Let P and Q be two statements, where P has truth table (T,T,F,F) and Q has truth table (T,F,T,F). As you might know, there are exactly 16 different truth tables of the form (w,x,y,z), all of which have been defined. For instance, the following truth tables have been given the following symbols:

• We define (T,F,F,F) to be the conjunction P∧Q
• We define (T,T,T,F) to be the disjunction P∨Q
• We define (F,T,T,F) to be the exclusive disjunction P⊻Q
• We define (T,F,F,T) to be the logical nor P⊽Q
• We define (T,F,T,T) to be the implication P⇒Q
• We define (T,T,F,T) to be the converse implication P⇐Q
• We define (T,F,F,T) to be the logical equality P≡Q (sometimes authors use a different symbol, the bicondition P⇔Q)
• etc...

Note that these compound statements are logically equivalent to infinitely many other expressions. For example,

• P⇒Q has the same truth table as Q⇐P
• P⇐Q has the same truth table as Q⇒P
• P⊽Q has the same truth table as ¬(P∨Q)
• P≡Q has the same truth table as (P⇒Q)∧(P⇐Q) and ¬(P⊻Q)

Note in each of the examples above, the compound statement is written at first in the form P(?)Q, and then in a more complicated expression(s). That is, the first expression writes the compound statement as a binary operation of P and Q. For instance:

• The implication is a binary operation mapping the statements P,Q to the statement P⇒Q
• The converse implication is a binary operation mapping the statements P,Q to the statement P⇐Q
• The logical nor is a binary operation mapping the statements P,Q to the statement P⊽Q
• The logical equality is a binary operation mapping the statements P,Q to the statement P≡Q

Now, we get to my question. What if we go back to our list of 16 possible truth tables, and consider the truth tables (T,T,F,F) or (T,F,T,F)? These should look familiar, because these are the truth tables for P and Q! (Note: These are essentially "unary operations"; for the purpose of my question we could also consider the other unary operations ¬P or ¬Q or even the two nullary operations ⊤ and ⊥.) My question is really quite simple: how to express these truth tables as a compound statement of the form P(?)Q.

It seems like the question might be already answered; Wikipedia shows calls these binary operations "logical projections". For instance, (T,T,F,F) is the projection binary operation mapping P,Q to P, and (T,F,T,F) is the other projection binary operation mapping P,Q to Q.

But the question still remains, what symbol can be used to represent these two projections as binary operations? How can we write projection #1 in the form P(?)Q, and projection #2 in the form P(¿)Q? Does this projection operation have a recognized symbol? Unfortunately the Wikipedia page for logical projection looks like it has a lot of room for expansion.