In previous posts, we had introduced the basic concept of Interpretability with numerous examples, before moving on to the Interpretation Translation and also the concept of Mutual and bi-interpretation of models.

In this post, I shall expand the interpretability concept beyond models to consider the interpretation of one theory in another. To interpret one theory in another just means to provide a uniform manner of interpreting a model of the interpreted theory inside any given model of the host theory.

Many logicians, to be sure, take this case of theory interpretation as the main case of interpretability, and these logicians focus principally or even exclusively on the interpretation of theories when investigating interpretability.

My perspective, however, in keeping with my general inclination toward semantical concepts in logic, is that it is the interpretation of models that is fundamental for understanding interpretability. And I view the interpretation of theories accordingly through the semantic lens—an interpretation of a target theory S inside a host theory T, as we said, is simply a uniform method of interpreting a model of S inside any given model of T.

So let us develop the theory of the interpretation of theories.

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The main concept is that we shall in order to interpret one theory S inside another theory T, one should provide a uniform way of interpreting the objects and structural features of the first theory S inside the host theory T.

The host theory T is thus able to tell us all about a certain simulated version of the interpreted theory S. The host theory provides us with a fully detailed account of objects and structural features that constitute a realm in which S is true, using only the language and structural features of the host theory.

Let me be more precise.

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In this post, we shall explore the situation arising when we can also interpret the second model conversely inside the first—each model is interpreted in the other.

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Mutual interpretation

Two structures are said to be mutually interpretable when each of them is interpretable in the other.

Inside each of them, we can define a definable copy of the other structure, using a definable class of k-tuples with definable atomic structure and a definable congruence relation, so that the interpreted model is isomorphic to the quotient of that definable structure. The model N is isomorphic via j to the model N* definable inside M, and M is isomorphic via i to M* defined inside N.

(Stay tuned for the notion of bi-interpretability, a strictly stronger notion.)

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In the post on Interpretability last week, we discussed numerous examples of this phenomenon, including:

• Interpreting the integer order ⟨ℤ, < ⟩ inside the natural numbers ⟨ℕ, + ⟩.

• Interpreting set-theoretic mereology ⟨V,⊆⟩ inside a given model of set theory ⟨V,∈⟩.

• Interpreting the countable random graph in a countable model of set theory ⟨M,∈⟩.

• Interpreting modular arithmetic ⟨ℕ, ≡ₙ⟩ in Presburger arithmetic ⟨ℕ, +, 0, 1⟩.

• Interpreting the lattice of finite sets ⟨P_fin(ℕ), ⊆ ⟩ in arithmetic ⟨ℕ, +, ·, 0, 1, < ⟩.

• Interpreting the integer ring ⟨ℤ, +, ·⟩ inside the semi-ring of natural numbers ⟨ℕ, +, ·⟩.

• Interpreting the rational field ⟨ℚ, +, ·⟩ in the ring of integers ⟨ℤ, +, ·⟩.

• Interpreting the complex field ℂ in the real field ℝ.

• Interpreting hereditarily finite set theory ⟨HF,∈⟩ in arithmetic ⟨ℕ,+,·,0,1,<⟩.

• Interpreting urelement set theory in pure set theory.

• and several others…

In this post, I should like to provide a fuller and more precise account of what it means to interpret one structure within another. After doing so, we shall discuss the interpretation translation, by which one can express all the features of the interpreted structure entirely in terms of the resources and structural features of the host structure.

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The formal definition of interpretation

Let us be a bit more precise with the interpretation terminology. We say that

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To interpret one model in another, we should define a simulated domain of suitable objects in the host structure to serve as the individuals of the proxy interpreted model—we allow ourselves to use finite tuples, not just points, and also we allow the convenience of a definable equivalence relation, simulating the interpreted model as a quotient by this congruence. On this new domain, we define analogues of the atomic structure of the interpreted model, always using only the language and structure of the host model. In this way, we produce a translation of the underlying language of the interpreted model into the language of the host.

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To get clear on the notion, let us explore a series of specific examples of interpretability. Eventually, we shall get clear on the distinctions between mutual interpretability, bi-interpretability, direct interpretations, faithful interpretations, synonymy, definitional equivalence, and more.

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In an earlier post we showed that the theory of endless dense linear orders admits elimination of quantifiers, and then in a follow-up post, we showed the same for the theory of the successor operation.

Here, we shall consider the theory of addition in the natural numbers.

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Presburger arithmetic

Presburger arithmetic is the theory of addition on the natural numbers, that is, the theory of the structure ⟨ℕ,+,0,1⟩. The numbers 0 and 1 are actually definable here from addition alone, since 0 is the unique additive identity, and 1 is the only number u that is not expressible as a sum x + y with both x ≠ u and y ≠ u. So we may view this model if desired as a definitional expansion of ⟨ℕ,+⟩, with addition only. The number 2 is similarly definable as 1 + 1, and indeed any number n is definable as 1 + ··· + 1, with n summands, and so this is a pointwise definable model and hence also Leibnizian.

Definable sets

The theory can define the order, since x ⩽ y ⇔ ∃z (x + z = y). And it knows that addition is commutative x + y = y + x and order-preserving and so forth. The induction principle also is valid here.

What are the definable subsets? We can define the even numbers, of course, since x is even if and only if ∃y (y + y = x). We can similarly define congruence modulo 2 by

More generally, we can express the relation of congruence modulo n for any fixed n as follows:

What I claim is that this exhausts what is expressible.

Elimination of quantifiers

Namely, every assertion in the theory of natural-number addition amounts to a Boolean combination of atomic congruency assertions.

Theorem. Presburger arithmetic admits elimination of quantifiers in the definitional expansion with all the modular congruence relations.

In particular, every assertion in the language of ⟨ℕ,+,0,1⟩ is equivalent to a quantifer-free assertion in the language with the congruence relations.

Let us prove it and draw several further conclusions as a consequence.

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We had said that this is quite a remarkable property when it occurs, because it reveals a severe limitation on the range of concepts that can be expressed in the theory—a quantifier-free assertion, after all, is able to express only combinations of the immediate atomic facts at hand. We are therefore generally able to prove quantifier-elimination results for a theory only when we already have a profound understanding of the theory and its models.

In this post, let us prove the quantifier elimination result for the theory of the successor operation in the natural numbers.

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Theory of successor

Consider the first-order theory of a successor function, as realized for example in the Dedekind model, ⟨ℕ,S,0⟩, where S is the successor operation Sn = n + 1.

The theory has the following axioms:

1. Zero is not a successor.

2. The successor function is one-to-one.

3. Every number is either zero or a successor.

4. There are no cycles in the successor operation.

The third axiom is also sometimes known as the very weak induction principle, for it expresses an elementary consequence of induction in the classical theory of Dedekind arithmetic. Namely, by induction every natural number is either 0 or a successor, since this property is had by 0 and is passed from every number to its successor. The no-cycles axiom continues with all axioms of the form ∀x (x ≠ Sⁿx) for each metatheoretic natural number n > 0.

In the Dedekind model ⟨ℕ,S,0⟩, every individual is definable, since x = n just in case x = SS···S0, where we have n iterative applications of S. So this is a pointwise definable model, and hence also Leibnizian. Note the interplay between the n of the object theory and n of the metatheory in the claim that every individual is definable.

What definable subsets of the Dedekind model can we think of? Of course, we can define any particular finite set, since the numbers are definable as individuals. For example, we can define the set {1, 5, 8 } by saying, “either x has the defining property of 1 or it has the defining property of 5 or it has the defining property of 8.” Thus any finite set is definable, and by negating such a formula, we see also that any cofinite set—the complement of a finite set—is definable. Are there any other definable sets? For example, can we define the set of even numbers? How could we prove that we cannot? The Dedekind structure has no automorphisms, since all the individuals are definable, and so we cannot expect to use automorphism to show that the even numbers are not definable as a set. We need a deeper understanding of definability and truth in this structure.

Quantifier-elimination for the theory of successor

Theorem. The theory of a successor function admits elimination of quantifiers—every assertion is equivalent in this theory to a quantifier-free assertion.

It will be a hands-on proof.

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We shall begin with the theory of dense linear orders.

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Endless dense linear orders

Consider first the theory of an endless dense linear order, such as the rational order ⟨ℚ,<⟩. In light of our work in an earlier post, including Cantor’s categoricity theorem that every two countable endless dense linear orders are isomorphic, we already have a fairly deep understanding of this theory and this particular model.

Consider any two rational numbers x, y in the structure ⟨ℚ,<⟩. What can one say about them? Well, we can certainly make the atomic assertions that x < y or that x = y or that y < x. Remarkably, however, this is all that can be said—every assertion φ(x,y) in the language of orders, I claim, is equivalent in ⟨ℚ,<⟩ to a Boolean combination of these assertions.

Theorem. The theory of the rational order ⟨ℚ,<⟩ admits elimination of quantifiers—every assertion φ(x,...) is logically equivalent in the rational order to a quantifier-free assertion.

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In the standard semantics of first-order logic, as we defined it in previous posts, the equality relation = gets special treatment—it is not just another binary relation symbol, to be interpreted differently in various models, but rather our semantics specifies in the atomic case of the satisfaction relation that it is to be interpreted as actual identity. Namely, a model satisfies the statement that two individuals are equal if and only if they are in fact equal. Thus, we have treated = in the disquotational manner, much like how we treat the other logical symbols such as ∧, ¬, and ∃. In this sense, we treat = as a logical primitive.

I should like to explore the question of whether this special treatment of equality was required. Can we somehow express true equality in the equality-free fragment of first-order logic? Let us define that an assertion φ in a first-order language is equality-free, if the equality symbol = does not appear in it.

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Models of clones

There is a standard construction in mathematics of the real line with two origins, which refers to the partially pre-ordered structure which is just like the real line order, except that there are two distinct number 0s—we might denote them 0 and 0*—which are related to all the other numbers just as 0 is, and which are ⩽ related to one another just as 0 is related to itself. This structure sometimes serves as an example or counterexample in connection with natural questions or conjectures.

More generally, let us consider for any given model all the various models of clones that we might construct over it. These are the models arising when we have duplicated some or perhaps all of the individuals of the original model with an assortment of exact copies, clones exhibiting all the same properties as the originals, as much as is possible except for them all being distinct from one another.

Specifically, I define that a structure M* is a model of clones over a model M, if

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If there had been no clown in the subway car, then my description wouldn't have succeeded—there would have been no referent, no unique individual falling under the description. My reference would similarly have failed if there had been a clown, but no juggling clown, or if there had been a juggling clown, but juggling very well, and so no badly juggling clown.

My reference would have failed in a different way if the subway car was packed full of badly juggling clowns, for in this case the description would not have succeeded in picking out just one of them. In each of these failing cases, there seems to be something wrong or insensible with my statement, “the badly juggling clown in the subway car has a sad expression.” What would be the meaning of this assertion, after all, if there was no such clown, if for example all the clowns were juggling very well?

Bertrand Russell emphasized that when one makes an assertion involving a definite description like this, then part of what is being asserted is that the definite description has succeeded. According to Russell’s account, when I say, “the book I read last night was fascinating,” then I am asserting first of all that indeed there was a book that I read last night and exactly one such book, and furthermore that this book was fascinating. For Russell, the assertion “the king of France is bald” asserts first, that there is such a person as the king of France and second, that the person fitting that description is bald. Since there is presently no such person as the king of France (and there wasn’t when Russell considered the assertion), Russell takes the statement to be false.

Iota expressions

Let us introduce a certain notational formalism, originating in Russell and Whitehead’s Principia Mathematica (1910-13), to assist with our analysis of definite descriptions. Namely, the inverted iota notation ℩x ψ(x) is a formal term denoting “the x for which ψ(x),” such as “the first prime number after 100” in the integer ring or “the house in which I lived as a child.” Such a reference succeeds in a model M precisely when there is indeed a unique x for which ψ(x) holds,

or in other words, when

The value of the term (℩x ψ(x)) interpreted in M is this unique object fulfilling property ψ. The use of iota expressions is perhaps the most meaningful when this property is indeed fulfilled, that is, when the reference succeeds, and we might naturally take them otherwise to be meaningless or undefined, a failed reference.

Because the iota expressions are not always meaningful in this way, their treatment in formal logic faces many of the same issues faced by a formal treatment of partial functions, functions that are not necessarily defined on the whole domain of discourse. According to the official semantics for first-order logic as we described it in an earlier post, after all, the interpretation of a function symbol f is a function defined on the whole domain of the model—in other words, in first-order logic we always interpret function symbols with total functions.

But partial functions commonly appear throughout mathematics, and we might naturally seek a formal treatment of them in logic. One immediate response to this goal is simply to point out that partial functions are already fruitfully and easily treated in first-order logic by means of their graph relations, the relation holding of a pair (x,y) for a given function g when it lies on the graph, that is, when y = g(x). One can already express everything one would want to express about a partial function g by reference to the graph relation—whether a given point is in the domain of the function and if it is, what the value of the function is at that point and so on. In this sense, first-order logic already has a robust relational treatment of partial functions.

In light of that response, the dispute here is not about the expressive power of the logic, but is rather entirely about the status of terms in the language, about whether we should allow partial functions to appear as terms. To be sure, mathematicians often form partial term expressions, such as the terms √(9 - x²) or 1/x in the context of the real numbers ℝ, which are not defined for all values of x in the domain. Allowing partial functions as terms would therefore seem to align with this aspect of mathematical practice.

But the semantics are a surprisingly subtle matter. The main issue is that when a term is not defined it may not be clear what meaning, if any, to ascribe to assertions formed using that term. To illustrate the point, suppose that e(x) is a term arising from a partial function or from an iota expression that is not universally defined in a model M, and suppose that R is a unary relation that holds of every individual in the model. Do we want to say that M ⊨ ∀x R(e(x))? Is it true that for every person, the elephant they are riding is self-identical? Well, some people are not riding any elephant, and so perhaps we might say, no, that shouldn't be true, since some values of e(x) are not defined, and so this statement should be false.

Perhaps someone else suggests that it should be true, because R(e(x)) will hold whenever e(x) does succeed in its reference—in every case where someone is riding an elephant, this elephant is self-identical. Or perhaps we want to say the whole assertion is meaningless? If we say it is meaningful, but false, however, then it would seem we would want to say M ⊨ ¬∀x R(e(x)) and consequently also M ⊨ ∃x ¬R(e(x)). In other words, in this case we are saying that in M that there is some x such that e(x) makes the always-true predicate R false—there is a person for which the elephant they are riding is not self-identical. That seems weird and probably undesirable, since it only works because e(x) must be undefined for this individual x. Furthermore, this situation seems to violate the Russellian injunction that assertions involving a definite description are committed to the success of that reference, for in this case, the truth of the assertion ∃x ¬R(e(x)) would be based entirely on the failure of the reference e(x).

Ultimately we shall face such decisions in how to define the semantics in the logic of iota expressions and more generally in the first-order logic of partial functions as terms.

The strong semantics for iota expressions

Let me begin by describing what I call the strong semantics for the logic of iota expressions. A bit later we shall also have the weak semantics, as well as what I call the natural semantics.

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Let us look into how this works more precisely. The basic fact is that in every first-order theory, we may introduce new notions by stipulative definition, defining a new concept in terms of concepts already understood.

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Nondefinability

How could we ever come to know that an element is not definable?

Consider for example discrete integer order ⟨ℤ,<⟩. Can we define the number 0? Perhaps someone points out that 0, of course, is the unique additive identity, the only number z for which x + z = x for all numbers x. This would indeed define the number 0 using the addition operation + on the integers, but our structure here has only the order relation <. So that proposal is a definition taking place in the wrong language. The question is whether we can define 0 using only order-theoretic properties, properties expressible in terms of the order structure <.

Perhaps someone suggests that we try to use the fact that 0 is somehow in the “middle” of the integers. Will that work?

No, the number 0, it turns out, is not definable in the integer order ⟨ℤ,<⟩, and indeed no integer is definable in this structure. In fact all points in the discrete integer order look alike—they have all the same properties as each other. This is a consequence of the fact that this structure is isomorphic to itself by translation—the map x ↦ x + k is an order isomorphism of the integers with itself, an automorphism, for any fixed k, because x < y ↔ x + k < y + k. By such translations, we can thus map any given point isomorphically to any desired target point, moving all points up or down by the same corresponding amount. It follows from this by the theorem that isomorphisms preserve truth that any two integers have exactly the same order-theoretic properties, and in particular, none are definable, nor even discernible.

The general method can be expressed as follows:

Theorem. Every definable element of a structure is fixed by every automorphism of the structure. That is, if π : M ≅ M is an automorphism of structure M and a is definable in M, then π(a) = a. Consequently, if an automorphism moves an individual, then it is not definable. Indeed, a and π(a) are indiscernible.

Proof. We have observed by the isomorphisms-preserve-truth theorem that if π:M ≅ M is an automorphism, then M ⊨ φ[a] if and only if M ⊨ φ[π(a)] for every assertion φ. If φ is a defining property of a, therefore, then it must be that π(a) = a, since this property holds only of a. Equivalently, by contrapositive, if π(a) ≠ a, then individual a cannot be definable, as π(a) has all the same properties as a. In short, a and π(a) are indiscernible. □

Is this an equivalence? That is, is being definable in a structure equivalent to being fixed by every automorphism?

Interlude

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Definable elements

Specifically, an individual is definable in a structure if it satisfies a defining property, that is, if it has a property in the structure that amongst all individuals there only it has. More precisely, an individual a is definable in a structure M if there is a formula φ(x), expressible in the language of the structure, such that M ⊨ φ[a], while M ⊨ ¬φ[b] for all other objects b. We thus refer determinately to such a definable object a by mentioning this defining property φ, distinguishing this object from all the others. The structure is pointwise definable if indeed every individual is definable—every individual has a property that only it has.

Consider the directed-graph structure shown here—there are six individuals in the domain and the language consists of the binary pointing-at relation Pab, indicated by the arrows.

Node 1 at the right has the property that it points at another node, but is not pointed at itself by any node. This is a defining property of node 1 in this structure, because only node 1 has this combination of features. And the property is expressible in the language of pointing-at, namely, node 1 is the only node x for which (∃y Pxy) ∧ ¬(∃y Pyx).

Thus, node 1 is definable in this structure. Node 4 is similarly definable, since it is the only node that is pointed at, but does not itself point at any node. Node 6 is the only node that neither points at nor is pointed at by any node. In fact, each of the nodes in this structure is definable in terms of the points-at relation. Can you find defining properties for the other nodes? Think about it before continuing.

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With tongue in cheek, let me tell you that historians everywhere are shocked by the recent discovery that many or even most of our greatest thinkers and poets had first expressed their thoughts and ideas in the language of first-order predicate logic, rather than in natural language. As illustrated below, virtually all of our cultural leaders have first expressed themselves in the language of first-order predicate logic, before having been compromised by translations into the vernacular.

Can you identify the common translations of each of these famous quotations? I have provided a photo or drawing in each case as a hint. Please post your answers in the comments.

There are many more. Post your solutions in the comments.

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Isomorphism and elementary equivalence

An isomorphism of one structure M with another N in the same language is a way of realizing the two structures as copies of one another—it is a one-to-one correspondence of the individuals in the first structure with those in the second structure that respects all the atomic relations and operations between the two structures. An isomorphism of one order ⟨A, ◁^A⟩ with another ⟨B, ◁^B⟩, for example, is a bijective map π : A → B such that

More generally, an isomorphism of structure M with structure N is a bijective function π : M → N of the domains of these structures such that:

1. Every atomic relation instance holding in the structure M also holds for the corresponding instance in the structure N and vice versa:

2. The function operations of the language, if any, are respected by the isomorphism:

3. And finally, the interpreted constants of the language, if any, are respected by the isomorphism:

Two structures are isomorphic, written M ≅ N, if there is such an isomorphism. Isomorphic structures are thus copies of each other with respect to all the fundamental structure that has been deemed important enough to include in the signature of the structure.

Mathematicians typically give enormous importance to their isomorphism concepts. According to the philosophy of structuralism, the genuinely mathematical ideas and properties are precisely those that are preserved by isomorphism. According to this view, properties of a mathematical structure that are not preserved by isomorphism are deemed inessential—if we had thought them to be essential we would have incorporated those features into the fundamental structure; we would have expanded the language to include suitable new relations or operations capable of expressing those features, and in this case they would have been preserved by isomorphisms in the expanded context.

This perspective can be seen as an abstract analogue of the Erlangen program in geometry, originating with Felix Klein, by which one specifies a geometry by providing a group of transformations, regarding a concept as geometrical in that geometry exactly when it is preserved by all those maps. The circumference of a polygon is a geometrical notion with respect to the group of all isometries of the plane, but not if we allow dilation. Chirality in three-space is a geometric notion when we allow only orientation-preserving isometries—such as the manner that a molecule might move about in a vacuum or fluid suspension—but not when we also allow reflections.

A key feature of the isomorphism concept is that

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Substructures

One mathematical structure M is a subsubstructure of another N, written simply M ⊆ N, if the domain of M is contained in the domain of N, and the two structures agree on the interpretations of all the relations, functions and constants on that domain. For example, here is a chain of substructures:

This is a chain of substructures because every natural number is an integer, every integer is a rational number, every rational number is a real number and all these structures agree on addition and multiplication on their common domains, on the meaning of 0 and 1, and on the order <.

Elementary substructures

A substructure M of a structure N is an elementary substructure, written M ≺ N, if M is a substructure of N and also they agree on the truth of every assertion, so that M ⊨ φ[a₁,...,a_n] if and only if N ⊨ φ[a₁,...,a_n], for every a₁, ..., a_n in M and every assertion φ in the language of these structures.

None of the substructures pictured in the chain of substructures above are elementary substructures, nor even elementarily equivalent. To see this, observe that amongst these structures, only ℝ has a solution to x² = 2; only ℕ thinks x + y = 0 → x = 0; only ℚ is 2-divisible, but does not have √2; and only ℤ satisfies the negations of all those properties. So none of these substructures is an elementary substructure.

Absoluteness of truth

Although an assertion can have a different truth value in a substructure than it does in the parent structure, as we have just observed, nevertheless in certain situations one can find an agreement—some kinds of truth values will be absolute. Let us begin by showing that term evaluation is absolute between a substructure and its parent.

Lemma. Term evaluation is absolute between a substructure and its parent. That is, evaluating a term t at points a₁, ..., a_n in a substructure M ⊆ N gives the same value as evaluating it in the parent structure N,

Proof. We prove this by induction on terms. Assume we have a substructure M ⊆ N. The claim is true for constant symbols, c^M = c^N, simply because this is part of what it means to be a substructure—constants must be interpreted the same in a substructure as in the parent structure. The claim is also true for variables, since we are using the same valuation [a₁,...,a_n] in the substructure as in the parent structure, so each variable x_i is interpreted as a_i in each case. Finally, we consider terms of the form f(t₁,...,t_k) and assume inductively that the claim is true for terms t₁,...,t_k. Observe that

We used the induction hypothesis that t_i^M(a₁,...,a_n) = t_i^N(a₁,...,a_n) in the second equality, together with the fact that f ^M agrees with f ^N on points in M by the definition of substructure. Thus, by induction, we conclude that the claim holds for all terms. □

Absoluteness Theorem. Assume M is a substructure of N.

1. Quantifier-free truth is absolute between a substructure and its parent—the models agree on the truth of any quantifier-free assertion ψ at individuals a₁,...,a_n from the substructure M:

2. Existential assertions are upward absolute from a substructure to its parent—if ψ is quantifier-free, then

3. Universal assertions are downward absolute from a parent structure to any substructure—if ψ is quantifier-free, then

Proof. We know by the lemma above that term evaluation is absolute between a substructure M ⊆ N and its parent, and this implies that every equality assertion s = t of terms will have the same truth value in M as in N. Similarly, every relational atomic assertion Rt₁···t_k will have the same truth value in M as in N, because the terms evaluate the same and similarly the relation in M agrees with N on points in M. Thus, the claim of statement (1) is true for atomic assertions.

We may extend from the atomic assertions to all quantifier-free assertions by induction on formulas. If the truth equivalence of statement (1) holds for assertions φ and ψ, then it also holds, I claim, for φ ∧ ψ, φ ∨ ψ, φ → ψ, φ ↔ ψ, and ¬φ.

Let me illustrate for conjunction:

First, we use the definition of satisfaction to break up the conjunction, and then we use the induction hypothesis to move from M to N, and finally we use the definition of satisfaction again to reassemble the conjunction. The argument follows a similar pattern for all the other logical connectives. Since every quantifier-free assertion is built in this way from atomic assertions via logical connectives, this establishes statement (1) for all quantifier-free assertions.

For statement (2), suppose that the substructure satisfies an existential statement M ⊨ (∃x ψ)[a₁,...,a_n]. So there is an individual b in M for which M ⊨ ψ[a₁,...,a_n,b]. Since ψ is quantifier-free, it follows from statement (1) that this is absolute to the parent model N ⊨ ψ[a₁,...,a_n,b] and consequently N ⊨ (∃x ψ)[a₁,...,a_n], as desired.

The reader will prove statement (3) in the exercises. □

Elementary chains

Let us next consider the concept of a chain of models, a tower of structures M_n, each a substructure of the next.

The union or limit of such a chain is the model M = ∪_n M_n, whose domain is the union of the domains of the models appearing in the chain, upon which we interpret the structural elements, the functions, relations, and constants of the signature. These interpretations are well defined on limit model precisely because the models in the tower cohere with one another on this atomic structure.

An elementary chain is a special kind of chain, where each model is an elementary substructure of the next.

In this case, we claim, the limit model is an elementary extension of all the models in the chain.

Elementary chain theorem. The limit model of an elementary chain is an elementary extension of every model in the chain.

Proof.

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A theory is simply any set of sentences in a given language. A model of such a theory T is a structure M in which every sentence of the theory is true, that is, M ⊨ σ for every sentence σ in T, and in this case we write M ⊨ T.

Logical consequence

We say that a theory T entails a sentence σ, written

if every model of T is a model of σ. In this situation we also say that the theory T logically implies the sentence σ, that σ is a logical consequence of T, or that σ is valid in the theory T—all synonymous terminology for this central concept. One theory entails another, written T ⊨ S, if T entails every sentence σ in S. Two theories are equivalent if each entails the other—each theory has all sentences in the other theory as logical consequences.

I find it insightful to notice that this notion of entailment or validity has nothing to do with proof or argument or one's reasoning process—it is not about our knowledge of the entailment or about any epistemological concern, but rather a notion entirely of the semantic realm, about the nature and range of logical possibility, about what kinds of models there are, including perhaps infinite models of vast uncountable size, and whether they all satisfy σ. Namely, if it happens to be the case that every model of T is also a model of σ, then T entails σ and otherwise not. In this sense, logical consequence belongs not to epistemology, but rather perhaps to ontology or metaphysics.

Let us explore several natural logical operators on theories and their models.

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Truth in a structure

Having introduced the formal language, let us now define the accompanying semantics, a notion of truth for first-order logic. We aim to define what it means to say that a given sentence σ is true in a given structure M, which we write as follows:

We shall define truth by recursion on σ, but in doing so, the recursion will reduce from sentences to the slightly more general case of assertions with free variables, with temporary assignments of those variables to individuals of the model.

Namely, for any assertion φ(x₁,...,x_n) and individuals a₁, ..., a_n from M, we shall define the satisfaction relation

which expresses that assertion φ is true in M of those individuals. This relation is commonly read in various ways:

• “M satisfies φ(a₁,...,a_n )”

• “M satisfies φ at (a₁,...,a_n )”

• “M is a model of φ(a₁,...,a_n )”

• “M thinks that φ(a₁,...,a_n )”

• “φ(a₁,...,a_n) is true in M”

The definition will thus make use of the concept of a valuation in a model M of variable symbols x₁, ..., x_n, which is simply a function, conveniently denoted by [a₁,...,a_n], assigning each variable symbol x_i to a particular individual a_i of M. Such a valuation is sometimes written more elaborately as

in order to make the functional relation x_i ↦ a_i more clear, and it is good to keep in mind the functional meaning of this more elaborate notation, which is what we intend implicitly with the simpler notation [a₁,...,a_n].

In effect, a valuation amounts to a temporary expansion of the language treating those particular variable symbols as constant symbols, interpreted in M according to the valuation. Ultimately we shall define what it means for an assertion φ to be true by appealing recursively to the truth of the direct subformulas of φ, and it is naturally the quantifier case that will require us to make adjustments to the valuation, making new assignments to the variable of quantification.

Disquotational theory of truth

The recursive definition of truth that we shall provide, due to Alfred Tarski, instantiates his disquotational theory of truth, according to which an assertion is true exactly when the proposition that it asserts is the case.

Thus,

“Snow is white” is true if and only if snow is white.

This disquotational idea is the foundation of Tarski’s definition of truth in a model—it is a compositional theory of truth, by which the truth of a compound assertion is defined in terms of the truth of the constituents of which it is composed. Let me explain.

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We shall present a formal language for first-order predicate logic. Let us begin by being a little more precise with our semantic notions.

Structures in a given signature

One specifies a mathematical structure by specifying the intended domain of individuals and providing on that domain all the various relations, functions, and constants that are called for by the signature of the structure, which details the number and kinds of these structural features to be included. Order structures such as ⟨ℚ, ⩽ ⟩ or ⟨P(ℕ), ⊆ ⟩, for example, have a signature calling for one binary relation, while a linearly graded order ⟨A, ≼ ,⩽ ⟩, in contrast, has two binary relations, the partial order ≼ and the linear grading preorder ⩽. The signature of the standard model of arithmetic ⟨ℕ, +, ·, 0, 1, <⟩ specifies two binary functions, two constants and one binary relation. Different structures, even extremely different structures, can have the same signature.

In the general case, a signature has a family of relation symbols { R_i | i ∈ I }, a family of function symbols { f_j | j ∈ J }, and a family of constant symbols { c_k | k ∈ K}, each of them possibly empty, and the signature specifies for each relation and function symbol the intended arity. A model or structure ℳ in this signature provides an intended domain and interpretations on that domain for these structural features.

Thus, the structure is specified by the following information:

• The intended domain of objects M.

• For each relation symbol R_i, an actual relation R_i^ℳ on the domain M

• For each function symbol f_j, an actual function f_j^ℳ on the domain M

• For each constant symbol c_k, an actual element c_k^ℳ of the domain M

The relations R_i^ℳ and functions f_j^ℳ should have in each case the arity specified by the signature for that relation or function symbol.

In this way, the structure provides interpretations of the various language symbols appearing in the signature. We use superscripts to emphasize that R_i^ℳ is the structure ℳ's interpretation of the relation symbol R_i—another structure may have a different interpretation—and similarly with f_j^ℳ and c_k^ℳ. Later, we shall often omit this heavy notation when the meaning is clear. For example, in the rational order ⟨ℚ,⩽ ⟩ we might use ⩽ ambiguously to denote both the symbol for the relation and for the actual relation itself, which might otherwise be denoted ⩽^ℚ. But also, with the reader's permission I should like to drop the finicky font distinction between the structure ℳ and its domain M, and henceforth for simplicity I shall often simply denote them both by M—the meaning will be clear from context.

The key idea is that the various symbols R_i, f_j, c_k of the signature, which are meaningless on their own, are given an interpretation in the structure and thereby provided with meaning in that structure. This is an instance of the syntax/semantics dichotomy, for we distinguish between the symbols of the language and their interpretation in a structure. We shall presently develop the signature into a full corresponding language capable of expressing truths and meaning in a given structure.

Mathematics overflows with first-order structures—orders of all kinds, graphs, digraphs, groups, rings, fields, categories, diverse algebraic structures of every imaginable sort. Indeed, many mathematical fields of study are identified by the class of mathematical structures that are the principal focus of investigation: graph theory, group theory, order theory, ring theory, category theory and so on. The concept of a mathematical structure itself is extremely general and unifies all these mathematical investigations.

Let me illustrate the enormous range of the structure concept by mentioning the structure of chess, denoted 𝕮h, whose domain consists of…

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How are we to distinguish our mathematical structures from one another—how can we tell them apart? Often even very similar-seeming mathematical structures can be distinguished by subtle features differing amongst them, features expressible in the underlying formal language of the structures.

Consider, for an easy initial example, the following three order structures:

We have the familiar order relations on the domain of natural numbers, the integers, or on the set of rational numbers. Although we have used the same symbol ⩽ in each case, we interpret this symbol differently in the three structures as the usual order relation intended for the domain of that structure. These relations are each reflexive, transitive, and anti-symmetric, and so we have three partial orders here.

But these orders differ in their order-theoretic properties—let me challenge you to find statements expressible in the language of orders that distinguish them. For each structure, find a statement expressed in the language of ⩽ that is true in that order structure and false in the other two structures. Think about it on your own before continuing further.

Interlude

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