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Propositional calculus

In mathematical logic, a propositional calculus or logic (also called sentential calculus or sentential logic) is a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of inference rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions. Such a constructed sequence of formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem.

Usually in Truth-functional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false.Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic.
Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus and expanded by the Stoics. The logic was focused on propositions. This advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood. Consequently, the system was essentially reinvented by Peter Abelard.
Propositional logic was eventually refined using symbolic logic. Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were reachieved by logicians like George Boole and Augustus De Morgan completely independent of Leibniz.
Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic was an advancement from the earlier propositional logic. Predicate logic has been described to be combining "the distinctive features of syllogistic logic and propositional logic." Consequently, it ushered a new era in the history of logic. However, advances in propositional logic were still made after Frege. These include Natural Deduction, Truth-Trees and Truth-Tables. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz. Truth-Trees were invented by Evert Willem Beth. The invention of truth-tables, however, is of controversial attribution.
The ideas preceding truth tables have been found in both Frege and Bertrand Russell whereas the actual 'tabular structure' (i.e. being formed in a table format) is generally credited to either Ludwig Wittgenstein, Emil Post or both (independently of one another). Besides Frege and Russell, others credited for having preceding ideas of truth-tables include Philo, Boole, Charles Sanders Peirce, and Ernst Schröder. And besides Post and Wittgenstein, others credited with the tabular structure include Łukasiewicz, Schröder, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis. Ultimately, some, like John Shosky, have concluded "It is far from clear that any one person should be given the title of 'inventor' of truth-tables.".

In general terms, a calculus is a formal system that consists of a set of syntactic expressions, a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted to be logical equivalence, on the space of expressions.
When the formal system is intended to be a logical system, the expressions are meant to be interpreted to be statements, and the rules, known to be inference rules, are typically intended to be truth-preserving. In this setting, the rules (which may include axioms) can then be used to derive ("infer") formulæ representing true statements from given formulæ representing true statements.
The set of axioms may be empty, a nonempty finite set, a countably infinite set, or be given by axiom schemata. A formal grammar recursively defines the expressions and well-formed formulæ (wffs) of the language. In addition a semantics may be given which defines truth and valuations (or interpretations).
The language of a propositional calculus consists of a set of primitive symbols, variously referred to be atomic formulae, placeholders, proposition letters, or variables, and a set of operator symbols, variously interpreted to be logical operators or logical connectives.
A well-formed formula (wff) is any atomic formula, or any formula that can be built up from atomic formulæ by means of operator symbols according to the rules of the grammar.
Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata, however, range over all propositions. It is common to represent propositional constants by A, B, and C, propositional variables by P, Q, and R, and schematic letters are often Greek letters.

Other logical calculi

Propositional calculus is about the simplest kind of logical calculus in current use. It can be extended in several ways. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler – but in other ways more complex – than propositional calculus.) The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used.
First-order logic (aka first-order predicate logic) results when the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, all keeping the rules of propositional logic with some new ones introduced. (For example, from "All dogs are mammals" we may infer "If Rover is a dog then Rover is a mammal".) With the tools of first-order logic it is possible to formulate a number of theories, either with explicit axioms or by rules of inference, that can themselves be treated as logical calculi. Arithmetic is the best known of these; others include set theory and mereology. Second-order logic and other higher-order logics are formal extensions of first-order logic. Thus, it makes sense to refer to propositional logic as "zeroth-order logic", when comparing it with these logics.
Modal logic also offers a variety of inferences that cannot be captured in propositional calculus. For example, from "Necessarily p" we may infer that p. From p we may infer "It is possible that p". The translation between modal logics and algebraic logics concerns classical and intuitionistic logics but with the introduction of a unary operator on Boolean or Heyting algebras, different from the Boolean operations, interpreting the possibility modality, and in the case of Heyting algebra a second operator interpreting necessity (for Boolean algebra this is redundant since necessity is the De Morgan dual of possibility). The first operator preserves 0 and disjunction while the second preserves 1 and conjunction.
Many-valued logics are those allowing sentences to have values other than true and false. (For example, neither and both are standard "extra values"; "continuum logic" allows each sentence to have any of an infinite number of "degrees of truth" between true and false.) These logics often require calculational devices quite distinct from propositional calculus. When the values form a Boolean algebra (which may have more than two or even infinitely many values), many-valued logic reduces to classical logic; many-valued logics are therefore only of independent interest when the values form an algebra that is not Boolean.

http://en.wikipedia.org/wiki/Propositional_calculus

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Seventh Level of Hell
Guarded by the Minotaur, who snarls in fury, and encircled within the river Phlegethon, filled with boiling blood, is the Seventh Level of Hell. The violent, the assassins, the tyrants, and the war-mongers lament their pitiless mischiefs in the river, while centaurs armed with bows and arrows shoot those who try to escape their punishment. The stench here is overpowering. This level is also home to the wood of the suicides- stunted and gnarled trees with twisting branches and poisoned fruit. At the time of final judgement, their bodies will hang from their branches. In those branches the Harpies, foul birdlike creatures with human faces, make their nests. Beyond the wood is scorching sand where those who committed violence against God and nature are showered with flakes of fire that rain down against their naked bodies. Blasphemers and sodomites writhe in pain, their tongues more loosed to lamentation, and out of their eyes gushes forth their woe. Usurers, who followed neither nature nor art, also share company in the Seventh Level.


For the violent ones 

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