Question

Show (a) \(\varphi \models \varphi\); (b) \(\varphi \models \psi\) and \(\psi \models \sigma \Rightarrow \varphi \models \sigma\); (c) \(\models \varphi \rightarrow \psi \Rightarrow \varphi \models \psi\).

Short answer

(a) Reflexive; (b) Transitive; (c) Follows from Deduction Theorem.

Step-by-step solution

Understand Reflexivity

We need to show \( \varphi \models \varphi \). This is known as the reflexivity of entailment. An interpretation that makes \( \varphi \) true will necessarily make \( \varphi \) true, because it is the same formula.

Apply Reflexivity

For any interpretation \( I \), if \( I \models \varphi \), then it is trivially true that \( I \models \varphi \). Therefore, \( \varphi \models \varphi \) holds for any formula \( \varphi \).

Understand Transitivity

We need to show \( \varphi \models \psi \) and \( \psi \models \sigma \Rightarrow \varphi \models \sigma \). This is known as the transitivity of entailment. If \( \varphi \) entails \( \psi \) and \( \psi \) entails \( \sigma \), then \( \varphi \) must also entail \( \sigma \).

Apply Transitivity

Assume \( \varphi \models \psi \) and \( \psi \models \sigma \) hold. For any interpretation \( I \) such that \( I \models \varphi \), it follows that \( I \models \psi \). Likewise, since \( I \models \psi \), it follows that \( I \models \sigma \). Thus, \( \varphi \models \sigma \).

Understand Deduction Theorem

We need to show \( \models \varphi \rightarrow \psi \Rightarrow \varphi \models \psi \). This leverages the deductive relationship between propositions, meaning that if \( \varphi \rightarrow \psi \) is a tautology, then \( \varphi \) entails \( \psi \).

Apply Deduction Theorem

Consider \( \models \varphi \rightarrow \psi \). For any interpretation \( I \) such that \( I \models \varphi \), the implication \( \varphi \rightarrow \psi \) must also hold since \( \varphi \rightarrow \psi \) is a tautology. Hence, \( I \models \psi \). Therefore, \( \varphi \models \psi \).

Key concepts

Reflexivity in Logic

Reflexivity in logic is a foundational concept that states a logical formula is always entailed by itself. This idea is often expressed as \( \varphi \models \varphi \). When we talk about entailment, we mean that if something holds true in one scenario, it should also hold true in related scenarios. In the case of reflexivity, it's about the simplest scenario: the same proposition.

Let's break it down: If you have a formula \( \varphi \), and an interpretation (or a situation) where \( \varphi \) is true, then obviously, in that same interpretation, \( \varphi \) will still be true. This may seem redundant, but it underlines the consistency of logic. No new conditions need to apply in this interpretation for it to be true again.

• It's like saying, "If it's raining, then it's raining." This is inherently true without needing further demonstration.
• It establishes a basis for more complex logic principles by showing logical reliability.

Reflexivity is pivotal in formal logic systems, highlighting the self-evident nature of truths within a single proposition.

Transitivity in Logic

Transitivity in logic helps us connect different propositions logically. It expresses the idea that if one thing leads to a second, and that second thing leads to a third, then the first thing should naturally lead to the third. Formally, we write this as: if \( \varphi \models \psi \) and \( \psi \models \sigma \), then \( \varphi \models \sigma \).

This principle is very important in proving that logical arguments hold true across different steps.

• It allows us to extend the reasoning process by chaining entailments.
• By understanding that each step pushes the implication further, it simplifies complex reasoning by breaking it into manageable parts.

For instance, imagine you have three facts: A implies B, and B implies C. With transitivity, you can immediately conclude that A implies C.

Thus, transitivity is a powerful tool in logic, assisting in the process of deduction, ensuring that conclusions remain consistent and valid over a chain of statements.

Deduction Theorem

The Deduction Theorem is a significant principle in mathematical logic that relates implications to entailment. In essence, it shows how we can derive one proposition from another by using logical implications. Specifically, it states if \( \models \varphi \rightarrow \psi \) is valid, then \( \varphi \models \psi \).

What this means is when the implication \( \varphi \rightarrow \psi \) is a tautology – universally true – we can deduce that wherever \( \varphi \) holds true, \( \psi \) must also be true.

• This principle is useful for deriving conclusions from assumptions.
• It bridges the gap between proof and implication, illustrating how a provable implication can be translated into a logical entailment.

For example, if we can prove that "If George is in the kitchen, then the light is on" is always true, then whenever George is indeed in the kitchen, we can confidently say that the light must be on.

This theorem empowers the proof processes in mathematics and computer science by linking assumptions and derived conclusions.