Entailment and Inference in Predicate Logic

A valid argument form is one in which the conclusion follows logically from the premises. In the given argument form, if P is true, then Q must also be true. Therefore, it is a valid argument form.

A sound argument is one in which both the premises and the conclusion are true. In the given argument, if it is raining, then the ground is indeed wet. Therefore, it is a sound argument.

A complete argument system is one in which every valid argument can be proven from the axioms of the system. Classical propositional logic is a complete argument system, meaning that every valid propositional argument can be proven from its axioms.

Modus ponens is a rule of inference that allows us to infer the conclusion Q from the premises P and If P, then Q. In the given example, we have the premise P and the conditional statement If P, then Q. Therefore, we can infer the conclusion Q.

Modus tollens is a rule of inference that allows us to infer the conclusion not P from the premises If P, then Q and not Q. In the given example, we have the conditional statement If P, then Q and the premise not Q. Therefore, we can infer the conclusion not P.

Hypothetical syllogism is a rule of inference that allows us to infer the conclusion If P, then R from the premises If P, then Q and If Q, then R. In the given example, we have the conditional statements If P, then Q and If Q, then R. Therefore, we can infer the conclusion If P, then R.

Affirming the consequent is a fallacy that occurs when we infer the truth of the antecedent from the truth of the consequent. For example, if we have the conditional statement If P, then Q and we know that Q is true, we cannot conclude that P is also true. This is because there may be other conditions that can also lead to Q being true.

Universal generalization is a rule of inference that allows us to infer the conclusion For all x, P(x) from the premises For all x, P(x) and P(a), where a is an arbitrary member of the domain of discourse. In the given argument form, we have the premise For all x, P(x) and the premise P(a). Therefore, we can infer the conclusion For all x, P(x).

Existential generalization is a rule of inference that allows us to infer the conclusion There exists an x such that P(x) from the premises There exists an x such that P(x) and P(a), where a is an arbitrary member of the domain of discourse. In the given argument form, we have the premise There exists an x such that P(x) and the premise P(a). Therefore, we can infer the conclusion There exists an x such that P(x).

Universal instantiation is a rule of inference that allows us to infer the conclusion P(a) from the premises For all x, P(x) and a is an arbitrary member of the domain of discourse. In the given argument form, we have the premise For all x, P(x) and the premise a is an arbitrary member of the domain of discourse. Therefore, we can infer the conclusion P(a).

Existential instantiation is a rule of inference that allows us to infer the conclusion P(a) from the premises There exists an x such that P(x) and a is an arbitrary member of the domain of discourse. In the given argument form, we have the premise There exists an x such that P(x) and the premise a is an arbitrary member of the domain of discourse. Therefore, we can infer the conclusion P(a).

Hypothetical syllogism is a rule of inference that allows us to infer the conclusion If P, then R from the premises If P, then Q and If Q, then R. In the given argument form, we have the premises If P, then Q and If Q, then R. Therefore, we can infer the conclusion If P, then R.

Disjunctive syllogism is a rule of inference that allows us to infer the conclusion Q from the premises P or Q and Not P. In the given argument form, we have the premises P or Q and Not P. Therefore, we can infer the conclusion Q.

Constructive dilemma is a rule of inference that allows us to infer the conclusion Either R or S from the premises Either P or Q, If P, then R, and If Q, then S. In the given argument form, we have the premises Either P or Q, If P, then R, and If Q, then S. Therefore, we can infer the conclusion Either R or S.

Destructive dilemma is a rule of inference that allows us to infer the conclusion Not Q from the premises Either P or Q, If P, then R, If Q, then S, and Not R. In the given argument form, we have the premises Either P or Q, If P, then R, If Q, then S, and Not R. Therefore, we can infer the conclusion Not Q.