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The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a necessary consequence of its premises. An argument that is not valid is said to be "invalid". An example of a valid (and sound) argument is given by the following well-known syllogism:
This resolution technique uses proof by contradiction and is based on the fact that any sentence in propositional logic can be transformed into an equivalent sentence in conjunctive normal form. [4] The steps are as follows. All sentences in the knowledge base and the negation of the sentence to be proved (the conjecture) are conjunctively ...
Being a valid argument does not necessarily mean the conclusion will be true. It is valid because if the premises are true, then the conclusion has to be true. This can be proven for any valid argument form using a truth table which shows that there is no situation in which there are all true premises and a false conclusion. [2]
A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values : as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
A logical argument, seen as an ordered set of sentences, has a logical form that derives from the form of its constituent sentences; the logical form of an argument is sometimes called argument form. [6] Some authors only define logical form with respect to whole arguments, as the schemata or inferential structure of the argument. [7]
In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, [2] and reductio ad impossibile. [3] A mathematical proof employing proof by contradiction usually proceeds as follows: The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies ...
Traditionally, a proof is a platform which convinces someone beyond reasonable doubt that a statement is mathematically true. Naturally, one would assume that the best way to prove the truth of something like this (B) would be to draw up a comparison with something old (A) that has already been proven as true. Thus was created the concept of ...
In propositional logic, modus ponens (/ ˈ m oʊ d ə s ˈ p oʊ n ɛ n z /; MP), also known as modus ponendo ponens (from Latin 'mode that by affirming affirms'), [1] implication elimination, or affirming the antecedent, [2] is a deductive argument form and rule of inference. [3] It can be summarized as "P implies Q. P is true. Therefore, Q ...