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1.1 Basic logical notions

argument, premises, conclusion Definition. An ARGUMENT is a pair of things:
· a set of sentences, the PREMISES;
· a sentence, the CONCLUSION.
Comment. All arguments have conclusions, but not all arguments have premises: the set of premises can be the empty set! Later we shall examine this idea in some detail.
Comment. If the sentences involved belong to English (or any other natural language), we need to specify that the premises and the conclusion are sentences that can be true or false. That is, the premises and the conclusion must all be declarative (or indicative) sentences, such as `The cat is on the mat' or `I am here', and not sentences such as `Is the cat on the mat?' (interrogative) or `Close the door' (imperative). We are going to construct some formal languages in which every sentence is either true or false. Thus this qualification is not present in the definition above.
validity Definition. An argument is VALID if and only if it is necessary that if all its premises are true, its conclusion is true.
Comment. The intuitive idea captured by this definition is this: If it is possible for the conclusion of an argument to be false when its premises are all true, then the argument is not reliable (that is, it is invalid). If true premises guarantee a true conclusion, then the argument is valid.
Alternate formulation of the definition. An argument is VALID if and only if it is impossible for all the premises to be true while the conclusion is false.
entailment Definition. When an argument is valid we say that its premises ENTAIL its conclusion.
soundness Definition. An argument is SOUND if and only if it is valid and all its premises are true.
Comment. It follows that all sound arguments have true conclusions.
Comment. An argument may be unsound in either of two ways: it is invalid, or it has one or more false premises.
Comment. The rest of this book is concerned with validity rather than soundness.