1. Introducuction: This article gives a truth-valuational semantics for Classic Propositional Logic, briefly CPL.

This allows a very simple presentation of classical propositional logic in terms of basic mathematics and English.

Part of the reason for this is that there is a far-going parallel between propositional logic and ordinary algebra that was first noticed by George Boole.

Ordinary algebra is a formal language with statements about numbers and variable terms like "x", "y" for numbers, that enable one to write statements like "x >= x" i.e. "x is greater or equal than x" etc.. Constant terms in algebra are terms for numbers like "0", "1" etc. If the the term "x" is uniformly by respectively "y" and "0" in the statement "x>=x", the statements "y>=y" and "0>=0" result.

It should be noted that variables are terms that do not occur normally in ordinary language, for that contains no distinction between variables and constants.

A formal language is characterized by its containing variables, and by having an explicitly assumed precise grammar, that allows one to decide what is and what is not a well-formed expression of that formal language.

Formal logic is here characterized as a formal language with explicitly assumed rules of inference that all have the property of leading from true assumptions to only true conclusions. And

Again, in ordinary algebra one such a rule of inference is known as the symmetry of identity, and says that if it is true that x=y, then it may be concluded y=x. Another such rule in algebra is the asymmetry of smaller than: if it is true x<y, then it is true that not y<x. This last rule allows one to conclude in ordinary algebra that if 0<1, then it is not true 1<0.

Ordinary algebra also shows the reason for the introduction of variables: with them one can write statements and rules that are true of any numbers, such as "x>=x" and "if x>y then not y>x", in the end simply on the basis of the assumption that one may uniformly replace number-variables in algebraic statements by any number-constants.

So a propositional logic differs from ordinary algebra in being about propositions, where propositions are statements that are either true or false, and the "being about" is because a propositional logic is a formal language with variables for propositions, in the same sort of way algebra has variables for numbers, and hence is about numbers.

The property that characterizes propositions - that a proposition is a statement that is either true or false - makes propositions differ from other kind of statements (or sentences) like questions, exclamations, exhortations, orders, excuses etc.

One interesting feature of basic propositional logic is that it presumes the notion of truth and falsity, and does not define them: all that is assumed is that each and any proposition has the property of being true or false, and that these properties can be used to define further aspects of propositions.

The fundamental constants of ordinary algebra are +, *, -, : and =, <, allowing statements like "x+y=y+x", "x+0=x" etc. that the reader must have some familiarity with. And in algebra, spaces and brackets are also used to mark grouping, as in "(x+(y+z)) = ((x+y)+z)".

2. Propositional Logic: The fundamental constants of CPL are as in the following table, in which "iff" means "if and only if":

So we get the possibility of writing "p&q" reading it as "p is true and q is true" (presuming that "p" and "q" may be uniformly replaced by any proposition), and similarly as indicated in the above table.

It is possible to add other logical constants, such as "==>" which is named implication and "p==>q" is read as "if p then q", but one can define implication in terms of & and ~ or V and ~, as will be shown below, and in any case the notions of and terms for "and", "or" and "not" are very fundamental for any language.

Now just as with ordinary algebra, we need a syntax to write our formulas and know which expressions are, and which are not, formulas of CPL. This comes about as follows, and has been illustrated and used already:

One reason to be so explicit about this is to show what CPL is not about:

This abstracts from (in the sense: knowingly disregards) non-statements, that are not true or false, like questions and exclamations, and it also abstracts from other operators, like "it may be the case that", "if", and many others.

What remains is very simple and very general indeed, and consists of

Again, all of these compound statements are assumed to be either true or false, as in Natural Language, and as indeed will follow from the assumptions we will make.

What we now have abstacted may be considered as a rather natural core of any reasoning with statements, at least when this happens using the logical constants "and", "or" and "not" to form statements, that may be extended by further assumptions, and that may be further clarified and made amenable to a formal treatment by the following assumptions, that introduce notation and terminology:

"~" corresponds to "it is not true that". It is said to be 1-place because when it is written before a statement "X" a new statement "~X" is formed. "&" corresponds to "and" and is said to be 2-place because when it is written between two statements "X" and "Y" a new statement "X&Y" is formed, and simlarly for "V" that corresponds to "or".

One reason to introduce a new notation is to be able to write formulas that are as brief as possible, and are free of any irrelevant symbols.

Another reason is that the new terms will receive a more precise meaning by further assumptions than the terms in natural language to which they correspond.

It is possible, and for some purposes convenient, to write "&" and "V" not between but before the two statements they are meant to connect into one new proposition, but as the notation adopted is much easier to read than the alternative considered in this paragraph I merely mention it.

Next, we introduce notation for statements or propositions:

A propositional variable is a term that may be uniformly substituted in a formula by a proposition. A proposition is a statement that is either true or false. Formulas are built from propositional variables and operators according to the rules given below. A uniform substitution of a propositional variable "P" in a propositonal schema S by a proposition such as "it rains" is a replacement of ALL occurences of "P" in S by "it rains": uniform substitition means that copies of the same variable are replaced by copies of the same constants (or other variable: "re-lettering").

According to C and D there are the propositional variables X,X',X'',X''' etc. It is often convenient to use any letter of the alphabet, and write (A&(B&C)) rather than (X&(X'&X'')), and this convention will be used without mention from now on. The reason to make this a convention rather than a postulate that e.g. a,b,...z are propositional variables is to have simple postulates not cluttered by what may be made conventional.

According to 1..4 and using conventional relettering, ((~(A)&(B))V~(C)) is a formula. This can be seen as follows: (X) is a PS by (1), and therefore ~(X) by (2). (X') is a PS by (1) and so (~(X)&(X')) by (3). ~(X'') is a PS as before, and therefore ((~(X)&(X'))V~(X'') is, by (3). Relettering X by A, X' by B and X'' by C yields the desired result.

The example shows that we can delete the brackets produced by rule (1) without ambiguity, i.e. write ((~A&B)V~C) instead, and also that we can delete the outermost brackets, resulting in (~A&B)V~C. These simplifications will be made in what follows without mention. Rules for the systematic deletion of brackets without introducing ambiguity may be given, but this will not be done since we will not deal with very long formulas.

Note that (4) has a positive function: It enables us to conclude that whatever can not be formed by (1)..(3) is not a formula, which we could not do without it.

3. Formal Semantics of CPL: A semantics for a language states the meanings of its terms. A formal semantics for CPL states the conditions under which its formulas are true or false. Accordingly, it involves a functional assigments of the terms "true" and "false", or abbreviations thereof, to propositions, and one of  the things that makes a formal semantics differ from a semantics for a natural language is that in a formal semantics the non-logical terms may  be  -  and usually are - variables.

We shall use the following:

This abbreviates the more normal functional expression "value(p)" and is introduced merely for convenience.

In semantical postulates the "p" in "[p]" is ANY formula, whether simple or compound, i.e. with or without operators. Uniform substitution is presumed i.e. if one substitutes e.g. "q" for "p" in a formula and "p" occurs elsewhere in the formula it too must be substituted by "q". (If one wanted to invoke set-theory at this point, one would say that it is assumed here that there is a function from the set of propositions to the set of truth-values, and that if "p" is any proposition, "[p]" is the value the function assigns to "p".)

The basic postulate is BV:

This corresponds to "every proposition is true or not", but with the difference that using the numbers 1 and 0 instead of the predicate "is true" and its denial enables us to use arithmetic and algebra later on.

The reason it is said "every proposition is true or not" (rather than: "true or false") is to be logical and to reserve the use of "false" for possible other uses than a synonym of "not true".

The use of arithmetic can be avoided, but to do so tends to complicate arguments, and is less clear about a definite analogy between arithmetic and propositional logic that was first noted and developed by Boole. 

Besides, in the present treatment propositional logic is founded on one's knowledge of English and basic mathematics. The reader is assumed to know his or her "three Rs": reading, writing and arithmetic, and propositional logic is based on that knowledge.

The postulate of bivalence can be formulated as follows, introducing algebra, and it being presumed that [p] may be any proposition of PL, with or without operators, and of arbitrary (finite) length:

The three postulates for CPL deal with the three operators ~, & and V. The postulates are as follows, with the same presumption as before:

P~ enables us to find the value of a denial of a proposition from the value of the proposition and conversely, because - using arithmetical laws, which we assumed known at the outset - it amounts to [~p]=1-[p]. P& defines a conjunction as the product of the values of the conjuncts. PV defines a disjunction as the sum of the values of disjuncts diminished by the product of the values of the disjuncts.

The reason for the occurence of "-[p][q]" in PV can be seen when considering all possible values:

The diminuend makes no difference in the last three lines, and takes care that the sum does not exceed 1 in the first line, thus preserving BV.

Note that PV can be avoided by using instead De Morgan's definition of V:

This also shows how we can reason about the values of propositions, which propositions have by our axiom and postulates, simply using the definitions, uniform substitutions and algebra: Here in fact we reduce CPL to a small and simple part of standard algebra, that is presumed known to the reader. UP

4. Semantical Theorems of CPL:  We have explained the grammar and semantical postulates of CPL, and can proceed to give some theorems with proof.

Indeed, what we in fact have achieved at this point - barring some unclarities, the avoidance of which would require too much fuss - is in fact a relation between formulas of CPL and algebra, that allows us to reason algebraically about formulas of CPL.

So the proofs that follow are again simply arguments in English and algebra about the values of the propositions of CPL. And established theorems (in fact algebraic identities) are used later on in the same way as original assumptions, again according to common algebraic conventions.

Also, I shall supply commonly used names in many cases, but leave most comments out as CPL is very well known. The reason to list these theorems is to have a foundation for the system of Extended Propositional Logic, to be founded on top of CPL and indeed extending it.

T1 says that the double denial of a proposition has the value the proposition has.

T2 says that the conjunction of a statement and itself has the value of the statement.

T3 says that the conjunction of a proposition and its denial is always false

T4 says that the disjunction of a proposition and its denial is always true

T5 is a version of De Morgan's theorem in one variable. It is proved by PV, P~ and T3, and says when a conjunction and disjunction amount to the same.

Using these theorems about only one propositional variable we can easily restate their import in a table:

The first row apart from the first column displays propositional schemes. The first column - "Possibilities" - is unusual in this kind of tabular representations of propositional logic, since it is usually absent and replaced by a copy of the second and third columns that list the possible truth-values of p. However, explicitly stating the possibilities - that may be conceived in the present case as respectively the fact represented by the formula p and the fact represented by the formula ~p - will turn out to be very helpful later (since it preempts the need for a third truth-value in such tables, as we shall see).

In any case, the first column(s) in all the tables that follow will intuitively display all possible facts that may be represented by the propositional variables used, and truth-values accordingly are supposed to be given by a function from facts represented by simple propositions to values of propositional schemes that contain some or all of these simple propositions and no other propositions.

T6 says that any proposition can be expanded in terms of any other proposition.

T7 says that the order of conjuncts doesn't matter for the truth of a conjunction.

T8 extends T7.

T9 is used in the proof of T10, that follows:

T10 analyses a disjunction of p and q into one of precisely three conjunctions. (T7 is needed to get from (3) to (1).)

T11 gives all four possibilities involved in two alternatives.

T12 is the law of De Morgan (already known in Antiquity, incidentally) mentioned earlier.

Here are some of the theorems just proved in terms of truth tables:

Finally, here are two fundamental theorems about & and V in CPL:

T13 is the law of distributing & over V. This is fairly intuitive, but in CPL V also distributes over &:

At this point it is possible to prove that all rules of propositional logic stated in the previous section are valid, provided one  uses the following:

The proofs all follow the same format: Take the premisses of the assumed rule as a conjunction and show that given the semantical assumptions each assumed rule that starts with a true premiss must have a true conclusion. Of course, the reason valid rules of inference are so desirable is that once one knows a rule of inference is valid, one knows one can use it to infer new truths from known or assumed truths, precisely because a valid rule of inference can infer only truths from truths.