 As we've been talking about, logic is a form of reasoning conducted according to a strict systematic set of principles. It is generally understood to be the systematic study of the form of arguments, the science that investigates the principles governing correct or reliable inferences. Logic arose from a concern with the correctness of argumentation and a desire to find a solid basis for the verification of an argument. The great logician Gottlob Brege described the function of logic as such. To discover truth is the task of all sciences. It falls to logic to discern the laws of truth. I assigned to logic the task of discovering the laws of truth, not of assertion or thought. How logic should be properly defined is a somewhat controversial matter. For some, it is interpreted in a narrow sense as simply dealing with deductive reasoning. Such a narrow conception controversially excludes much of what is called informal logic and processes of inductive reasoning. It is important to distinguish deductive validity and inductive validity. As we previously talked about, an inference is deductively valid if and only if there is no possible situation in which all the premises are true but the conclusion is false. An inference is inductively strong if and only if its premises give some degree of probability to its conclusion. Informal logic is a study of natural language arguments. Informal logic is associated with fallacies, critical thinking, general thinking skills, and the interdisciplinary inquiry known as argumentation theory. Informal logic may include both deductive and inductive reasoning. Formal logic involves deductive reasoning, typically using a formal language or formal system. Logic is generally considered formal when it is translated from natural language into a formal language based upon a well-defined set of rules. For example, the expression, all x's are y's, shows the underlining logical form common to the sentences, all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. Throughout history it has been recognised that in all reasoning there are certain common patterns, three of which have come to be known as the laws of thought. They are the law of identity, whatever is, is, or all a's are a's. The law of contradiction, a thing cannot both be and not be. And the law of excluded middle, a thing must either be or not be. Deductive thinking is largely reducible to a form such as all men are mortal, socrates is a man, therefore socrates is mortal. Or more exactly, if all men are mortal, and if socrates is a man, socrates must be mortal. Such a form is known as a syllogism. The investigation of deduction and the elaboration of the syllogism are the work of Aristotle. Aristotle defines the syllogism as, quote, a discourse in which certain, specific things, having been supposed, something different from the things supposed, results of necessity, because these things are so. The syllogism is the basis of deductive reasoning and Aristotelian logic, forming the backbone to logic for many centuries to come, building the rationale behind Euclidean geometry and scholastic philosophy. With the rise of the modern era, the inextricable development of science brought new systems of thought and reasoning. It was the English philosopher Francis Bacon that challenged the orthodox of Aristotelian thinking that placed priority on deductive reasoning, seeing inductive reasoning as a temporary solution. Modern science was built on the logic of induction as a method for understanding the world. The old logic of the syllogism was sound for clarifying what we already knew, but it was not useful for discovering new knowledge. Francis Bacon proposed the method of inductive reasoning as the foundations for modern science, whereby empirical data would be collected and analyzed to infer general conclusions. Francis Bacon described this empirical inductive process of acquiring knowledge as such, quote, Now my method, though hard to practice, is easy to explain, and it is this, I propose to establish progressive stages of certainty. The evidence of the sense helps and guarded by a certain process of correction I retain, but the mental operation which follows the act of sense I for the most part reject, and instead of it I open and lay out a new and certain path for the mind to proceed in, starting directly from the simple sensuous perception. In the 19th century arose a new logical field called logistic or symbolic logic. Its characteristic form is the application of mathematical symbols to logic and its substance is the analysis of relations. The fundamental inadequacy of Aristotelian logic, according to the logicians, arises in the use of language rather than symbols. In 1854, George Boo created the first algebra of logic to express natural language arguments in algebraic form. An example of how language can fail the logician is the alleged ambiguity of the copula. For example, in the statements a is b and all a are b, the is and are seem to express different relations. The application of mathematical symbols to logic not only removes any such possible ambiguity, but also greatly simplifies logical processes and admits the extension of their application far beyond the province of the Aristotelian logic. This classical form of logic discussed above is a form of bivalent or two-value logic that is that it most naturally divides propositions into true or false. Non-classical logics are those systems that reject various rules of classical logic, most notably the creation of this dichotomy. Classical logic only permits conclusions which are either true or false. However, there are also propositions with variable answers, such as one might find when asking a group of people to identify a colour. In such instances the truth appears as the result of reasoning from inexact or partial knowledge in which the sampled answers are mapped onto a spectrum of possibilities.