Logical consequence

Sometimes, one true fact leads to another. We call this a consequence.

If you know two things are true, you can find a third. This third thing must also be true. It follows the first two facts.

This works because of how the ideas are built. It does not depend on what the words mean. It only needs the shape of the idea.

Think about a pattern. If all frogs are green, and Kermit is a frog, then Kermit must be green. You can see the truth without even looking at him.

This helps us think in a clear way. It is a rule for our minds to follow.

Sometimes, one true fact leads to another. This is called a logical consequence. It means if some ideas are true, then a new idea must be true too. This new idea follows from the others.

Logicians look at the shape of an idea. They call this the logical form. A good argument works because of this shape. It does not matter what the words mean. For example, if all X are Y, and all Y are Z, then all X are Z. This rule works for any X, Y, or Z. This is called a formal consequence.

A Polish thinker named Alfred Tarski studied this. He found three main parts of this idea. First, it relies on the logical form. Second, it is a priori. This means you can know it is true without seeing it in the real world. Third, it has a modal part. This means it is impossible for the first facts to be true and the last fact to be false.

Some people study this using proofs. They call this proof theory. Others use models to check if ideas fit. They call this model theory. These ways help us keep our thinking clear and true.

Have you ever noticed how one true fact can lead you to another? This special link is called a logical consequence. It happens when a new statement must be true because other statements are true. We call the starting statements premises. The new statement that follows is the conclusion. If the conclusion is guaranteed by the premises, the argument is valid. This concept is a building block for all of logic. It helps us understand how truth travels from one idea to the next.

Logicians study the shape of these ideas. They call this shape the logical form. A formal consequence depends only on this shape. It does not care about the specific meaning of the words. For example, if all X are Y and all Y are Z, then all X must be Z. This works no matter what X, Y, or Z are. This is different from a material consequence. A material consequence depends on what the words actually mean. For instance, knowing Fred is Mike's nephew depends on knowing what a nephew is. A formal rule works even if you use nonsense words.

Many thinkers have explored these rules over time. A Polish logician named Alfred Tarski identified three important features of this idea. First, it relies on the logical form of sentences. Second, it is a priori. This means you can know it is true without using your senses or seeing evidence. Third, it has a modal component. This means it is impossible for the premises to be true while the conclusion is false. These ideas help us define how logic works in a precise way.

There are two main ways to study these connections. The first is called proof theory. This is the study of syntactic consequence. It uses formal rules to show how one formula follows from another. The second is called model theory. This is the study of semantic consequence. It looks at whether a statement is true in every possible model. Some people even use modal accounts. These accounts use the idea of possible worlds to check for truth. They say a conclusion follows if it is impossible to imagine a world where the premises are true and the conclusion is false.

Logical consequence helps us keep our thinking steady. It is a way to preserve truth. Most thinkers believe a good argument never lets you move from truth to a lie. Some people call this truth-preservation. Others suggest a different way called warrant-preservation. This focuses on whether an idea is justifiable. You can see this in everyday life when you follow a recipe or a set of directions. If the steps are true and you follow them, the result must follow too.

Logical consequence is a fundamental concept in the field of logic. It describes the relationship between statements where one statement must be true if others are true. These starting statements are called premises. The statement that follows from them is called the conclusion. When a conclusion is guaranteed by its premises, the argument is considered valid. Philosophers ask deep questions about this relationship. They wonder in what sense a conclusion truly follows from its premises. They also seek to define what it actually means for a conclusion to be a consequence.

To understand this, we must look at the logical form of an argument. A formal consequence depends entirely on the structure of the statements. It does not care about the specific meaning of the words used. For example, consider the scheme: All X are Y; All Y are Z; Therefore, all X are Z. This argument is formally valid. It remains valid no matter what you substitute for X, Y, or Z. This is different from a material consequence. A material consequence relies on the actual meanings of words. For instance, saying "Fred is Mike's nephew" because "Fred is Mike's brother's son" requires knowing what a nephew is.

Logicians use different methods to define these connections. One way is through syntactic consequence, which is studied in proof theory. In this approach, a formula is a consequence if there is a formal proof for it within a system. This is often shown using a special symbol called a turnstile. The turnstile symbol was first introduced by Gottlob Frege in 1879. However, its modern use began with Rosser and Kleene between 1934 and 1935. Syntactic consequence does not rely on any outside interpretation. It stays strictly within the rules of the formal system.

Another approach is semantic consequence, which is studied in model theory. This method looks at how statements behave in different models or interpretations. A formula is a semantic consequence if there is no model where the premises are true and the conclusion is false. This means the set of interpretations making the premises true is a subset of those making the conclusion true. Some thinkers also use modal accounts to explain this. Modal accounts use the ideas of logical necessity and possibility. They suggest a conclusion follows if it is impossible for the premises to be true while the conclusion is false. This can be imagined through "possible worlds." If you cannot imagine a world where the premises are true and the conclusion is false, the consequence holds.

The Polish logician Alfred Tarski identified three essential features of entailment. First, the relation must rely on the logical form of the sentences. Second, the relation must be a priori. This means it can be determined without using empirical evidence or sense experience. You do not need to observe the world to know a logical truth. Third, the relation has a modal component. This connects back to the idea of necessity mentioned in modal accounts.

Most accounts of logical consequence are truth-preservational. This means the main goal is to ensure truth is never lost. A good inference should never move from true premises to an untrue conclusion. However, some thinkers propose warrant-preservational accounts instead. This idea is favored by intuitionists. They believe a good inference should move from justifiable premises to a justifiable conclusion. This shifts the focus from absolute truth to what can be properly asserted.

Logical consequence also behaves in specific ways regarding new information. Most standard accounts are monotonic. This means if a conclusion follows from a set of premises, it will also follow if you add more premises to that set. However, non-monotonic consequence relations exist. These allow for conclusions to change when new information is added. For example, the idea that "Tweety can fly" might follow from knowing Tweety is a bird. But if you add the information that Tweety is a penguin, that conclusion may no longer follow.