Inference in First-Order Logic (FOL) involves deriving new sentences or conclusions from given premises using logical reasoning, relying on the formal structure of FOL, including variables, constants, predicates, functions, and quantifiers to construct valid arguments. Key inference rules like Modus Ponens, Universal Elimination, Existential Introduction, and Universal Introduction are applied, with techniques such as direct proof, proof by contradiction, and resolution aiding in systematically deriving conclusions or proving theorems. Substitution is a fundamental operation in FOL, involving the replacement of variables with constants or terms. The use of equality indicates that two terms refer to the same object. Universal Generalization infers universal statements from particular instances, while Universal Instantiation allows substituting ground terms for variables in universal statements. Existential Instantiation simplifies existential quantifiers by introducing a new constant, and Existential Introduction generalizes specific instances to existential statements. Generalized Modus Ponens is a lifted version of Modus Ponens used to derive conclusions by applying the rule to quantified variables and substitutions, making it crucial for logical arguments and proofs.
Inference in First-Order Logic (FOL) involves deriving new sentences or conclusions from a set of given premises using logical reasoning. It relies on the formal structure of FOL, including variables, constants, predicates, functions, and quantifiers, to construct valid arguments. Key inference rules, such as Modus Ponens, Universal Elimination, Existential Introduction, and Universal Introduction, are applied to manipulate these sentences. Techniques like direct proof, proof by contradiction, and resolution help systematically derive conclusions or prove theorems. The process of unification is essential in automated reasoning systems, enabling the matching and inference of new information. Automated theorem proving systems leverage these principles to systematically apply inference rules and derive logical conclusions, making inference in FOL a fundamental aspect of logical reasoning and AI. This process of AI inference ensures robust decision-making and logical consistency within artificial intelligence systems.
Substitution is a fundamental operation in first-order logic, applied to terms and formulas across all inference systems. The process becomes more complex with the presence of quantifiers in FOL. For instance, writing F[a/x] indicates that the constant "a" replaces the variable "x".
Note: First-order logic can express facts about some or all objects in the universe.
Equality: In first-order logic, atomic sentences are constructed not only using predicates and terms but also by incorporating equality. Equality symbols are used to indicate that two terms refer to the same object.
Example: Father(John) = Smith.
In this example, the term Father(John) refers to the same object as Smith. The equality symbol can also be combined with negation to show that two terms refer to different objects.
Example: ￢(x=y), which is equivalent to x ≠ y.
First-Order Logic Inference Rules for Quantifiers:
First-order logic (FOL) includes a set of inference rules that govern how we can manipulate and infer new information from statements involving quantifiers. Here are the primary inference rules for quantifiers in FOL, explained in detail:
Universal Generalization is a rule of inference used in formal logic, particularly in first-order logic. It allows us to infer a universal statement from a particular instance. The rule states that if we can prove that a certain property P(c) holds for an arbitrary element c in the universe of discourse, then we can conclude that the property P holds for all elements x in the universe. In other words, we can generalize from a specific instance to a universal statement.
The rule can be formally represented as follows:
Suppose we want to prove that every number greater than 0 is positive. We can use universal generalization as follows:
Formally:
Universal generalization is widely used in mathematical proofs, especially in proving theorems that apply to all members of a set. It is a powerful tool that helps in establishing general truths from specific cases.
Universal Instantiation (also called as universal elimination or UI) is a valid inference rule. It can be applied multiple times to add new sentences. The new knowledge base (KB) is logically equivalent to the previous KB.
As per UI, we can infer any sentence obtained by substituting a ground term for the variable. The UI rule states that we can infer any sentence P(c) by substituting a ground term c (a constant within domain x) from ∀x P(x) for any object in the universe of discourse.
It can be represented as:
Suppose we know that all humans are mortal, i.e., ∀x (Human(x) → Mortal(x)). By using universal instantiation, we can infer that if Socrates is human, then Socrates is mortal.
Universal instantiation is frequently used in proofs and reasoning in first-order logic, allowing us to apply general rules to specific instances. This rule helps in breaking down universal statements into specific cases that can be further analyzed or utilized.
Existential Instantiation (also called as Existential Elimination) is a valid inference rule in first-order logic. It can be applied only once to replace the existential sentence. The new knowledge base (KB) is not logically equivalent to the old KB, but it will be satisfiable if the old KB was satisfiable.
This rule states that one can infer P(c) from the formula given in the form of ∃x P(x) for a new constant symbol c. The restriction with this rule is that c used in the rule must be a new term for which P(c) is true.
It can be represented as:
From the given sentence: ∃x (Crown(x) ∧ OnHead(x, John)),
So we can infer: Crown(K) ∧ OnHead(K, John), as long as K does not appear in the knowledge base.
The above used K is a constant symbol, which is called Skolem constant. The existential instantiation is a special case of the Skolemization process.
Existential instantiation is used to simplify existential quantifiers in logical expressions by introducing a specific constant that satisfies the existential condition. This technique is useful in proofs and reasoning within first-order logic.
Existential Introduction (also known as existential generalization) is a valid inference rule in first-order logic. This rule states that if there is some element c in the universe of discourse which has a property P, then we can infer that there exists something in the universe which has the property P.
It can be represented as:
Let's say that,
"Priyansu got good marks in English."
"Therefore, someone got good marks in English."
Existential introduction is used to generalize a specific instance to an existential statement. It is useful in logical proofs and reasoning, where we need to demonstrate that at least one element in the universe satisfies a given property.
For the inference process in first-order logic (FOL), we have a single inference rule which is called Generalized Modus Ponens. It is a lifted version of Modus Ponens.
Generalized Modus Ponens can be summarized as, "P implies Q and P is asserted to be true, therefore Q must be true."
style="text-align: justify;"According to Modus Ponens, for atomic sentences pi, pi', q, where there is a substitution θ such that SUBST(θ, pi') = SUBST(θ, pi), it can be represented as:
We will use this rule for "Kings are evil." We will find some x such that x is king, and x is greedy so we can infer that x is evil.
Here, let’s say:
Therefore, SUBST(θ, q) gives us evil(John).
The Generalized Modus Ponens rule is powerful for drawing conclusions in first-order logic by generalizing the Modus Ponens rule to work with quantified variables and substitutions. It is a fundamental rule for constructing logical arguments and proofs in FOL.