substitutional interpretation of quantifiers: examples?

Question

About the differences between propositional logic and (first order) predicate logic, given that if my basis is propositional logic I have to substitute the universal and existential quantifiers with conjonctions and disjunctions (substitutional interpretation of quantifiers), let's suppose that A is a statement of predicate logic and B is a statement of propositional logic ->

• can you give me one or more remarkable examples which underly the effects of this method while moving from A to B?

(p.s. Also the relation between this method and the problem of infinity is not clear for me)

Answer 1

Let $A \equiv \forall x \in \lbrace a,b,c \rbrace x \leq 5$, let: $$\alpha \equiv a \leq 5$$ $$\beta \equiv b \leq 5$$ $$\gamma \equiv \leq 5$$ then $B \equiv \alpha \land \beta \land \gamma$. Of course if you want to do the same with an existential proposition you just need to change conjunctions by disjunctions. Is this the kind of example you are looking for? Note that if you want to do the same with an infinite set you will not be able to do it using conjunctions in propositional logic, that's part of the utility of quantifiers

Edit (based on the discussion we had in the other question): The thing is you can't build a proposition using an infinite amount of propositions (try to do it!),so if you wanted to say something like "every natural number is the precesor of another natural number" using propositional logic you will not be able to do it using a proposition for each natural number and putting all those prpositions toogether with conjunctions, as in the example before. In this case the prposition "every natural number is the precesor of another natural number" must be an atom, when you translate this same proposition in the language of predicate logic you "break" it into smaller pieces (constant, variables, functions, relations, quantifiers) which you can use time and time again to talk about completely different things. My conclution is that the expressive power of predicate logic is much higher (you can do much more things with less amount of "pieces"). Just like in physics, we are interested in finding the most elementary components of the universe we are studing.

Answer 2

By "the problem of infinity", I take it you mean the a big problem that infinite models (in particular, uncountably infinite ones) pose for the substitutional interpretation of quantifiers.

Those who espouse this interpretation of quantifiers have philosophers and not mathematicians — philosophers of a nominalist bent, as well.

For theories whose intended models are uncountable, e.g. the theory of real closed fields, with intended model $\Bbb R$, it's basically silly to say the $\forall x$ means "for every name that can be substituted for $x$", or that $\forall x \varphi(x)$ really means the conjunction of all $\varphi(\mathbf{c})$ for every name $c$. This would have to be a conjunction of $2^\aleph_0$-many formulas, each instantiated with a different name for a real, and every real having a name.

Unless we artificially manufacture those "names", in the usual language of ordered fields there simply aren't enough names (closed terms): there are far more reals than there are names (countably many). Where would all these names come from? Well, one good source would be the model itself: granting that $\Bbb R$ exists, add every real $r\in \Bbb R$ as a "name" (for itself) to the first order language of ordered fields.

Furthermore, it's a big leap from first order logic, with its finite formulas, to infinitely long formulas. Logics with infinitely long formulas have been studied, for decades, and some are well-understood (well, one or two), but they lack any of the nice properties of first-order logic such as a proof procedure (!) and completeness. It's no explanation at all to "reduce" $\exists x x > 0$ to an uncountably infinite disjunction.

The substitutional interpretation of quantifiers makes sense, but I have to conclude, as have others, that it's misguided. It moves the line between syntax and semantics, expanding the realm of syntax in order to define the semantic notion of satisfaction (truth of a sentence/formula in a model).