The study of logic is the effort to determine the conditions under which one is justified in passing from given statements, called premises, to a conclusion that is claimed to follow from them. Logical validity is a relationship between the premises and the conclusion such that if the premises are true then the conclusion is true.
The major task of logic is to establish a systematic way of deducing the logical consequences of a set of sentences, or argument. The logical concept of an argument is: a collection of two or more propositions, all but one of which are the premises supposed to provide support for the conclusion.
It is necessary first to identify or characterize the logical consequences of a set of sentences. The procedures for deriving conclusions from a set of sentences then need to be examined to verify that all logical consequences, and only those, are deducible from that set.
A deductive argument is said to be valid when the inference from premises to conclusion is perfect. Validity is a property of the argument's form. It doesn't matter what the premises and the conclusion actually say. It just matters whether the argument has the right form. So, in particular, a valid argument need not have true premises, nor need it have a true conclusion.
Any deductive argument that is not valid is invalid: it is possible for its conclusion to be false while its premises are true, so even if the premises are true, the conclusion may turn out to be either true or false.
If we don't accept the premises of an argument, we don't have to accept its conclusion, no matter how clearly the conclusion follows from the premises. Also, if the argument's conclusion doesn't follow from its premises, then we don't have to accept its conclusion in that case, either, even if the premises are obviously true.
The combination of true premises and a valid inference is a sound argument; it is a piece of reasoning whose conclusion must be true. The trouble with every other case is that it gets us nowhere, since either at least one of the premises is false, or the inference is invalid, or both. The conclusions of such arguments may be either true or false, so they are entirely useless in any effort to gain new information.
In recent times, the question has been raised whether all the truths regarding some domain of interest can be contained in a specifiable deductive system.
In a deductive argument, the truth of the premises is supposed to guarantee the truth of the conclusion; in an inductive argument, the truth of the premises merely makes it probable that the conclusion is true.
Scientific method involves the interplay of two kinds of reasoning: inductive, and deductive.
Both classical logic and modern logic are systems of deductive logic. In a sense, the premises of a valid argument contain the conclusion, and the truth of the conclusion follows from the truth of the premises with certainty. Efforts have also been made to develop systems of inductive logic, such that the premises are evidence for the conclusion, but the truth of the conclusion follows from the truth of the evidence only with a certain probability. The most notable contribution to inductive logic is that of the British philosopher John Stuart Mill, who in his System of Logic (1843) formulated the methods of proof that he believed to characterize empirical science. This inquiry has developed in the 20th century into the field known as philosophy of science. Closely related is the branch of mathematics known as probability theory.
Both classical and modern logic in their usual forms assume that any well-formed sentence is either true or false. In recent years efforts have been made to develop systems of so-called many-valued logic, such that an assertion may have some value other than true or false. In some this is merely a third neutral value; in others it is a probability value expressed as a fraction ranging between 0 and 1 or between -1 and +1. Another development in recent years has been the effort to develop systems of modal logic, to represent the logical relations between assertions of possibility and impossibility, necessity and contingency.
These areas consider the framework in which mathematics itself is carried out. To the extent that this considers the nature of proof and of mathematical reality, it borders on philosophy.
Mathematical logic lies at the heart of mathematics, but a good understanding of the rules of logic came only after their first use. Besides basic propositional logic used formally in computer science and philosophy as well as mathematics, this field covers general logic and proof theory, leading to Model theory. Here we find celebrated results such as the Gödel's incompleteness theorem and Church's thesis in recursion theory. Applications to set theory include the use of forcing to determine the independence of the Continuum hypothesis. Applications to analysis include Nonstandard analysis, an alternate perspective for calculus. Undecidability issues permeate algebra and geometry as well. This heading includes Set Theory as well: axiomatizations of sets, cardinal and ordinal arithmetic, and even Fuzzy Set theory.