# What does model-theoretic semantics have to do with the problem of multiple generality?

I was reading the Wikipedia article on logic, and in the section of semantics, it says:

The introduction of quantification, needed to solve the problem of multiple generality, rendered impossible the kind of subject-predicate analysis that underlies medieval semantics. The main modern approach is model-theoretic semantics, based on Alfred Tarski's semantic theory of truth.

And then it says what Model-Theoretic Semantics is about:

The approach assumes that the meaning of the various parts of the propositions are given by the possible ways we can give a recursively specified group of interpretation functions from them to some predefined domain of discourse: an interpretation of first-order predicate logic is given by a mapping from terms to a universe of individuals, and a mapping from propositions to the truth values "true" and "false".

Also, I read an article on model theory

As well as the article on the problem of multiple generality in wikipedia

I'm still confused as to what does Model-Theoretic Semantics have anything to do with the Problem of Multiple Generality.

From my experience with the use of models in first-order logic, these can be a way of expressing truth-functional entailment.

Think of a model for first-order logic as equivalent to the truth-table for sentential logic.

Say we have the famous case of multiple generality: 1. All humans are mortal 2. Tyrone is a human 3. Therefore Tyrone is a mortal

Expressed in first-order logic, we have: {∀x(Hx→Mx),Ht}⊨Mt

We can form a model of this as a sort of Venn-Diagram, where single place predicates are expressed as circles forming sets. It would be impossible for us to form a model where ∀x(Hx→Mx) is true and Ht is true whilst Mt is false. This system can be used to show truth-functional entailment, rather than using a deductive system of derivation. or: A set of sentences can be shown to truth-functionally entail the conclusion so long as there does not exist a model capable of satisfying the premises whilst the confusion fails to obtain. This proof of truth-functional entailment (⊨) is distinct from the deductive entailment (⊢) which might standardly be used in a derivation.

• so, in short, you are telling me that model-theoretic serves as representing sentences that have multiple generalities? Sorry if I am not understanding you. Apr 25 '18 at 18:08
• This is only my limited understanding of model theory (limited by the minimal extent to which I've used it. What I mean to say is that it is a way of showing that a set of sentences in first-order logic truth-functionally entail the conclusion, rather than just deductively prove (imply) it. Apr 25 '18 at 18:25

Wikipedia's example of the problem of multiple generality is that in the traditional logic (Aristotelian syllogistic) one can not derive from "Some cat is feared by every mouse" that "All mice are afraid of at least one cat stems from the possibility".

There are two connected problems that preclude it, the absence of multi-place predicates, in this case we need Fears(x,y), and the possibility of nested quantifiers on different variables, which raises the dependence and scope issues. In the classical subject-predicate analysis all predicates were one-place, monadic, i.e. "properties", which allowed to build interpretations from individuals by attaching properties to them. P(a) is true if P happens to be a predicate attached to the individual a.

In the polyadic calculus this is no longer possible if we want to take account of "relations" also. In addition to attaching properties to individuals one needs to introduce Skolem functions that reflect quantifier dependence, e.g. ∀x∃yR(x,y) is reflected in a function y=f(x). Starting from some individuals we will not get a full interpretation until iteratively applying Skolem functions to them, this is how the Skolem model is built. Tarski's definition of interpretation is in some sense dual to this process, it takes the domain of individuals as already built and gives a recursive procedure for assigning truth values to sentences, which is a far reaching complication of monadic predication.