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Modus ponens is a mixed hypothetical syllogism and is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of modus ponens. The history of modus ponens goes back to antiquity. [4]
The column-11 operator (IF/THEN), shows Modus ponens rule: when p→q=T and p=T only one line of the truth table (the first) satisfies these two conditions. On this line, q is also true. Therefore, whenever p → q is true and p is true, q must also be true.
This is the modus ponens rule of propositional logic. Rules of inference are often formulated as schemata employing metavariables . [ 2 ] In the rule (schema) above, the metavariables A and B can be instantiated to any element of the universe (or sometimes, by convention, a restricted subset such as propositions ) to form an infinite set of ...
Another form of argument is known as modus tollens (commonly abbreviated MT). In this form, you start with the same first premise as with modus ponens. However, the second part of the premise is denied, leading to the conclusion that the first part of the premise should be denied as well. It is shown below in logical form. If A, then B Not B
The rule of hypothetical syllogism holds in classical logic, intuitionistic logic, most systems of relevance logic, and many other systems of logic.However, it does not hold in all logics, including, for example, non-monotonic logic, probabilistic logic and default logic.
The history of the inference rule modus tollens goes back to antiquity. [4] The first to explicitly describe the argument form modus tollens was Theophrastus. [5] Modus tollens is closely related to modus ponens. There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent.
Affirming the antecedent (modus ponens) and denying the consequent (modus tollens) are valid. Affirming the consequent and denying the antecedent are invalid. [ 7 ]
When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false. Material implication can also be characterized inferentially by modus ponens , modus tollens , conditional proof , and classical reductio ad absurdum .