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Quantifiable logic or in the cotext of this app First-Order Logic, often abbreviated as FOL, is a branch of logic that extends Propositional Logic. It deals with propositions (statements) that are either true or false, similar to Propositional Logic. However, it goes further by introducing variables, predicates, quantifiers, and functions, allowing for more complex and expressive statements. In this app, you can generate natural deduction proofs for FOL containing universal and existential quantifiers.
For more detailed information, see Wikipedia
See Quantificational Logic Calculator here
All uppercase alphabets are allowed as predicates. Lowercase letters are treated as variables Use of empty premises is not allowed.
Modus Ponens is a valid rule of inference that states if you have a conditional statement (p → q) and the antecedent (p) is true, then you can infer that the consequent (q) is also true.
Modus Tollens is a valid rule of inference that states if you have a conditional statement (p → q) and the consequent (q) is false, then you can infer that the antecedent (p) must also be false.
Material Implication is a logical equivalence rule that simplifies conditional statements. It states that (p → q) is equivalent to (¬p ∨ q), where ¬ represents negation and ∨ represents disjunction.
De Morgan's Laws are a set of rules that describe how to negate conjunctions and disjunctions. They state that ¬(p ∧ q) is equivalent to (¬p ∨ ¬q) and ¬(p ∨ q) is equivalent to (¬p ∧ ¬q).
Simplification is a rule that allows you to simplify a conjunction (p ∧ q) to its individual components (p and q).
Hypothetical Syllogism is a valid rule of inference that states if you have two conditional statements (p → q) and (q → r), you can infer the conditional statement (p → r).
Biconditional Elimination is a rule that allows you to extract the implications from a biconditional statement (p ↔ q). It results in two conditional statements: (p → q) and (q → p).
The Biconditional Introduction is a fundamental rule in propositional logic. It allows you to establish a biconditional statement (p ↔ q) when you have both implications: (p → q) and (q → p). In other words, if you have 'p implies q' and 'q implies p,' you can infer the biconditional 'p if and only if q.' This rule is used to express that two propositions are logically equivalent and can be interchanged in logical reasoning.
Disjunctive Syllogism is a valid rule of inference that states if you have a disjunction (p ∨ q) and the negation of one disjunct (¬p or ¬q), you can infer the other disjunct.
The Addition Rule is a logical inference rule that allows you to introduce a disjunction (OR) by asserting one of its disjuncts. In other words, if you have a proposition p, you can derive the statement (p ∨ q), where q represents any arbitrary proposition. This rule is useful when you want to expand your set of premises or conclusions by asserting a new possibility.
Double Negation is a rule that states that double negating a proposition (¬¬p) is equivalent to the original proposition (p).
The Conjunction Rule, also known as Simplification, is a fundamental rule of propositional logic. It states that if we have two individual propositions 'A' and 'B,' then we can infer the conjunction 'A and B' (A ∧ B). This rule allows us to simplify complex propositions by breaking them down into their constituent parts for analysis and inference.
The Transposition Rule, also known as the Contrapositive, is a fundamental rule in propositional logic. It allows us to transform an implication (p → q) into its contrapositive form (¬q → ¬p). In other words, if we have an implication 'p implies q,' we can infer its contrapositive, which states that 'not q implies not p.'
Negation, often denoted as ¬, is a fundamental logical operation that negates a proposition. If a proposition p is true, then ¬p is false, and if p is false, then ¬p is true.
Existential Generalization, often denoted as EG, is a valid rule of inference in predicate logic. It allows you to infer the existence of an element that satisfies a predicate.
Universal Generalization, often denoted as UG, is a valid rule of inference used in predicate logic. It allows you to generalize a statement from a specific instance to all elements of a set.
Existential Instantiation, often denoted as EI, is a valid rule of inference in predicate logic. It allows you to instantiate an existential quantifier (∃) with a specific element that satisfies the existential statement.
Universal Instantiation, often denoted as UI, is a valid rule of inference in predicate logic. It allows you to instantiate a universal quantifier (∀) with a specific element from a set.