What is the difference between propositional constants, propositional variables and schematic letters in propositional calculus?

Question

I am about to start the logic module in my first year undergraduate computer science course so have been reading through the propositional logic Wikipedia page.(https://en.wikipedia.org/wiki/Propositional_calculus).

Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata, however, range over all propositions. It is common to represent propositional constants by A, B, and C, propositional variables by P, Q, and R, and schematic letters are often Greek letters, most often φ, ψ, and χ.

I understand the gist of it, I am just stuck on this difference between mainly the (constants and the variables) and the schematic letters.

Additionally, if there are better resources to learn this I am more than happy to take suggestions :)

Answer 1

First, we need to define a few terms:

$ \begin{array}{lll} 1.& \text{Statement}& \text{A sentence that is either true or false.}\\ && \text{For example, "Georgia is located north of Florida."}\\ 2.& \text{Proposition}& \text{The meaning, or information content, of a sentence.}\\ && \text{For example, "Georgia is located north of Florida"} \\ && \text{and "Florida is located south of Georgia" are two}\\ && \text{distinct statements but one and the same proposition.} \\ && \text{Note not all texts distinguish between statements and}\\ && \text{propositions.}\\ 3.& \text{Complex Proposition}& \text{A proposition consisting of one or more parts that are}\\ && \text{themselves propositions. For example, "The sky is blue}\\ && \text{and the grass is green."}\\ 4.& \text{Atomic Proposition}& \text{A proposition consisting of one and only one part that}\\ && \text{is a proposition. For example, "The sky is blue." Note}\\ && \text{all atomic propositions are complex propositions, but not}\\ && \text{all complex propositions are atomic propositions.}\\ \end{array} $

In the language of propositional logic, letters such as $A,B,C,...,P,Q,R,...$ and schematic letters such as $φ, ψ, χ$ all represent propositions.

The first few letters $A,B,C,...$ of the alphabet are generally used to represent particular propositions. In other words, they are used to represent specific propositions. For instance, $A$ may be used to represent the sepcifcally identified proposition "My brother is tall," and anywhere I see $A$ I know that specific proposition is being referred to. For this reason, the letters $A,B,C$ are known as propositional constants because the meaning of the letters is specific and constant.

The letters $P,Q,R,...$ of the alphabet range over the set of all atomic propositions. This means the letter $P$ represents any arbitrary proposition that is comoposed of one and only one proposition. For instance, $P$ can mean "My brother is tall" or "My neighbor is an alien" or ... In other words, it is a place holder for any atomic proposition analogous to how $x$ and $y$ are place holders for real numbers in the expression $x+1=y$. Since these letters can represent any proposition in general, they are known as propositional variables. Note that propositional variables and\or constants can be joined by logical connectives to construct formulas that represent complex propositions. For instance, $(P \wedge Q) \to R$ means "If $P$ and $Q$, then $R$."

The schematic letters $φ, ψ, χ, ...$ range over the set of all complex propositions. This means $φ$ represents any arbitrary proposition that is comoposed of one or more propositions. For instance, $φ$ can mean "The sky is blue during the day and it becomes mostly black at night" or "The dog is inside the house or the dog is outside the house" or ... In other words, it is a place holder for any complex proposition in the same way $P$ is a place holder for any atomic proposition.

NOTE: The choice of letters, symbols, and even vocabulary may vary from one text to the next, but the underlying concepts do not. For instance, some texts will utilize uppercase letters for propositional constants while using lowercase letters for propositional variables. Or, some texts may use the utilize the letters $P,Q,R$ to represent complex propositions and abandon the use of schematic letters altogether. Whatever text you're working with, make sure you understand the corresponding definitions and notation.

Answer 2

A propositional constant is an example of a statement with a definite truth value (even if you don't know what it is), such as ‘the first decimal digit of $ 2 ^ { 9 9 9 } $ is $ 6 $’ or ‘it will be sunny tomorrow’ (assuming that you can approximate this as being definitely either true or false). Since we don't want to write that out all the time, we might write it as $ A $. In some more abstract context, we might call $ A $ a propositional constant without saying what it is, much as we might write $ a x ^ 2 + b x + c $ and say that $ a $ (but not $ x $) is a constant, to show where the focus will be.

A propositional variable is a symbol that stands for a statement that can't be analysed any further using logic, but we don't know or care which statement. So maybe its something about the decimal digits of powers of $ 2 $, or maybe it's something about the weather, or maybe it's something else; it doesn't matter. It just stands for a statement. So we'll just say ‘Let $ P $ be a propositional variable’, and leave it at that.

Using propositional variables and constants, together with logical connectives like conjunction and disjunction, we can create more propositions, such as $ A \wedge P $ or $ P \mathbin \to Q $. Unlike $ A $, we don't know or care what they mean exactly; but unlike $ P $, we can analyse them; they're not atomic. These are the analogues of algebraic expressions such as $ a x ^ 2 + b x + c $.

A propositional schematic variable stands for one of the propositions above. But now we don't know or care which one; maybe it's $ A \wedge P $, maybe it's $ P \mathbin \to Q $, maybe it's something else. So we'll just call it $ \varphi $. (Although I also see things like $ \mathcal A $ for this, and overloading $ P $ is not uncommon either.)