Logical consequence

Logical consequence, also known as entailment or logical implication, is an important idea in logic. It shows how one statement can come from another or from many statements. When we say that one statement is a logical consequence of others, it means that if the first statements are true, then the next statement must also be true.

A good argument is one where the ending idea fits naturally with the ideas we start with, called premises. This happens when the ending idea is a logical consequence of those starting ideas. Philosophers study logical consequence to understand better how ideas are linked and what it means for one idea to follow from others.

Logical consequence is needed and follows fixed rules. It doesn’t depend on personal feelings or opinions. Instead, it depends only on the shape of the statements and the rules of logic. This helps us see clearly if ideas must be true based on other true ideas.

Formal accounts

Logical consequence is about how statements relate to each other. It looks at the form of statements, not what they are about, to see if one statement follows from another.

Some arguments follow a pattern that works every time. For example, if we say "All X are Y" and "All Y are Z," then we know "All X are Z." This is a formal argument because it works for any X and Y. Other arguments depend on specific meanings, like family relationships, and are not formal in the same way.

A priori property

If we know that one idea, called Q, follows logically from another idea, called P, then understanding what P or Q means won’t change this fact. Knowing that Q is a logical result of P doesn’t need us to look at real-world examples. We can know this just by thinking carefully, without needing experience. But being formal in our thinking doesn’t always mean our logical result isn’t affected by real-world knowledge. So, the idea that logical consequence can be known just by thinking is separate from how formal our thinking is.

Proofs and models

There are two main ways to understand how one idea can come from another in logic. These are called proofs and models.

The study of logical ideas using steps or rules is called proof theory. The study of logical ideas using examples or situations is called model theory.

Syntactic consequence

A statement is a syntactic consequence in a special system if we can follow exact steps or rules to show it must be true from other statements.

Semantic consequence

A statement is a semantic consequence in a special system if there is no example where all the starting statements are true but this new statement is false. This means that whenever the starting statements are true, the new one must also be true.

Modal accounts

Modal accounts of logical consequence are based on the idea that one statement follows from others if it must always be true when the others are true.

For example, if we know all frogs are green and Kermit is a frog, we can say for sure that Kermit is green. This is because there is no way to imagine a situation where all frogs are green, Kermit is a frog, and Kermit is not green at the same time.

These ideas use special words like "must" and "impossible" to describe how one statement leads to another.