# Propositional Logic System

A Propositional Logic System is a deductive logic system composed of a propositional logic language (likely generated by a propositional logic grammar), and a propositional logic operation set.

**AKA:**Sentential Calculus/Theory.**Context:**- It can formulate a small set of a real-world problem.
- It can require many variables in order to express real-world problems.

**Example(s):**- …

**Counter-Example(s):****See:**Boolean Logic System, Predicate Logic System, P Language, Propositionalization; Zeroth-Order Logic, Mathematical Logic, Logical Connective, Logical Conjunction, Logical Disjunction, Negation, Material Conditional, Modus Ponens, Inference Rule, Formal System, Well-Formed Formula, Formal Language, Zeroth-Order Logic, Logical Connective.

## References

### 2017

- (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/propositional_calculus Retrieved:2017-6-15.
**Propositional calculus**(also called**propositional logic**,**sentential calculus**,**sentential logic**, or sometimes**zeroth-order logic**) is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components.

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Propositional_calculus Retrieved:2015-2-25.
- … The following is an example of a very simple inference within the scope of propositional logic:
:Premise 1: If it's raining then it's cloudy.

:Premise 2: It's raining.

:Conclusion: It's cloudy.

Both premises and the conclusions are propositions. The premises are taken for granted and then with the application of modus ponens (an inference rule) the conclusion follows.

As propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed anymore by logical connectives, this inference can be restated replacing those

*atomic*statements with statement letters, which are interpreted as variables representing statements::Premise 1: [math] P \to Q [/math] :Premise 2: [math] P [/math] :Conclusion: [math] Q [/math] The same can be stated succinctly in the following way: : [math] P \to Q, P \vdash Q [/math] When is interpreted as “It's raining” and as “it's cloudy” the above symbolic expressions can be seen to exactly correspond with the original expression in natural language. Not only that, but they will also correspond with any other inference of this

*form*, which will be valid on the same basis that this inference is.Propositional logic may be studied through 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. A constructed sequence of such 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.When a formal system is used to represent formal logic, only statement letters are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself.

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.

- … The following is an example of a very simple inference within the scope of propositional logic:

- (Russell, 2015) ⇒ Stuart Russell. (2015). “Unifying Logic and Probability.” In: Communications of the ACM Journal, 58(7). doi:10.1145/2699411
- QUOTE: For example, the rules of chess occupy 100 pages in first-order logic, 105 pages in propositional logic, and 1038 pages in the language of finite automata. The power comes from separating predicates from their arguments and quantifying over those arguments: so one can write rules about On (p, c, x, y, t) (piece p of color c is on square x, y at move t) without filling in each specific value for c, p, x, y, and t.

### 2013

- (Genesereth & Kao, 2013) ⇒ Michael Genesereth, and Eric Kao. (2013). “Introduction to Logic." Morgan & Claypool Publishers. doi:10.2200/S00432ED1V01Y201207CSL005

### 2011

- (Sammut & Webb, 2011) ⇒ Claude Sammut, and Geoffrey I. Webb. (2011). “Propositional Logic.” In: (Sammut & Webb, 2011) p.812

### 2009

- WordNet. http://wordnetweb.princeton.edu/perl/webwn?s=propositional%20logic
- S: (n) propositional logic, propositional calculus (a branch of symbolic logic dealing with propositions as units and with their combinations and the connectives that relate them)