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Let P and Q be two statements, each having two possible truth values: true (T) or false (F). Then there are exactly 16 unique compound statements involving P and Q with corresponding truth tables of the form (w,x,y,z). We have given symbols to some of these truth tables; for example, when P has truth table (T,T,F,F) and Q has truth table (T,F,T,F), then

• We write the statement corresponding to (T,F,T,T) as P⇒Q
• We write the statement corresponding to (T,F,F,T) as P⊽Q
• We write the statement corresponding to (T,F,F,T) as P⇔Q

Note that these compound statements are logically equivalent to other expressions:

• The statement P⇒Q has the same truth table as ¬P∨Q
• The statement P⊽Q has the same truth table as ¬(P∨Q)
• The statement P⇔Q has the same truth table as (P⇒Q)∧(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. That is, the first expression writes the compound statement as a binary operation of P and Q. For instance:

• The implication maps (P,Q) to P⇒Q
• The logical nor maps (P,Q) to P⊽Q
• The biconditional maps (P,Q) to P⇔Q

Now the binary operation which maps (P,Q) to P is called the logical projection onto P, and the binary operation which maps (P,Q) to Q is called the logical projection onto Q. My question is: What symbol can be used to represent these two logical projections as binary operations? That is, how can we write these two logical projections in the form P(?)Q and P(¿)Q? Do these two projection operations have recognized symbols? Unfortunately the Wikipedia page for logical projection looks like it has a lot of room for expansion.

(Note: P and Q are essentially "unary operations"; for the purpose of my question we could have considered the other unary operations ¬P or ¬Q or even the two nullary operations ⊤ and ⊥.)

Edit:

After some time, I seem to have found a possible answer: Donald Knuth, in his section on Boolean Basics in The Art of Computer Programming Volume 4A, Pre-Fascicle 0B, writes the logical projections as P L Q and P R Q. That is, we could express projection onto P as the binary operationL(P,Q) and express projection onto Q as the binary operation R(P,Q). See this StackExchange question for how to typeset these symbols.

Let P and Q be two statements, each having two possible truth values: true (T) or false (F). Then there are exactly 16 unique compound statements involving P and Q with corresponding truth tables of the form (w,x,y,z). We have given symbols to some of these truth tables; for example, when P has truth table (T,T,F,F) and Q has truth table (T,F,T,F), then

• We write the statement corresponding to (T,F,T,T) as P⇒Q
• We write the statement corresponding to (T,F,F,T) as P⊽Q
• We write the statement corresponding to (T,F,F,T) as P⇔Q

Note that these compound statements are logically equivalent to other expressions:

• The statement P⇒Q has the same truth table as ¬P∨Q
• The statement P⊽Q has the same truth table as ¬(P∨Q)
• The statement P⇔Q has the same truth table as (P⇒Q)∧(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. That is, the first expression writes the compound statement as a binary operation of P and Q. For instance:

• The implication maps (P,Q) to P⇒Q
• The logical nor maps (P,Q) to P⊽Q
• The biconditional maps (P,Q) to P⇔Q

Now the binary operation which maps (P,Q) to P is called the logical projection onto P, and the binary operation which maps (P,Q) to Q is called the logical projection onto Q. My question is: What symbol can be used to represent these two logical projections as binary operations? That is, how can we write these two logical projections in the form P(?)Q and P(¿)Q? Do these two projection operations have recognized symbols? Unfortunately the Wikipedia page for logical projection looks like it has a lot of room for expansion.

(Note: P and Q are essentially "unary operations"; for the purpose of my question we could have considered the other unary operations ¬P or ¬Q or even the two nullary operations ⊤ and ⊥.)

Edit:

After some time, I seem to have found a possible answer: Donald Knuth, in his section on Boolean Basics in The Art of Computer Programming Volume 4A, Pre-Fascicle 0B, writes the logical projections as P L Q and P R Q. That is, we could express projection onto P as the binary operationL(P,Q) and express projection onto Q as the binary operation R(P,Q). See this StackExchange question for how to typeset these symbols.

Let P and Q be two statements, each having two possible truth values: true (T) or false (F). Then there are exactly 16 unique compound statements involving P and Q with corresponding truth tables of the form (w,x,y,z). We have given symbols to some of these truth tables; for example

• We write the statement corresponding to (T,F,T,T) as P⇒Q
• We write the statement corresponding to (T,F,F,T) as P⊽Q
• We write the statement corresponding to (T,F,F,T) as P⇔Q

Note that these compound statements are logically equivalent to other expressions:

• The statement P⇒Q has the same truth table as ¬P∨Q
• The statement P⊽Q has the same truth table as ¬(P∨Q)
• The statement P⇔Q has the same truth table as (P⇒Q)∧(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. That is, the first expression writes the compound statement as a binary operation of P and Q. For instance:

• The implication maps (P,Q) to P⇒Q
• The logical nor maps (P,Q) to P⊽Q
• The biconditional maps (P,Q) to P⇔Q

Now the binary operation which maps (P,Q) to P is called the logical projection onto P, and the binary operation which maps (P,Q) to Q is called the logical projection onto Q. My question is: What symbol can be used to represent these two logical projections as binary operations? That is, how can we write these two logical projections in the form P(?)Q and P(¿)Q? Do these two projection operations have recognized symbols? Unfortunately the Wikipedia page for logical projection looks like it has a lot of room for expansion.

(Note: P and Q are essentially "unary operations"; for the purpose of my question we could have considered the other unary operations ¬P or ¬Q or even the two nullary operations ⊤ and ⊥.)

Edit:

After some time, I seem to have found a possible answer: Donald Knuth, in his section on Boolean Basics in The Art of Computer Programming Volume 4A, Pre-Fascicle 0B, writes the logical projections as P L Q and P R Q. That is, we could express projection onto P as the binary operationL(P,Q) and express projection onto Q as the binary operation R(P,Q). See this StackExchange question for how to typeset these symbols.

Let P and Q be two statements, each having two possible truth values: true (T) or false (F). Then there are exactly 16 unique compound statements involving P and Q with corresponding truth tables of the form (w,x,y,z). We have given symbols to some of these truth tables; for example, when P has truth table (T,T,F,F) and Q has truth table (T,F,T,F), then

• We write the statement corresponding to (T,F,T,T) as P⇒Q
• We write the statement corresponding to (T,F,F,T) as P⊽Q
• We write the statement corresponding to (T,F,F,T) as P⇔Q

Note that these compound statements are logically equivalent to other expressions:

• The statement P⇒Q has the same truth table as ¬P∨Q
• The statement P⊽Q has the same truth table as ¬(P∨Q)
• The statement P⇔Q has the same truth table as (P⇒Q)∧(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. That is, the first expression writes the compound statement as a binary operation of P and Q. For instance:

• The implication maps (P,Q) to P⇒Q
• The logical nor maps (P,Q) to P⊽Q
• The biconditional maps (P,Q) to P⇔Q

Now the binary operation which maps (P,Q) to P is called the logical projection onto P, and the binary operation which maps (P,Q) to Q is called the logical projection onto Q. My question is: What symbol can be used to represent these two logical projections as binary operations? That is, how can we write these two logical projections in the form P(?)Q and P(¿)Q? Do these two projection operations have recognized symbols? Unfortunately the Wikipedia page for logical projection looks like it has a lot of room for expansion.

(Note: P and Q are essentially "unary operations"; for the purpose of my question we could have considered the other unary operations ¬P or ¬Q or even the two nullary operations ⊤ and ⊥.)

Edit:

After some time, I seem to have found a possible answer: Donald Knuth, in his section on Boolean Basics in The Art of Computer Programming Volume 4A, Pre-Fascicle 0B, writes the logical projections as P L Q and P R Q. That is, we could express projection onto P as the binary operationL(P,Q) and express projection onto Q as the binary operation R(P,Q). See this StackExchange question for how to typeset these symbols.

Let P and Q be two statements, each having two possible truth values: true (T) or false (F). Then there are exactly 16 unique compound statements involving P and Q with corresponding truth tables of the form (w,x,y,z). We have given symbols to some of these truth tables; for example

• We write the statement corresponding to (T,F,T,T) as P⇒Q
• We write the statement corresponding to (T,F,F,T) as P⊽Q
• We write the statement corresponding to (T,F,F,T) as P⇔Q

Note that these compound statements are logically equivalent to other expressions:

• The statement P⇒Q has the same truth table as ¬P∨Q
• The statement P⊽Q has the same truth table as ¬(P∨Q)
• The statement P⇔Q has the same truth table as (P⇒Q)∧(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. That is, the first expression writes the compound statement as a binary operation of P and Q. For instance:

• The implication maps (P,Q) to P⇒Q
• The logical nor maps (P,Q) to P⊽Q
• The biconditional maps (P,Q) to P⇔Q

Now the binary operation which maps (P,Q) to P is called the logical projection onto P, and the binary operation which maps (P,Q) to Q is called the logical projection onto Q. My question is: What symbol can be used to represent these two logical projections as binary operations? That is, how can we write these two logical projections in the form P(?)Q and P(¿)Q? Do these two projection operations have recognized symbols? Unfortunately the Wikipedia page for logical projection looks like it has a lot of room for expansion.

(Note: P and Q are essentially "unary operations"; for the purpose of my question we could have considered the other unary operations ¬P or ¬Q or even the two nullary operations ⊤ and ⊥.)

Edit:

After some time, I seem to have found a possible answer: this page writes the logical projections as P L Q and P R Q. That is, we could express projection onto P as the binary operationL(P,Q) and express projection onto Q as the binary operation R(P,Q).

Let P and Q be two statements, each having two possible truth values: true (T) or false (F). Then there are exactly 16 unique compound statements involving P and Q with corresponding truth tables of the form (w,x,y,z). We have given symbols to some of these truth tables; for example

• We write the statement corresponding to (T,F,T,T) as P⇒Q
• We write the statement corresponding to (T,F,F,T) as P⊽Q
• We write the statement corresponding to (T,F,F,T) as P⇔Q

Note that these compound statements are logically equivalent to other expressions:

• The statement P⇒Q has the same truth table as ¬P∨Q
• The statement P⊽Q has the same truth table as ¬(P∨Q)
• The statement P⇔Q has the same truth table as (P⇒Q)∧(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. That is, the first expression writes the compound statement as a binary operation of P and Q. For instance:

• The implication maps (P,Q) to P⇒Q
• The logical nor maps (P,Q) to P⊽Q
• The biconditional maps (P,Q) to P⇔Q

Now the binary operation which maps (P,Q) to P is called the logical projection onto P, and the binary operation which maps (P,Q) to Q is called the logical projection onto Q. My question is: What symbol can be used to represent these two logical projections as binary operations? That is, how can we write these two logical projections in the form P(?)Q and P(¿)Q? Do these two projection operations have recognized symbols? Unfortunately the Wikipedia page for logical projection looks like it has a lot of room for expansion.

(Note: P and Q are essentially "unary operations"; for the purpose of my question we could have considered the other unary operations ¬P or ¬Q or even the two nullary operations ⊤ and ⊥.)

Edit:

After some time, I seem to have found a possible answer: Donald Knuth, in his section on Boolean Basics in The Art of Computer Programming Volume 4A, Pre-Fascicle 0B, writes the logical projections as P L Q and P R Q. That is, we could express projection onto P as the binary operationL(P,Q) and express projection onto Q as the binary operation R(P,Q). See this StackExchange question for how to typeset these symbols.