Proposition

What is a proposition? A proposition is an expression of the meaning of a sentence, especially in the case where a sentence is considered either TRUE or FALSE. It is can be used as a synonym for a sentence or statement or an independent clause.

What is a sentence?. A "simple sentence" - a sentence that is not composed of any sentences smaller than itself. A "compound sentence" - a sentence that is composed of two or more simple sentences.

"sentential connectives" - operators that join sentences. "sentential logic" - propositional logic (possibly)

"Propositional Languages" are often constrasted with "FOL" as a means of separating of the languages.

Basic properties of Opaque Propositional Languages

Atoms are opaque.

Basic properties of Transparent Propositional Languages

Atoms are transparent.

Transparent Propositional Languages

Logic	Language		Vocabulary	Notes
States of the System	-1 - Views	-		We are interested in systems and the states that they have.  The states that they have are described by systems of languages.  In the very simple case of a light, the light is on or it  is off, and we describe those two states in terms of our description of the state of the light.  The light is a noun, and it's state is described with a verb.
Opaque Atoms	0 - Atoms	0		Opaque Transparent Language
Domain	0 - Atoms	0		Opaque Transparent Language
Language	0 - Atoms	0		Opaque Transparent Language.  Opaque Propositional Languages are a special case of transparent propositional languages, one where there are no constants (cons={}) and the predicates have an arity of zero {P,0}.
Explicit Representation of the sameness of concepts across sentences and propositions.	1 - Constants Predicates	*	Predicates and Constants	Our verbs become predicates and our nouns are constants.
Domain	1 - Constants Predicates	*
Predicates	1 - Constants Predicates	*	Predicates	Arity
Constants	1 - Constants Predicates	*
Sentences	1 - Constants Predicates	*
Variables	2 - Constants Predicates Variables	* *		To build a language with variables is not hard.  How many variables?  Undiagrammable systems may be either finite or infinite, but in either case there is usually no neat upper limit on the number of components.  Inorder to ensure that there are enough variables, it is customary to take the countable infinite set Var = {x1, x2, ...}
Term	2 - Constants Predicates Variables	* *	Term = Constants and Variables Term = Cons U Var	Suppose Cons = {a,b} then Term = {a,b,x1,x2,...}
Well-Formed Formula (wff)	2 - Constants Predicates Variables	* *		The difference between a sentence and a wff is that a wff divides the set of interpretations into three disjoint  subsets: 1. The models of the wff 2. The models of the negation of the wff 3. those that are "undecided"
Pointed Interpretation	2 - Constants Predicates Variables	* *		Introduced when the
Valuation determined by the pointed interpretation	2 - Constants Predicates Variables	* *
Pointed Model	2 - Constants Predicates Variables	* *
Function Symbols	3 - Constants Predicates Variables Function Symbols	* * *
Sorts	4 - Constants Predicates Variables Function Symbols Sorts	* * * *
Opaque				Opaque Propositional Languages are a special case of transparent propositional languages, one where there are no constants (cons={}) and the predicates have an arity of zero {P,0}.
Transparent			Domain: this is a map of our system, not the system itself.	Step 1 - Domain --- Step 2 - Constants --- Step 3 - Predicates