"A formal proof or derivation is a finite sequence of sentences, each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference."
A formal proof is a step-by-step demonstration of the truth of a statement or theorem in a formal language. It is used in mathematics and logic to establish the validity of arguments.
Propositional Logic: This is the study of propositions and their logical relationships, including conjunction, disjunction, implication, and negation.
Predicate Logic: Predicate logic is used for reasoning about properties and relations between objects in the world. It involves quantifiers such as universal and existential quantifiers.
Natural Deduction: Natural deduction is a system of formal logic that is used to derive conclusions from premises by applying rules of inference.
Proof Theory: Proof theory is the study of formal proofs and their properties. It involves the study of the structure of proofs, the relations between proofs, and the properties of the rules of inference used in the proofs.
Set Theory: Set theory is the study of sets and their properties, including relations such as inclusion and intersection.
Axiomatic Systems: An axiomatic system is a set of axioms and rules that are used to derive proofs. It is used to establish the foundations of mathematical logic.
Model Theory: Model theory is the study of mathematical structures such as groups, fields, and spaces. It involves the study of the properties of these structures and the relationships between them.
Formal Specification: Formal specification is the process of defining a system using formal methods. It involves the specification of requirements, the design of the system, and the verification of the system.
Program Verification: Program verification is the process of proving that a program is correct with respect to a specification. It involves the use of formal methods to verify the behavior of the program.
Type Theory: Type theory is the study of the properties of types and their relationships. It is used in computer science to ensure that programs are well typed and free from type errors.
Direct Proof: Direct proof is an approach in which we start with a hypothesis and proceed to prove that the conclusion is true.
Indirect Proof: This type of proof is also known as reductio ad absurdum, which means "reduction to the absurd." In indirect proof, you assume the opposite of what you want to prove and show that it leads to a contradiction. This contradiction implies that the original hypothesis is true.
Proof by Contradiction: A proof by contradiction starts with the assumption that the conclusion is false and then uses logic to show that this leads to a contradiction.
Proof by Mathematical Induction: Mathematical induction is used to prove statements that involve integers. It consists of two parts: a base case and an inductive step. The base case is usually proved directly, and the inductive step uses the assumption that the statement is true for some integer to show that it is true for the next integer.
Proof by Cases: In this type of proof, we break the hypothesis into several cases, and then prove that the conclusion holds true for each one of them.
Proof by Exhaustion: Proof by exhaustion is used when there are only a finite number of cases for a hypothesis. This type of proof involves checking every single case to show that the conclusion holds true in all of them.
Constructivist Proof: This type of proof is used when the conclusion is a statement that asserts the existence of a particular object. To prove the statement, the proof has to construct the object that satisfies the property involved.
Non-Constructive Proof: This type of proof is used when the conclusion is a statement that asserts the existence of a particular object but does not require constructing it. The proof typically involves showing that such an object exists by means of contradiction.
Reductio Ad Absurdum: In this type of proof, a hypothesis is assumed to be false, and then this leads to a contradiction. This contradiction shows that the hypothesis is true.
Proof by Contraposition: In this type of proof, the conclusion is proved by showing that its contrapositive is true. The contrapositive of a statement is its inverse with the negation of the conclusion.
"It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable."
"If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system."
"The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists."
"The concepts of Fitch-style proof, sequent calculus, and natural deduction are generalizations of the concept of proof."
"For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus (of some formal system) to the previous well-formed formulas in the proof sequence."
"Formal proofs often are constructed with the help of computers in interactive theorem proving."
"Significantly, these proofs can be checked automatically, also by computer."
"Checking formal proofs is usually simple."
"The problem of finding proofs (automated theorem proving) is usually computationally intractable and/or only semi-decidable, depending upon the formal system in use."
"Checking formal proofs is usually simple, while the problem of finding proofs (automated theorem proving) is usually computationally intractable and/or only semi-decidable, depending upon the formal system in use."
"Yes, a formal proof can include assumptions."
"Each sentence in a formal proof is either an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. So, axioms and assumptions serve as distinct types of sentences in a formal proof."
"A formal proof or derivation is a finite sequence of sentences... It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable."
"No, a formal proof or derivation is a finite sequence of sentences."
"The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof."
"These proofs can be checked automatically, also by computer."
"It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable."
"Yes, the concepts of Fitch-style proof, sequent calculus, and natural deduction are generalizations of the concept of proof."
"The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists."