Propositional logic (propositional calculus, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic) is part of the mathematical logic. Its purpose is the study of the logical relations between “propositions” and defines the formal laws according to which the complex propositions are formed by assembling simple propositions by means of the logical connectors and these are chained to produce valid reasonings. He is one of the formal systems, pillars of the mathematical logic of which he helps in the formulation of concepts. It is considered the modern form of Stoic logic.
The notion of proposition has been the subject of much debate during the history of logic; the consensual idea is that a proposition is a syntactic construct supposed to speak of truth. In mathematical logic, the calculation of propositions is the first step in the definition of logic and reasoning. It defines the deduction rules that link the propositions to each other without examining the content; it is thus a first step in the construction of calculus of predicates, which is interested in the content of propositions and which is a formalization of mathematical reasoning. The calculation of propositions is sometimes called the logic of propositions, propositional logic or calculation of statements, and sometimes the theory of truth functions.
Definition of a proposition
Although the calculation of the propositions does not concern itself with the content of the propositions, but only with their relations, it may be interesting to discuss what this content might be. A proposition gives information on a state of affairs. Thus “2 + 2 = 4” or “the book is open” are two propositions. In classical logic (bivalent logic), a proposition can take only true or false values. An optative phrase (which expresses a wish like “God protect us!”), an imperative sentence (“Come!”, “Shut up!”) or a query, is not a proposition. “May God protect us!“ cannot be true or false: it expresses only a wish of the speaker. On the other hand, a phrase like “In this calculation, all the computer variables are strictly positive” is a proposition whose content has been modified by the quantizer all and which is supposed to prove in the duration. This type of proposition is studied in the modal logic, more precisely in the temporal logic in this case, because of the assertion of its durability.
Proposition and predicate
If a proposition is an assertion with a value of truth, a predicate is an expression whose truth value depends on variables it contains. The predicate “My country is in Europe” will be true, false or indeterminate depending on the value of the variable “My country”. If the reader is French, we will get the proposition “France is in Europe”, which is true; if the reader is Canadian, we will get the proposition “Canada is in Europe”, which is false; if the reader is Russian, we will get the proposition “Russia is in Europe” which is indeterminate because, as we know Russia is astride Europe and Asia.
Definition of a deductive system
A calculus or a deductive system is, in logic, a set of rules allowing in a finite number of steps and according to explicit rules to determine if a proposition is true. Such a process is called a demonstration. We also associate with the propositions a mathematical structure which makes it possible to guarantee that these reasonings or demonstrations have sense, one says that one gave him a semantics. In the calculus of classical proposition, this semantics uses only two values, true and false (often denoted 1 and 0). A fully determined proposition (ie whose values of elementary constituents are determined) takes only one of these two values.
In theories of mathematical logic, we therefore consider two so-called syntactic and semantic points of view, as is the case in the calculus of propositions.
- Syntactic aspect: it is a question of defining the language of the calculus of the propositions by the rules of writing of the propositions.
- Semantic aspect: here it is a matter of giving meaning to the symbols representing the logical connectors according to the truth value of the basic propositions (thus ¬ means no). This meaning is given, for example, by truth tables or by Kripke models. An aspect halfway between syntax and semantics describes the rules of inference allowing the deduction of propositions from others. These rules of deduction generate specific propositions called theorems.
If the deduction corresponds perfectly to the semantics the system is said to be complete.