An argument consists of a claim, which is called a conclusion, together with a set of propositions, called premises, which are given to support the truth of the conclusion.
The relationship between the conjunction of an argument's premises and the argument's conclusion is called the argument's inference.
Let P be the set of premises for a given argument, and let C be the argument's conclusion. If P implies C, then we say the argument is valid. In other words a valid argument is an argument whose inference is one of implication. Equivalently, if an argument has the property that it is impossible to have all true premises and a false conclusion, then the argument is said to be valid. When the inference of an argument is one of implication, then the inference is said to have the validity property and the argument is said to be valid.
An argument that is valid by virtue of the fact that it is impossible to have all true premises or because the conclusion is always true, is called trivially valid.
If it is possible for an argument to have all true premises but still have a false conclusion, then the argument is invalid, which is to say it is not valid. Equivalently, if the inference from the premises to the conclusion of an argument is not one of implication, then we call the argument invalid.
An argument is sound if it is valid and has all true premises.
If an argument is valid, but has at least one premise which is false, then the argument is valid but not sound.
When the inference between the premises of an argument and the conclusion is judged to be relevant and in addition the judgment is that of strong relevance, then the argument is known as a strong argument.
Additionally when an argument is considered to be strong, and actually has all true premises, then the argument is called cogent. On the other hand if the argument is considered strong, but at least one premise is false, then the argument is classified as being strong but not cogent.
When the premises of an argument are only relevant to the conclusion, and in addition they are judged to be weakly relevant, then the argument is called a weak argument.
An argument whose inference is invalid but which claims or is intended to be valid is said to be an erroneous proof, or an invalid deductive argument.
We considered how to combine propositions into larger propositions using the word 'and' and termed the process of combining any set of propositions 'conjunction', where we noted that the truth of a conjunction implies the truth of all of the individual propositions combined using the term 'and'.
Since our goal was to classify arguments - to aide us in that goal we introduced term inference as the relationship between the conjunction of all the premises of an argument and the conclusion and stated that our classification of arguments would be based on the argument's inference.
Finally we introduced the concepts of validity, invalidity, soundness, valid but not sound, relevance, strong and weak arguments, cogent and strong but not cogent and erroneous proof as a list of argument classifications based on an argument's inference.