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Mathematical logic, also called 'logistic', 'symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. [5]
A variety of basic concepts is used in the study and analysis of logical reasoning. Logical reasoning happens by inferring a conclusion from a set of premises. [3] Premises and conclusions are normally seen as propositions. A proposition is a statement that makes a claim about what is the case.
First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables.
Modern philosophers reject quantum logic as a basis for reasoning, because it lacks a material conditional; a common alternative is the system of linear logic, of which quantum logic is a fragment. Mathematically, quantum logic is formulated by weakening the distributive law for a Boolean algebra, resulting in an orthocomplemented lattice.
Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups).
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
One experiment revolving around the Wason four card problem found many influences on people's selection in this task experiment that were not based on logic. The non-logical inferences made by the participants from this experiment demonstrate the possibility and structure of extra logical reasoning mechanisms. [5]
Frequently, the proposed solutions led to further questions that were often simultaneously of philosophical and mathematical nature. All these questions led, at the end of the 19th century and the beginning of the 20th century, to debates which have been called the foundational crisis of mathematics. The following subsections describe the main ...