Predicate Logic and Quantifiers

A propositional variable is a variable that can take on the values true or false. A predicate variable is a variable that can take on the values true or false for a given object. For example, the propositional variable "P" can take on the values true or false, while the predicate variable "F(x)" can take on the values true or false for a given object x.

A universal quantifier asserts that a property holds for all objects in a domain. An existential quantifier asserts that a property holds for at least one object in a domain. For example, the universal quantifier "(∀x)Fx" asserts that all objects in the domain have the property F, while the existential quantifier "(∃x)Fx" asserts that at least one object in the domain has the property F.

A free variable is a variable that is not within the scope of a quantifier. A bound variable is a variable that is within the scope of a quantifier. For example, in the proposition "(∀x)Fx", the variable "x" is a bound variable, while the variable "F" is a free variable.

A singular proposition is a proposition that asserts something about a particular object. A general proposition is a proposition that asserts something about all objects in a domain. For example, the proposition "Socrates is a philosopher" is a singular proposition, while the proposition "All men are mortal" is a general proposition.

A categorical proposition is a proposition that asserts something about a class of objects. A hypothetical proposition is a proposition that asserts something about a conditional relationship between two propositions. For example, the proposition "All men are mortal" is a categorical proposition, while the proposition "If it is raining, then the ground is wet" is a hypothetical proposition.

A deductive argument is an argument in which the conclusion follows logically from the premises. An inductive argument is an argument in which the conclusion is supported by the premises but does not follow logically from them. For example, the argument "All men are mortal. Socrates is a man. Therefore, Socrates is mortal" is a deductive argument, while the argument "All of the swans I have seen are white. Therefore, all swans are white" is an inductive argument.

A sound argument is an argument in which the premises are true and the conclusion follows logically from the premises. A valid argument is an argument in which the conclusion follows logically from the premises. For example, the argument "All men are mortal. Socrates is a man. Therefore, Socrates is mortal" is a sound argument, while the argument "All swans are white. Socrates is a swan. Therefore, Socrates is white" is a valid argument but not a sound argument.

A categorical syllogism is a syllogism in which the premises and conclusion are all categorical propositions. A hypothetical syllogism is a syllogism in which at least one of the premises or the conclusion is a hypothetical proposition. For example, the syllogism "All men are mortal. Socrates is a man. Therefore, Socrates is mortal" is a categorical syllogism, while the syllogism "If it is raining, then the ground is wet. It is raining. Therefore, the ground is wet" is a hypothetical syllogism.

A major premise is a premise that is more general than the conclusion. A minor premise is a premise that is more specific than the conclusion. For example, in the syllogism "All men are mortal. Socrates is a man. Therefore, Socrates is mortal", the major premise is "All men are mortal" and the minor premise is "Socrates is a man".

A conclusion is a proposition that follows logically from the premises of an argument. A corollary is a proposition that follows logically from a theorem. For example, in the argument "All men are mortal. Socrates is a man. Therefore, Socrates is mortal", the conclusion is "Socrates is mortal". In the theorem "The sum of the angles of a triangle is 180 degrees", the corollary is "The sum of the angles of a right triangle is 90 degrees".

A lemma is a proposition that is used to prove a theorem. A postulate is a proposition that is assumed to be true without proof. For example, in the proof of the Pythagorean theorem, the lemma "The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides" is used to prove the theorem "The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides".

A theory is a set of propositions that explains a natural phenomenon. A hypothesis is a proposition that is put forward as a possible explanation for a natural phenomenon. For example, the theory of evolution is a set of propositions that explains how different species of organisms have evolved over time. The hypothesis that the universe began with a Big Bang is a proposition that is put forward as a possible explanation for how the universe came into being.

A law is a statement of a relationship between two or more variables that is based on empirical evidence. A principle is a statement of a general truth that is not based on empirical evidence. For example, the law of gravity is a statement of the relationship between the mass of an object and the force of gravity acting on it. The principle of non-contradiction is a statement of the general truth that it is impossible for something to be both true and false at the same time.

An axiom is a proposition that is assumed to be true without proof. A theorem is a proposition that is proved from other propositions. For example, in the axiomatic system of Euclidean geometry, the axioms are the propositions that are assumed to be true without proof, and the theorems are the propositions that are proved from the axioms.