a term. Other "molecular" formulas are built up by combining two sentences using ∧,∨,→, or ↔, or by putting ¬ in front of a sentence. Finally, "quantified formulas" can be built up by putting a quantifier in front of a sentence. So the formulas are:
Atomic formulas: formulas which are made by applying a predicate to a term, for example F(a), F(x), F(b)...
Molecular formulas, which can be made by
i. Sticking two formulas together using $\land, \lor,\rightarrow, \leftrightarrow$, and wrapping in parentheses, for example $(F(a)\land G(x))$
- Putting a negation in front of a formula, for example ¬(Fa ∧ Gx)
Quantified formulas, which can be made by taking a formula and putting a quantificational phrase in front of it, for example ∀x(F(a) ∧ G(x)).
The above specification of the language tells us to put in parentheses whenever we built up a formula out of two other formulas. But as in the case of our original propositional language, we can sometimes afford to drop parentheses. In particular, we can still drop outermost parentheses, use the lefty-rule to drop parentheses that only serve to group a series of junctions left-to-right (where these junctions are of the same type), and use the JI-rule to drop parentheses that only serve to tell us that a certain junction comes directly below a certain if-connective in the parsing tree.
Quantification and English
The second clue above shows us that we can learn something about natural language, by thinking about our formal language. In this section, I'll give some examples of natural language phenomena that our formal language can help us understand, and I'll use these examples as an opportunity to introduce two important concepts that we will need when we begin to spell our our deductive system for quantifiers. Those two concepts are the idea of the scope of a quantifier, and the idea of a quantifier binding a variable.
First, though, we need to have some idea of how to understand our quantified formulas. Just like with formulas in our previous language, the best way to do this will be to think about the parsing tree of the formula---how it was built up using the rules described above. As before, we'll always make our translation by using a scheme of abbreviation, which tells us which symbols in our formal language we are using to stand for expressions in natural language. The only difference will be that our scheme of abbreviation will now give the meanings of predicates and names, rather than of whole sentences. We can translate all of the logical connectives ∧, ∨, →, ↔ and ¬ as we did before. So the only new things we really need to deal with are quantificational phrases, and variables. To translate the quantificational phrase ∀x we will usually want to use the English phrase "Everything is such that", and to translate the quantificational phrase ∃x, we will usually want to use the English phrase "Something is such that." When we do this, we'll then want to translate the variable x with an English pronoun that refers back in an appropriate way to the English translation of the quantificational phrase ∀x or ∃x.
If we're given the scheme of abbreviation:
D = "is a dog"
C = "is cute"
then we should translate as follows:
- ∀x(Dx → Cx) should be "Everything is such that, if it is a dog, then it is cute.", by first translating the main connective ∀x, giving us "Everything is such that (Dx → Cx)", then translating the main connective of the remaining formal part, giving us "Everything is such that if Dx then Cx", and finally translating the two atomic formulas, giving us "Everything is such that if it is a dog, then it is cute."
- ∃x(Dx ∧ Cx) should be "Something is such that it is a dog, and it is cute.", which we can get by doing the same thing as above: translating the main connective at each stage.
English sentences which are the translations of formal sentences whose main connective is a universal quantifier, we call "universal generalizations." English sentences which are translations of formal sentences whose main connective is a existential quantifier, we call "existential generalizations."
- Universal and Existential Generalizations
- A universal generalization is a sentence which is the result of translating a formal sentence whose main connective is a universal quantifier. An existential generalization is a sentence which is the result of translating a formal sentence whose main connective is an existential quantifier
Restricted and Unrestricted Quantification
The two translations above, "Everything is such that, if it is a dog, then it it is cute", and "Something is such that it is a dog, and it is cute" are both awkward, to say the least. We could express the same ideas considerably more nicely in English by saying "Every dog is cute", and "Some dog is cute."
It turns out that English sentences like this, where we do not talk about absolutely all of the objects that we're presently concerned with, but instead talk about just some of the objects---those to which a certain predicate, like "is a dog" can truly be applied---are very common; they're perhaps the primary way that quantifiers are used in informal English. We can call a sentence where we talk about every member of some restricted class of objects---say, the dogs---a "restricted universal generalization." A sentence which asserts the existence of a member of some restricted class of objects, we can call a "restricted existential generalization."
The way that we express restricted universal generalization in our formal language is by making an assertion about absolutely all the objects. Suppose we wish to say that every object of which some predicate A is true has some property B. We then say, of every object, that if that object belongs to the restricted class---if the predicate A is true of it, then it has the property B. So we would translate the pseudo-English sentence "All As are B" into our formal language as ∀x(Ax → Bx).
We use a similar trick to express restricted existential generalization. Suppose we wish to say, that some object of which the predicate A is true has the property B. We can say this by asserting, in our formal language, that there's some object of which A is true, which has the property B. So we would translate the pseudo-English sentence "Some As are B" into our formal language as ∃x(Ax ∧ Bx).
In general, an English language sentence of the form "All As are Bs" can be viewed as a stylistic variant for the notion of a stylistic variant, take a look back to the earlier chapters on translation from symbols into English. on an English sentence of the form "Everything is such that if it is an A, it is a B", and similarly, sentences "Some As are Bs" can be viewed as a stylistic variants on sentences "Something is such that it is an A and it is a B."
Problem Set 15
Please translate the following sentences using this scheme of abbreviation
F = "is a frog"
H = "is happy"
I = "is ignorant"
One feature of natural language, which our symbolic language makes clearer, is the way in which certain kinds of ambiguities depend on the way in which a sentence is "parsed". For example, an ambiguous sentence like "Jack the ripper was seen by the guard or he was arrested and he was thrown in jail" can be parsed as either "(Jack the ripper was seen by the guard or he was arrested) and he was thrown in jail"---in which case the sentence implies that Jack was thrown in jail---or as "Jack the ripper was seen by the guard or (he was arrested and he was thrown in jail)"---in which case it does not imply that Jack was thrown in jail.
Because our formal language is careful to always make the parsing structure of a sentence explicit, these types of ambiguities can't arise. When we translate into symbols, we automatically disambiguate by inserting parentheses.
Knowing how to disambiguate a sentence by disambiguating its parsing structure can let us see certain "patterns" of in how sentences are parsed. A good example of this is the second clue, from above.
The clue was the fact that, while the sentences (7): "Every person is tall" and (8): "Any person is tall" seemed to have the same meaning, they behaved differently when they were combined with a negation. The English-language negation of the first sentence, (9): "It's not the case that every person is tall" seemed to have the logical form ¬∀xTx, where the predicate T abbreviates "is tall". The English language negation of the second sentence, (10): "It's not the case that any person is tall" seemed to have the logical form ∀x¬Tx.
This may seem like an isolated phenomenon. In fact, it is part of a larger pattern. Consider the sentences:
If anyone is tall, then we'll need a bigger boat.
If everyone is tall, then we'll need a bigger boat.
The first of these seems to mean that, if any person at all is tall, then we can infer that we need a bigger boat. So in particular, if I am tall, or if you are tall, or if Shaquille O'Neal is tall, then we'll need a bigger boat. In fact, if (14) is true, then we know that, for every person x, the following conditional is true: if x is tall, then we'll need a bigger boat. So (14) seems to have the following logical form: ∀x(Tx → P), where P abbreviates the sentence "We need a bigger boat". You may have the intuition that a good translation of (14) would be ∃xTx → B. That would be a good intuition: that sentence is logically equivalent to ∀x(Tx → B). Once we have the rules for derivations with quantifiers, we'll see that each of these two sentences can be derived from the other. On the other hand, (15) seems only imply that we need a bigger boat if everyone is tall. So it has a logical form like this: ∀xTx → P, where the quantified formula that means "everyone is tall" is in the antecedent of the conditional.
Now, compare the translations of our two "any" sentences with the translations of our two "every" sentences:
|∀x(Tx → B)||∀xTx → B|
Now, imagine writing out a parsing tree for each of these sentences. The important observation to make is that with the "any" sentences, the quantifier in the appropriate formalization seems to float "to the top of the parsing tree". If we were to parse the sentences under "any", we'd find that the quantifier is the main connective in both cases. On the other hand, when we deal with "every" sentences, we find that the quantifier sinks as low as it can in the parsing tree---in each "every" sentence, the quantifier ends up underneath the other logical connectives in a parsing of the sentence.
We can put this observation concisely by using a new concept. Let us say that the scope of a given connective, or quantificational phrase, is the set of formulas that comes below that connective or quantificational phrase in the parsing tree.
- The scope of a connective (i.e. ∧,∨,→,↔, or ¬, or quantificational phrase (i.e. ∀x or ∃x) is the set of formulas which come below that connective in the parsing tree.
For example, we can see that in the formula ∀x¬Tx that translates the sentence (10) ("It's not the case that any person is tall"), the formulas ¬Tx, and Tx are both within the scope of the quantificational phrase ∀x. In the formula ¬∀xTx, which translates (9), on the other hand, only Tx is within the scope of ∀x. Similarly, in ∀x(Tx → B), all the formulas Tx → B, Tx, and B are within the scope of ∀x, while in ∀xTx → B, the only formula within the scope of ∀x is Tx.
Let us say that a given quantificational phrase Q has narrow scope in a given formula if Q has just one formula in its scope. And let's say that a quantificational phrase Q has wide scope if every sub-formula of the formula in which Q occurs is within the scope of Q.
- Narrow and Wide Scope
A quantificational phrase Q in a formula ϕ has wide scope if every subformula of ϕ (other than ϕ itself) is within the scope of Q.
a quantificational phrase Q in a formula ϕ has narrow scope if Q has just one subformula of ϕ in its scope.
Using the concept of wide and narrow scope, we can formulate a general hypothesis, Of course, this is just a first hypothesis. It's easy to find cases where it doesn't quite work---so it needs some modifications before it is a really serious piece of semantic theory. which would explain the logical behavior of "any" and "every": the word "any" is used to express sentences whose quantificational phrase is intended to have narrow scope. The word "every" is used to express sentences whose quantificational phrase is intended to have wide scope.
So the concept of scope lets us express interesting generalizations about the way that certain pieces of language work. The concept of scope will also be important to formulate the next interesting feature of quantifiers: the way in which quantifiers serve to "bind" variables.
To understand variable binding, it will again help to start from an example in natural language. Consider the sentence
- For every person, there's some person such if that person is angry, then that person is happy.
Now, this sentence is clearly ambiguous. But the way in which it is ambiguous is interesting. When we say "that person", it's not clear if we're referring back to the quantificational phrase "every person" or to the quantificational phrase "some person". Let's disambiguate, by labeling the occurrences of "that person" in such a way that they match the quantificational phrases they refer back to. If we do this, then the two possible readings are something like this:
For every person1, there's some person2 such if that person1 is angry, then that person2 is happy.
For every person1, there's some person2 such if that person2 is angry, then that person1 is happy.
On the first reading, the reading given in (17), the sentence says that every person has a critic, somebody who's happy whenever that person is angry. But the sentence is not specific about whether or not two people might have the same critic, it just says that everybody has a critic. So, it's compatible with that first sentence that there's really just one critic---and that person is critical of everybody.
On the second reading, the reading given in (18), the sentence say that everybody is somebody's critic. That is to say: every person has someone of whom they're critical, someone who, if angry, makes that person happy. This clearly means something very different from the first sentence. For one thing, (18) implies that everyone is a critic, while (17) did not imply that there was more than one critic.
So there's a big difference in meaning between these two readings of the sentence. That's why the original sentence is ambiguous: because it admits more than one reading.
To eliminate this kind of ambiguity in our formal language, we do something analogous to labeling the different occurrences of the phrase "that person". But rather than distinguishing occurrences with labels, we instead use an entirely different variable for each occurrence. This, plus some careful specifications of the conditions under which a variable refers back to, or "is bound by" a quantifier, allows us to avoid the type of ambiguity that arises in a sentence like (16). Here are the rules for variable binding:
A quantificational phrase Q binds an occurance of a variable x if and only if
x is part of Q. So, Q is either ∃x or ∀x
x occurs in a formula within the scope of Q
There is no other quantificational phrase Q′ within the scope of Q such that x is a part of Q′ and x within the scope of Q′.
In the formula ∀y(Ax ∧ Ay), the variable x is not bound by the quantificational phrase ∀y, since x is not a part of ∀y. But, the variable y is bound by ∀y, since y is a part of ∀y, since ∀y has Ay in its scope, and since there's no other quantificational phrase that gets in the way. In the formula ∀x(Ax ∧ ∃xBx), the first quantificational phrase---the ∀x---binds the first occurrence of x, since x is part of ∀x, Ax is in the scope of ∀x, and there's no other quantificational phrase which has Ax in its scope. But the occurrence of x inside of Bx is not bound by ∀x; the ∃x "gets in the way", because x is a part of ∃x, Bx is within the scope of ∃x, and ∃x is part of a formula within the scope of ∀x.