Motivation for predicate logic.

Question

I am studying predicate logic,I came across this para-

Propositional logic, studied in Sections 1.1–1.3, cannot adequately express the meaning of all statements in mathematics and in natural language.

For example, suppose that we know that “Every computer connected to the university network is functioning properly.”

No rules of propositional logic allow us to conclude the truth of the statement “MATH3 is functioning properly",where MATH3 is one of the computers connected to the university network.

This is supposed to illustrate the motivation for predicate logic.

However,we can interpret the given statement as the conjunction of statements of the form "MATHx is connected to the university network and is working properly".

As this statement is true and is a conjunction the individual propositions have to be true;in terms of MATH3 too.

So propositional logic expresses the meaning of the statement adequately and we have also used the rules of propositional logic to conclude the truth about MATH3 contrary to whats been stated.

What am I missing?I have gone through the notes by Stephen Simpson and Wikipedia but they proved inadequate.

Answer 1

In terms of propositional logic, the statement:

“Every computer connected to the university network is functioning properly.”

can be symbolized only as $p$.

The statement: “(The computer) MATH3 is functioning properly" can be symbolized with $q$.

No propositional inference can licenses us to infer $q$ from $p$.

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In order to express the logical relation between the two, we need predicate logic.

We have to symbolize the first statement as:

$\forall x ( \text {Computer}(x) \to \text {Funct_prop}(x))$.

Now, using the auxiliary statement: $\text {Computer(MATH3)}$, we can apply Universal instantiation to derive:

$\text {Computer(MATH3)} \to \text {Funct_prop(MATH3)}$

and conclude by Modus Ponens with: $\text {Funct_prop(MATH3)}$.

Answer 2

If your domain is finite, then yes, we could do everything in propositional logic instead of predicate logic, but:

1. Not every domain is finite (think numbers, sets, graphs, etc.)

2. Even if the domain is finite, then:

2a. We may not know all of its elements (think all people in the world)

2b. There are just too many elements in the domain for this to be practical, efficient, or convenient

Answer 3

No rules of propositional logic allow us to conclude the truth of the statement “MATH3 is functioning properly",where MATH3 is one of the computers connected to the university network.

This is supposed to illustrate the motivation for predicate logic.

However,we can interpret the given statement as the conjunction of statements of the form "MATHx is connected to the university network and is working properly".

Yes, indeed, and this is the first step in moving from propositional logic to predicate logic.

You are no longer discussing a single statement about a single subject —a proposition—, rather you have moved to discussing a conjunction of similar statements about subjects that belong to a collection.

That is on the path to developing quantfied predicate statements.   Next is to move from discussing specific collections, to general collections, and formalise the rules of logic to manage the discussions.

Answer 4

If nullary relation symbols are allowed, then restricting the signature of a first-order language to nullary relation symbols results in propositional logic. Thinking of propositional logic as the result of restricting the signature to nullary relation symbols shows how in propositional logic we are constrained in referring to the elements of the domain in the logic.

Specifically, let $\mathcal{L}_0$ be our restricted signature, and let $D$ be the domain of an $\mathcal{L}_0$-structure. Since $D^0 = \{\emptyset\}$, each relation symbol may only be interpreted as $\emptyset$ (always false) or $D^0$ (always true) regardless of $D$. There are no terms in formulae; hence there is no use for quantifiers.

These restrictions prevent us from expressing many interesting statements about integers or humans, and result in large formulae in many other instances. It should be noted, though, that there are practical cases in which the "propositional approach" is surprisingly efficient: not in terms of the size of the formulae, but in terms of the time needed to decide satisfiability.

Answer 5

You can express a conjunction such as the following:

"Math1 is connected and working properly AND Math2 is connected and working properly AND ... MathX is connected and working properly",

in propositional logic. However, this does not say that "Every computer connected to the university network is functioning properly." as you have claimed. Why? Because there does not exist any guarantee that the conjunction above covers every computer in the network. You would at least need some clause that says something like "and the only computers on the network are math1, ... mathX".

Unfortunately though, this doesn't answer your question, but it seems too long for a comment, so I'll leave it as a so-called "answer".