Question

Let \(\mathfrak{A} \subseteq \mathfrak{B} . \varphi\) is called universal (existential) if \(\varphi\) is prenex with only universal (existential) quantifiers. (i) Show that for universal sentences \(\varphi \mathfrak{B} \models \varphi \Rightarrow \mathfrak{A} \models \varphi\). (ii) Show that for existential sentences \(\varphi \mathfrak{A} \models \varphi \Rightarrow \mathfrak{B} \models \varphi\). (Application: a substructure of a group is a group. This is one reason to use the similarity type \(\langle-; 2,1 ; 1\rangle\) for groups, instead of \(\langle-; 2 ; 0\rangle\), or \(\langle-; 2 ; 1\rangle\), as some authors do).

Short answer

(i) If \(\mathfrak{B} \models \varphi\), then \(\mathfrak{A} \models \varphi\) for universal \(\varphi\). (ii) If \(\mathfrak{A} \models \varphi\), then \(\mathfrak{B} \models \varphi\) for existential \(\varphi\).

Step-by-step solution

Understanding Universal Sentences

A universal sentence is a logical statement with the form that it is true for all elements of a structure. In formal logic, a universal sentence is expressed using the symbol \(\forall\), which means 'for all'. So a universal sentence \(\varphi\) is of the form \(\forall x_1 \forall x_2 \ldots \forall x_n \psi(x_1, x_2, \ldots, x_n)\), where \(\psi\) is a quantifier-free formula.

Expression for Universal Sentences and Models

If \(\varphi\) is a universal sentence, and we have \(\mathfrak{B} \models \varphi\), it means \(\varphi\) is true in the structure \(\mathfrak{B}\). Since \(\varphi\) has only universal quantifiers, it mandates all elements to satisfy \(\psi\). Given \(\mathfrak{A} \subseteq \mathfrak{B}\), all the elements of \(\mathfrak{A}\) are in \(\mathfrak{B}\), meaning \(\psi\) is true for \(\mathfrak{A}\) as well, thereby making \(\mathfrak{A} \models \varphi\).

Understanding Existential Sentences

An existential sentence is a logical statement with the form that there exists at least one element of a structure for which the statement is true. In formal logic, an existential sentence uses the symbol \(\exists\), which stands for 'there exists'. So an existential sentence \(\varphi\) is of the form \(\exists x_1 \exists x_2 \ldots \exists x_n \psi(x_1, x_2, \ldots, x_n)\), where \(\psi\) is a quantifier-free formula.

Expression for Existential Sentences and Models

If \(\varphi\) is an existential sentence and \(\mathfrak{A} \models \varphi\), it means there exists an element in \(\mathfrak{A}\) such that \(\psi\) holds. Since \(\mathfrak{A} \subseteq \mathfrak{B}\), this element will also exist in \(\mathfrak{B}\), making \(\psi\) true for at least that element in \(\mathfrak{B}\). Hence, \(\mathfrak{B} \models \varphi\) as well.

Key concepts

Universal Sentences

Universal sentences are the backbone of formal logic when it comes to expressions that hold true across an entire set of elements. The core idea behind a universal sentence is its declaration of a truth that applies universally—that is, for every single member within a structure. This type of sentence is represented with the symbol \( \forall \), which translates to 'for all'.

When constructing a universal sentence \( \varphi \), it appears in the form \( \forall x_1 \forall x_2 \ldots \forall x_n \psi(x_1, x_2, \ldots, x_n) \), with \( \psi \) being a formula without any quantifiers. This format ensures that its truth is tested against all possible elements \( x_1, x_2, ..., x_n \) present in the structure.

The strength of a universal sentence within a larger structure \( \mathfrak{B} \) is that if \( \varphi \) holds true in \( \mathfrak{B} \), it must also hold true in any substructure \( \mathfrak{A} \subseteq \mathfrak{B} \). This works because all elements of \( \mathfrak{A} \) are also within \( \mathfrak{B} \), inherently preserving the truth of \( \varphi \) across both.

Existential Sentences

Existential sentences pivot around the concept that at least one member of a structure abides by a certain truth. Characterized by the symbol \( \exists \), these logical declarations are expressed in the form \( \exists x_1 \exists x_2 \ldots \exists x_n \psi(x_1, x_2, \ldots, x_n) \). Here, \( \psi \) stands as a quantifier-free formula that is satisfied by one or more elements.

When an existential sentence \( \varphi \) is true in a structure \( \mathfrak{A} \), it signals the presence of at least one element within \( \mathfrak{A} \) that makes \( \psi \) hold. Since \( \mathfrak{A} \) is a subset of \( \mathfrak{B} \), any such element in \( \mathfrak{A} \) also exists in \( \mathfrak{B} \).

This subset relation ensures that if \( \mathfrak{A} \models \varphi \), then \( \mathfrak{B} \models \varphi \) must naturally follow, as the existential claim is supported by the broader context of \( \mathfrak{B} \). Existential sentences thus assert the existence of particular truths within possibly more extensive structures.

Mathematical Logic

Mathematical logic serves as the foundation for understanding the framework and interpretation of logical sentences, such as universal and existential ones, within mathematical structures. As both a tool and scientific discipline, it formalizes reasoning by encompassing areas like model theory, proof theory, and set theory.

With logical sentences, we delve into structures such as groups, fields, or rings, defining the rules and behaviors of elements within these spaces. In cases like the universal and existential statements discussed, we leverage logic to discern how truths are propagated or verified inside mathematical contexts.

A key application of mathematical logic is understanding why specific sentences behave a certain way across different structures, such as \( \mathfrak{A} \subseteq \mathfrak{B} \). It allows us to predict how a truth in one part of a system can extend to another, offering insights into the behavior of substructures and the overarching system. Thus, mathematical logic is indispensable as it puzzles out complex interrelations and properties of structures, empowering precise analytical processes.