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)
∃(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.