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In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid. A truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for ...
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:
It is an application of the general truth that if a statement is true, then so is its contrapositive. 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]
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]
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]
In particular φ is valid if and only if ¬φ is unsatisfiable, which is to say it is false that ¬φ is satisfiable. Put another way, φ is satisfiable if and only if ¬φ is invalid. For logics without negation, such as the positive propositional calculus, the questions of validity and satisfiability may be unrelated.
A truth table reveals the rows where inconsistencies occur between p = q delayed at the input and q at the output. After "breaking" the feed-back, [27] the truth table construction proceeds in the conventional manner. But afterwards, in every row the output q is compared to the now-independent input p and any inconsistencies between p and q are ...
A formula is logically valid (or simply valid) if it is true in every interpretation. [22] These formulas play a role similar to tautologies in propositional logic. A formula φ is a logical consequence of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.