MODULE: Predicate logic

TUTORIAL Q04: Interpreting WFFs in SPL

In this section we shall look at the interpretation of wffs in SPL. Although the syntax of SPL might seem rather complicated, it is actually not too difficult to understand the meanings of the wffs if you look at their syntax carefully.

Non-quantified sentences

Let us start by considering wffs that do not contain any quantifiers, but only predicates and names. The first thing you should remember is that a predicate letter followed by a name functions grammatically as a wff. An expression such as "Fa" says that a certain object a has the property F. Bearing this point in mind, when you come across a complicated wff, you can interpret the wff just as in SL. For example, "((Fa&Ga)→P)" is a conditional, and it says that if a is both F and G, then P. Similarly, "(Ga↔Gb)" is a biconditional, and it says that a is G if and only if b is also G.

Exercises

Question #1 Suppose we have this translation scheme :

a : Ann
b : Beth
Sx : x is a student Tx : x is a teacher

Use the above translation scheme to translate these wffs into English :

1. (~Ta&~Sa) [Show answer]
   Ann is neither a teacher nor a student.
2. (Ta→~Sa) [Show answer]
   If Ann is a teacher then Ann is not a student.
3. (Sa↔~Tb) [Show answer]
   Ann is a student if and only if Beth is not a teacher.
4. ((Sa&~Sb)→(~Ta&Tb)) [Show answer]
   If Ann is a student but Beth is not, then Beth is a teacher but Ann is not.

Question #2 Translate the following English sentences into SPL. Use "P" to mean "it is going to rain." :

1. If it is not going to rain, then Ann will go out. [Show answer]
   (~P→Ga)
2. Although it is going to rain, Ann and Beth both go hiking. [Show answer]
   (P&(Ha&Hb))
3. If it is not going to rain, and if Ann goes hiking, then Beth goes hiking too. [Show answer]
   ((~P&Ha)→Hb)

Quantified sentences

Now let us see what the quantified sentences mean. A quantified sentence in SPL is simply any wff in SPL that starts with either an existential or a universal quantifier. A wff that starts with an existential quantifier can be taken as saying that some object satisfies the condition given by the open sentence that follows the quantifier. So for example, consider this wff :

∃x((Fx&Gx)→P)

Now "((Fa&Ga)→P)" says that if a is both F and G, then P. It is a wff that talks about a particular object a. The difference with the quantified sentence is that the latter is not talking about the object a anymore. What the qauntified sentence says is that there is at least one object x such that if it is both F and G, then P. Maybe this object is a, or maybe it is some other object b, but the quantified sentence does not tell us which object it is.

Now consider also this wff :

∀x((Fx&Gx)→P)

What this wff says is that everthing is such that if it is F and G, then P. Notice that this is not the same as saying that if everthing is F and G, then P. That would be formalized using this wff :

(∀x(Fx&Gx)→P)

Do you know what is the difference between the two wffs? Roughly, what the first wff says is that if you take any object you can find, if it is both F and G, then "P" is true. So it implies that "P" is true even when there is only one thing in the whole world that is both F and G. But the second wff has no such implication. It says that "P" is true when everything is both F and G. So there is no guarantee that "P" is true when only one object is both F and G. Obviously, the first wff entails the second, but not vice versa.

Here are a few more examples :

∀y(By&Dy) : Everything is both B and D

∀y(~By&Dy) : Everything is not B but D

∀y(By&~Dy) : Everything is B but not D

∀y(~By&~Dy) : Everything is such that it is not B and and it is not D

∀y~(By&Dy) : Everything is such that it is not both B and D

∀y(By→Dy) : Everything is such that if it is B then it is D (Or, every B is D)

∀y(By→~Dy) : Every B is not D

∀y(~By→Dy) : Everything that is not B is D

∀y(~By→~Dy) : Everything that is not B is also not D

∀y~(By→Dy) : Everything is such that it is not the case if it is B it is also D

You can replace "∀" with "∃", and "every" by "some" (meaning "at least one") in these examples to obtain the translation of the corresponding existentially quantified wffs.

Exercises

Question #3

Translate the wffs below into English using this translation scheme :

a : Ann
b : Beth
Hx : x is heavy
Ox : x is old

1. ∃x(Hx→Ox) [Show answer]
   There is something that is old if it is heavy.
2. ~∃y(Hy&Oy) [Show answer]
   Nothing is both heavy and old. (No heavy thing is old, No old thing is heavy.)
3. ~∀z(Hz&Oz) [Show answer]
   Not everything is heavy and old.
4. ~∀z(Hz→Oz) [Show answer]
   Not every heavy thing is old.
5. (∃xHx∨Oa) [Show answer]
   Either something is heavy, or Ann is old
6. (Oa→∃yOy) [Show answer]
   If Ann is old, then something is old.
7. (∃xHx↔~∀x~Hx) [Show answer]
   Something is heavy iff not everything is not heavy.
8. ((∀xHx&∀x~Ox)→(Hb&~Ob)) [Show answer]
   If everything is heavy and everything is not old, then Ann is heavy and Ann is not old.
9. (∀x(Hx→Ox)→(Hb→Ob)) [Show answer]
   If every heavy thing is old, then if Beth is heavy then she is old.
10. ∀y(Hy∨Oy) [Show answer]
    Everything is either heavy or old
11. (∀yHy∨∀yOy) [Show answer]
    Either everything is heavy or everything is old

Domain of quantification

Notice that when we use the word "everything" in ordinary language, we do not always mean "absolutely everything in the whole universe". Suppose you are going on a trip and I ask, "Have you packed everything into your suitcase?" In such a situation, of course I am not asking whether you have got everything in the universe in your suitcase. Rather, "everything" here probably refers only to those portable items you need for the trip. This is an illustration of what is called restricting the domain of quantification. The domain of quantification is the class of things we are talking about when we use quantifiers such as "every", "all", or "some". To restrict the domain of quantification is to limit the class to a particular group of objects. In the example just given, the domain of quantification includes only the items you need for your trip. Objects such as the moon, or the Eiffel Tower, are therefore not included in the class. Whereas if a physicist proclaims that "everything is made up of elementary particles", the domain of quantification will be larger and presumably includes all the physical objects in the whole universe. Similarly, when a teacher says "everyone is here" in a lecture, it might be that the only people included in the domain are his students.

Restricting the domain of quantification can make formalization easier. For example, we might formalize "everyone is wicked" as " ∀x(Hx→Wx)". But if we restrict the domain of quantification so that we are only talking about human beings, then we can just write down "∀xWx", and leave it as understood that the domain includes all human beings only. But remember : if you do restrict the domain of quantification in formalization, you should define the domain explicitly. For example, consider this simple inference:

Everyone is wicked.
If everyone is wicked, then nobody goes to heaven.
So nobody goes to heaven.

By restricting the domain only to human beings, we can formalize the argument easily :

Translation scheme:

Domain : the set of all human beings
Wx : x is wicked
Gx : x goes to heaven.

Formalized sequent:

∀xWx, (∀xWx→~∃xGx) ~∃xGx

What if you do not restrict the domain? In such case the formalization can proceed as follows:

Translation scheme:

Domain : everything
Wx : x is wicked
Px : x is a person
Gx : x goes to heaven

Formalized sequent:

∀x(Px→Wx), (∀x(Px→Wx)→~∃x(Px&Gx)) ~∃x(Px&Gx)

So you can see that restricting the domain simplifies the formalization. But do remember that with any argument there should only be a single domain. That implies you should not use different domains to formalize the following argument:

Every human being is an animal.
Every animal can feel pain.
So every human being can feel pain.

The correct way to formalize this argument is not to restrict the domain at all, but to write down something like the following:

Translation scheme :

Hx : x is a human being
Ax : x is an animal
Cx : x can feel pain

Formalized sequent :

∀x(Hx→Ax), ∀x(Ax→Cx) ∀x(Hx→Cx)

It would be a mistake to have three domains, one for each premise and another one for the conclusion.

Exercises

Formalize these English sentences in SPL :

1. Everyone is happy. [Show answer]
   ∀xHx
2. If everyone is happy then nobody is sad. [Show answer]
   (∀xHx→~∃xSx)
3. If everyone is happy then someone is not sad. [Show answer]
   (∀xHx→∃x~Sx)
4. If someone is sad, then not everyone is happy. [Show answer]
   (∃xSx→~∀xHx)
5. Everybody is sad if someone is sad. [Show answer]
   (∃xSx→∀xSx)
6. Someone is not sad and not happy. [Show answer]
   ∃x(~Sx&~Hx)
7. Nobody is sad and happy. [Show answer]
   ~∃x(Sx&Hx)
8. If nobody is sad and happy, then everyone who is sad is not happy, and everyone who is happy is not sad. [Show answer]
   (~∃x(Sx&Hx)→(∀x(Sx→~Hx)&∀x(Hx→~Sx)))
9. Either Tom is happy, or Jane is sad. [Show answer]
   (Ht∨Sj)
10. Someone is either happy, or sad, but not both. [Show answer]
    ∃x((Hx∨Sx)&~(Hx&Sx))
11. If Tom is happy, then everyone is happy. [Show answer]
    (Ht→∀xHx)

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