## First order logic

**What is first-level predicate logic?**

**First-order logic (FOL) is a method based on mathematics for assigning unique properties to an object.** Here, each sentence/statement is decomposed into its subject and its predicate. The relationship between them is done in first-level predicate logic by P(x), where P stands for predicate and the variable x for the corresponding subject.

It should be noted that the **Predicates in First-Order Logic refer to only one subject at a time.**. Unlike in linguistics, a predicate is not necessarily a verb, but merely provides relevant information about the subject in question. The use of the **Predicates also allow relations to be established**; for example, through comparisons (greater/smaller than, equal to, etc.).

In the first-level predicate logic, the Quantifiers and represented by the symbols ∀ (universal quantifier; read: "for all") and ∃ (existential quantifier; read: "it exists" or "for some"). The representation is done in First-Order Logic by mathematical symbols and consists of:

- Terms: Human, animal, plant etc.
- Names of objects. In the linguistic sense, these can be both objects and subjects!
- Variables a, b, c, ..., x, y, z etc.

These stand for objects that are not yet known.

Predicates [red, fragrant, is a flower etc.] stand for properties and relations that are linguistically comparable to verbs or attributes.

Quantifiers [∀, ∃] allow statements about sets of objects for which the predicate applies.

Relations [∧ (and), ∨ (or), →(implies), ⇒ (follows from), ⇔ (is equivalent to), == (equality - operator)] give conclusions about relations.

**Example of first level predicate logic**

The rose is red.

P(x) = red(rose)

The rose is fragrant.

P(x) = fragrant(rose)

The rose is a flower.

P(x) = Flower(Rose)

We learn about the rose that it **red** is, **Smells** and a **Flower** is.

This results in ∀:

**All** Roses are **red**.

**All** Roses **fragrant**.

**All** Roses are **Flowers**.

However, not all roses are red and not every rose is fragrant.

That all roses are flowers, on the other hand, is a true statement.

∀(x) Rose(x) → Flower(x)

In order that the other two statements can be checked for their correctness, existential quantifiers are now used.

From the two statements:

"All the roses are red." and "All the roses are fragrant." are made by using ∃:

"Some roses are red." and "Some roses are fragrant."

To translate it into a first-order formula, we need to define a variable x:

A predicate A(x), where **x** the **Rose** and a predicate G(x), which corresponds for **x** is, **red** resp. **Smells**.

∃(x) Rose(x) → red(x)

resp.

∃(x) Rose(x) → fragrant(x)

This tells you that there are roses that are **red** are and roses exist that **fragrant**. It follows logically that there must also be roses that are not red or that are not fragrant.