In logic, the symbol |= represents a semantic concept known as semantic entailment or logical consequence. It signifies that a statement (or conclusion) is true in every possible situation where the assumptions or premises are also true. Let's break down its meaning and usage:
Semantic Entailment Explained
The |= symbol links two logical statements, typically represented as 'A |= B'. This is read as "A semantically entails B" or "B is a logical consequence of A." Here's a breakdown:
- A: Represents a set of premises, assumptions, or a logical formula.
- |=: Represents the semantic entailment relation.
- B: Represents the conclusion or statement that follows from 'A'.
The key idea is that in every possible model (or interpretation) where A is true, B is also true. It's a statement about truth in all possible scenarios where the premises hold.
Table: Syntactic vs. Semantic Entailment
| Symbol | Concept | Meaning | Focus |
|---|---|---|---|
| ** | -** | Syntactic Entailment | "There is a proof of B assuming A." (Focuses on rules of inference and proof systems). |
| ** | =** | Semantic Entailment | "B holds for every model of A." (Focuses on the truth of statements in all interpretations) |
Key Difference: According to our reference, |- is syntactic, while |= is semantic. The distinction is crucial. Syntactic entailment (|- ) involves showing that B can be derived from A using rules of inference. Semantic entailment (|=) asserts that if A is true, B must be true in every conceivable case.
Understanding Semantic Entailment with Examples
- Example 1: If A is "All dogs are mammals" and B is "My dog is a mammal," then A |= B, because in any situation where it's true that all dogs are mammals, then, my dog, being a dog, must also be a mammal.
- Example 2: If A is "It is raining" and B is "The ground is wet," then it’s not universally true that A |= B. The ground could be wet for other reasons. However, "It is raining AND there are no obstacles between the rain and the ground" |= "The ground is wet" is a true logical entailment.
- Example 3: If A is "All humans are mortal" and B is "Socrates is mortal", then the proposition "Socrates is a human", we would have {“All humans are mortal”, “Socrates is human”} |= "Socrates is mortal"
Practical Insights
- Semantic entailment is used to establish the validity of arguments. If a conclusion is a semantic consequence of the premises, the argument is considered logically sound.
- It's related to the concept of validity in logic, which is concerned with the truth preservation of inferences.
- The |= symbol is a cornerstone concept in formal logic, providing a framework for rigorous reasoning and proof.
