Rules of Inference and Proofs Flashcards
By the end of this deck, learners will have a solid understanding of the basic rules of inference within propositional logic and their application in various types of proofs. They will be equipped to use these inference rules methodically to construct coherent and logically sound proofs, enhancing their ability to reason and argue within the framework of formal logic.
What is a rule of inference in propositional logic?
A rule of inference is a logical form that identifies a valid segment of an argument, allowing one to deduce a conclusion from one or more premises.
What is modus ponens in propositional logic?
Modus ponens is a rule of inference that allows one to deduce the consequent from a conditional statement and its antecedent. If ‘p → q’ and ‘p’ are true, then ‘q’ can be concluded.
Define modus tollens.
Modus tollens is a rule of inference where, if ‘p → q’ is true and ‘q’ is false (‘¬q’), then ‘p’ must also be false (‘¬p’).
What is a proof in the context of propositional logic?
A proof is a logical argument that establishes the truth of a statement based on axioms, definitions, and previously established statements using rules of inference.
How does the rule of “and elimination” work?
“And elimination” allows you to infer either of the conjuncts as true if you know that ‘p ∧ q’ is true.
Explain the “and introduction” rule.
“And introduction” allows you to infer ‘p ∧ q’ if you know that both ‘p’ and ‘q’ are true independently.
What is the purpose of using inference rules in proofs?
Inference rules are used in proofs to derive conclusions from premises logically, ensuring the argument’s validity and soundness.
Define the rule of “or introduction.”
“Or introduction” allows you to infer ‘p ∨ q’ from the truth of either ‘p’ or ‘q’.
How does the rule of “or elimination” work in propositional logic?
“Or elimination” allows you to infer a conclusion from ‘p ∨ q’ if you can separately derive the same conclusion from both ‘p’ and ‘q’.
What is a direct proof?
A direct proof establishes the truth of a statement by a straightforward application of inference rules and logical axioms from the premises to reach the conclusion.
Explain the concept of proof by contradiction.
Proof by contradiction involves assuming the negation of the statement you want to prove and showing this assumption leads to a contradiction, thus establishing the statement’s truth.
What is conditional proof?
Conditional proof is a method where you assume the antecedent of a conditional statement and show that the consequent necessarily follows.
How is “reductio ad absurdum” related to proof by contradiction?
“Reductio ad absurdum” is another term for proof by contradiction, where a statement is proven by showing that its negation would result in an absurdity or contradiction.
What does the rule of “double negation” state?
Double negation states that negating a statement twice will return to the original statement’s truth value, i.e., ‘¬(¬p)’ is equivalent to ‘p’.
How do you use the rule of “biconditional introduction” in proofs?
Biconditional introduction allows you to infer ‘p ↔ q’ if you can prove ‘p → q’ and ‘q → p’.
