[PL06] Interpretation


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

§1. 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, "((Fb&Gb)→P)" is a conditional, and it says that if b is both F and G, then P. Similarly, "(Gc↔Gb)" is a biconditional, and it says that c is G if and only if b is also G.

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)
  2. (Ta→~Sa)
  3. (Sa↔~Tb)
  4. ((Sa&~Sb)→(~Ta&Tb))

Translate the following English sentences into MPL. Use "P" to mean "it is going to rain" :

  1. If it is not going to rain, then Ann will go out.
  2. Although it is going to rain, Ann and Beth both go hiking.
  3. If it is not going to rain, and if Ann goes hiking, then Beth goes hiking too.

§2. Quantified sentences

Now let us see what the quantified sentences mean. A quantified sentence in MPL is simply any wff in MPL 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 everything is such that if it is F and G, then P. Notice that this is not the same as saying that if everything 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.

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)
  2. ~∃y(Hy&Oy)
  3. ~∀z(Hz&Oz)
  4. ~∀z(Hz→Oz)
  5. (∃xHx∨Oa)
  6. (Oa→∃yOy)
  7. (∃xHx↔~∀x~Hx)
  8. ((∀xHx&∀x~Ox)→(Hb&~Oa))
  9. (∀x(Hx→Ox)→(Hb→Ob))
  10. ∀y(Hy∨Oy)
  11. (∀yHy∨∀yOy)
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