Question

Show that \(\mathfrak{A} \models \varphi \Rightarrow \mathfrak{A} \models \psi\) for all \(\mathfrak{A}\), implies \(\models \varphi \Rightarrow \models \psi\), but not vice versa.

Short answer

The implication holds forward: if all models of \(\varphi\) satisfy \(\psi\), then \(\varphi\)'s validity implies \(\psi\)'s. The reverse is not guaranteed.

Step-by-step solution

Understand the Definitions

We start by understanding the definitions: - \(\mathfrak{A} \models \varphi\) means that the structure \(\mathfrak{A}\) satisfies the formula \(\varphi\).- \(\models \varphi\) means that \(\varphi\) is valid, i.e., true in every structure.The implication \(\models \varphi \Rightarrow \models \psi\) means that if \(\varphi\) is valid, then \(\psi\) is also valid.

Analyze the Forward Implication

To show that \(\mathfrak{A} \models \varphi \Rightarrow \mathfrak{A} \models \psi\) for all \(\mathfrak{A}\) implies \(\models \varphi \Rightarrow \models \psi\), we consider: If every structure \(\mathfrak{A}\) that satisfies \(\varphi\) also satisfies \(\psi\), then when \(\varphi\) is valid (true in all structures), \(\psi\) must also be valid (true in all structures).

Consider Validity of Formulas

From Step 2, since \(\varphi\) is valid in all structures, and every structure satisfying \(\varphi\) satisfies \(\psi\), \(\psi\) must also be valid in all structures. Hence, \(\models \varphi \Rightarrow \models \psi\) must hold.

Analyze the Reverse Implication

Next, for the reverse, assume \(\models \varphi \Rightarrow \models \psi\) but not necessarily \(\mathfrak{A} \models \varphi \Rightarrow \mathfrak{A} \models \psi\) for all \(\mathfrak{A}\). Consider \(\models \varphi\) to mean that \(\varphi\) is true universally. If this implies \(\psi\) is also valid, it doesn't guarantee that in every single structure \(\varphi\) leading to \(\psi\) must hold. Examples can exist where some structures satisfy \(\varphi\) but don't satisfy \(\psi\), not affecting the overall validity of \(\varphi \Rightarrow \psi\).

Conclusion with Examples or Counterexamples (if needed)

Illustrate the concept with examples or theoretical constructs that show that even when \(\models \varphi \Rightarrow \models \psi\), some cases of \(\mathfrak{A} \models \varphi \land eg(\mathfrak{A} \models \psi)\) might exist, as long as \(\psi\) is correctly postulated wherever true universality is questioned.

Key concepts

Model Theory

Model theory is a branch of mathematical logic studying the relationships between formal languages and their interpretations or models. In this context, a model is a structure that gives meaning to the sentences of a language. These structures, denoted by \(\mathfrak{A}\), consist of a domain and interpretations of the symbols and operations found in the language. Understanding how these structures interpret formulas helps us analyze logical implications.
Model theory provides tools to evaluate how different formulas behave in various structures, especially how a formula like \(\varphi\) might imply another formula \(\psi\) within the same structure. When saying \(\mathfrak{A} \models \varphi\), it means that the model \(\mathfrak{A}\) satisfies \(\varphi\), i.e., every element of the domain behaves according to the specifications set out by the formula.

Validity in Logic

Validity in logic refers to a formula being true in every possible structure or model. A formula is valid if it holds in all interpretations offered by the universe of discourse. When we say \(\models \varphi\), it signifies that \(\varphi\) is universally true, no exceptions across all possible models.
In logical implications, if \(\varphi \Rightarrow \psi\) holds, meaning \(\models \varphi \Rightarrow \models \psi\), it suggests that the truth of \(\varphi\) universally necessitates the truth of \(\psi\). Hence, every scenario where \(\varphi\) is true, \(\psi\) must also be true across all logical structures. This universality is the cornerstone of logic validity, ensuring arguments hold regardless of individual model structures.

Logical Structures

Logical structures are the frameworks used to give meaning to expressions within a logic system. They consist of a non-empty set, known as the domain, and interpretations for constants, functions, and relations of a language. Each logical structure can vary, but they all fulfill the role of interpreting formulas and determining their truths.
Analyzing logical structures involves assessing how different elements within the domain apply rules or conditions specified by a formula. It's crucial for understanding how \(\mathfrak{A} \models \varphi \Rightarrow \mathfrak{A} \models \psi\) works, as it focuses on every structure itself and how it satisfies or does not satisfy certain formulas. These structures validate propositions by assessing individual, internal consistency rather than external universality.

Formulas in Logic

Formulas in logic are expressions that convey a statement or a set of operations using the symbols of a logical system. These can include variables, logical connectives, quantifiers, or both rules and hypotheses within an argument.
The essence of a formula is that it represents a well-defined logical statement, such as \(\varphi\) or \(\psi\); their relationship and interaction form the basis of logical analysis and implications. For a formula such as \(\varphi\) to imply another formula \(\psi\) within a structure, it must be confirmed that whenever \(\varphi\) holds, \(\psi\) also holds. These logical interdependencies in formulas help explain broader implications and are key to understanding complex logical assessments.