In Propositional Logic, can you explain something about SL/PL Logic Problems and what the SL/PL symbols mean, for example, like in the two unsolved problems and one unsolved theorem below?
a) ~(∃x) (Px ● Mx) Pr.
b) (∃x) (Mx ● Sx) Pr. /∴ ~(x) (Sx ⊃ Px)
a) (∃x) Ax ⊃ (x) (Cx ⊃ Bx) Pr.
b) (x) (Bx ⊃ ~Ax) Pr /∴ ~(∃x) (Ax ⊃~ (x)Cx
/∴ (x) (Ax ⊃ Bx) ⊃ ((x) Ax ⊃ (x) Bx)
SL/PL stands for Statement Logic (also Sentential Logic) and Propositional Logic. All are various ways to refer to the field of study examining language logic, most often called Propositional Logic. In other words, SL and PL in "SL/PL" refer to the same field of study: Propositional Logic.
Propositional logic is an ancient field of study that has had new insurgences of interest, development and expansion in various eras throughout history. One of the latest and generally familiar is the nineteenth century work of George Boole who expanded logic study by developing a "mathematical-style 'algebra' to replace Aristotelian syllogistic logic." Boolean algebras, or Boolean mathematical logic, became the basis of "truth-functional propositional logics utilized in computer design and programming" ("Propositional Logic," Internet Encyclopedia of Philosophy).
Other well known thinkers involved in developing propositional logic during the nineteenth century are mathematician and logician Charles L. Dodgson (famous in literature as Lewis Carroll, author of Alice in Wonderland) and philosopher and logician John Venn (famous for the Venn diagram used to show logical relationships within fixed sets).
Propositional logic (PL) considers language statements as wholes as opposed to semantic units such as noun and verb. These language statements are called propositional statements.
PL examines logical operators between propositional statements. Logical operators are words or phrases, e.g., if and only if, that modify or join propositional statements to either change them or make them more complex. "In English, words such as and, or, not, if ... then..., because, and necessarily, are all [logical] operators" ("Propositional Logic").
"Truth-functional" units--and, or, if - then, if and only if, not--are a class of logical operators that determine the truth-falsity relationship between propositional statements.
Propositional statements are represented by capital letters (A, B etc), while truth-functional units are represented by signs--&, v, →, ↔, ¬ (without italics)--corresponding respectively to the above list of truth-functional units. In addition, the sign ⊃ is sometimes used instead of → for material implication ("Propositional Logic").
The objective of PL is to understand the logical operations and the logical relationship of truth or falsity between propositional statements. The following is an example of PL given in "Propositional Logic":
Consider the English compound sentence, "Paris is the most important city in France if and only if Paris is the capital of France and Paris has a population of over two million." If we use the letter 'I' in language PL to mean that Paris is the most important city in France, this sentence would be translated into PL as follows:
I ↔ (C & P)
The parentheses are used to group together the statements 'C' and 'P' and differentiate the above statement from the one that would be written as follows:
(I ↔ C) & P
This latter statement asserts that Paris is the most important city in France if and only if it is the capital of France, and (separate from this), Paris has a population of over two million. The difference between the two is subtle, but important logically.
A theorem in PL is a well-formed formula (wff) that can be derived from PL axioms and rules of inference alone without the need for any additional premise ("Propositional Logic").