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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: All men are mortal. (True) Socrates is a man. (True) Therefore, Socrates is mortal. (True) What makes this a valid argument is not that it has true premises and a true conclusion.
An argument can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound. For example, John might be going to work on Wednesday. In this case, the reasoning for John's going to work (because it is Wednesday) is unsound. The argument is only sound on ...
The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where P and Q are called the statement's disjuncts. The rule makes it possible to eliminate a disjunction from a logical proof. It is the rule that
The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument. 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.
In this example, the first premise is a conditional statement in which "P" is the antecedent and "Q" is the consequent. The second premise "affirms" the antecedent. The conclusion, that the consequent must be true, is deductively valid. A mixed hypothetical syllogism has four possible forms, two of which are valid, while the other two are invalid.
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]
Forms of logical reasoning can be distinguished based on how the premises support the conclusion. Deductive arguments offer the strongest possible support. Non-deductive arguments are weaker but are nonetheless correct forms of reasoning. [28] [29] The term "proof" is often used for deductive arguments or very strong non-deductive arguments. [30]
Most valid arguments are not yet known to be valid. To determine validity in non-obvious cases deductive reasoning is required. There is no deductive reasoning in an argument per se; such must come from the outside. Every argument's conclusion is a premise of other arguments. The word constituent may be used for either a premise or conclusion.