Logical consequence

Logical consequence, also known as entailment or logical implication, is a key idea in logic. It describes how one statement can follow from another or from several statements. When we say that one statement is a logical consequence of others, we mean that if the first statements are true, then the following statement must also be true.

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

Logical consequence is necessary and follows set rules. It doesn’t depend on personal feelings or opinions. Instead, it depends only on the form of the statements and logical rules. This helps us see clearly whether ideas must be true based on other true ideas.

Formal accounts

Logical consequence is about how statements relate to each other based on their structure. It looks at the form of statements, not their specific content, to see if one statement follows from another.

Some arguments follow a pattern that works every time, like saying "All X are Y" and "All Y are Z", so "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. This does not need to think about what the statements mean in real life.

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. In other words, 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.