Question

Show that the connective \(\downarrow\), with valuation function \([\varphi \downarrow \psi]=1\) iff \([\varphi]=[\psi]=0\), forms a functionally complete set (neither \(\varphi\), nor \(\psi)\).

Short answer

The NAND operator is functionally complete because it can express NOT, AND, and OR.

Step-by-step solution

Understand the NAND connective

The NAND operator, denoted as \(\downarrow\), is defined such that \(\varphi \downarrow \psi = 1\) if and only if both \(\varphi = 0\) and \(\psi = 0\). Essentially, it is equivalent to neither \(\varphi\) nor \(\psi\) being true.

Implement Logical Negation using NAND

Since \(\varphi \downarrow \varphi = 1\) if \(\varphi = 0\), we can represent logical negation with NAND as follows: \(eg \varphi = \varphi \downarrow \varphi\). This means that you can negate a proposition by computing \(\downarrow\) with itself.

Implement Logical OR using NAND

An OR operation, \(\varphi \lor \psi\), can be implemented using NAND and previously defined negation: \(\varphi \lor \psi = (eg \varphi) \downarrow (eg \psi)\). Replace the negations using Step 2 results: \(= (\varphi \downarrow \varphi) \downarrow (\psi \downarrow \psi)\).

Implement Logical AND using NAND

Logical AND operation \(\varphi \land \psi\) can be implemented using De Morgan's Law: \(eg(eg\varphi \lor eg\psi)\). Using the definitions from Steps 2 and 3, this becomes: \((\varphi \downarrow \varphi) \downarrow (\psi \downarrow \psi)\) negated. Consequently, you execute AND as: \((\varphi \downarrow \psi) \downarrow (\varphi \downarrow \psi)\).

Verify Functional Completeness

Since we have shown how to express negation, conjunction (AND), and disjunction (OR) using only the NAND connective, and these form the basis of all logic expressions, NAND is functionally complete.

Key concepts

Logical Negation

Logical negation is a core principle in propositional logic. It allows us to invert the truth value of a given proposition. For instance, if a statement is true, the negation will make it false, and vice versa. This operation is crucial for constructing more complex logical expressions.

In many logical systems, the negation of a proposition \( \varphi \) is denoted by \( eg \varphi \). In terms of truth values:

• If \( \varphi = 1 \) (true), then \( eg \varphi = 0 \) (false).
• If \( \varphi = 0 \) (false), then \( eg \varphi = 1 \) (true).

The NAND operator offers an elegant way to represent negation. By applying NAND to a single proposition, you can achieve the negation of that proposition. For example, \( eg \varphi = \varphi \downarrow \varphi \). This shows that with NAND, you can perform logical negation, forming the basis for more complicated operations.

NAND Operator

The NAND operator is a fundamental concept in digital logic design and computing. It stands for "Not AND" and is denoted by the symbol \( \downarrow \). This operator outputs false only if all its inputs are true. In every other scenario, it returns true.

In a truth table format, the behavior of NAND can be expressed as:

• \( \varphi = 1 \), \( \psi = 1 \) then \( \varphi \downarrow \psi = 0 \)
• \( \varphi = 1 \), \( \psi = 0 \) then \( \varphi \downarrow \psi = 1 \)
• \( \varphi = 0 \), \( \psi = 1 \) then \( \varphi \downarrow \psi = 1 \)
• \( \varphi = 0 \), \( \psi = 0 \) then \( \varphi \downarrow \psi = 1 \)

The NAND operation is significant because it is functionally complete. This means you can derive any logical function or operation (like AND, OR, NOT) using only NAND operators, which is an invaluable property for designing electronic circuits.

Logical Operations

Logical operations like AND, OR, and NOT form the backbone of computational logic. They allow us to construct expressions that evaluate to true or false based on the input values.

1. **AND Operation**: This operation outputs true only if both operands are true. It's mathematically expressed as \( \varphi \land \psi \). With NAND, this becomes \((\varphi \downarrow \psi) \downarrow (\varphi \downarrow \psi)\).
2. **OR Operation**: Outputs true if at least one operand is true. It is written as \( \varphi \lor \psi \). Using NAND, it can be represented as \((\varphi \downarrow \varphi) \downarrow (\psi \downarrow \psi)\).
3. **NOT Operation**: As previously discussed, NOT changes the truth value of a single operand. Using NAND, it's simply \( \varphi \downarrow \varphi \).

These logical operations allow us to process binary information, critical for computing systems. By using the NAND operator, we can replicate all these functions and more, showcasing its utility in logical operations.

Propositional Logic

Propositional logic is a branch of logic that deals with propositions which can be either true or false. It employs connectives to form more complex expressions and is essential for various fields including mathematics, computer science, and philosophy.

Propositional logic is composed of primitives known as propositions. Connectives such as AND (\( \land \)), OR (\( \lor \)), and NOT (\( eg \)) are used to build compound propositions.

• A **proposition** is a declarative statement that can either be true or false but not both.
• **Connectives** are used to combine propositions, allowing us to express logical relationships and construct logical formulas.

Functional completeness is an important concept within propositional logic. It indicates the ability to form all possible logical functions or operations using a limited set of operators. The realization that the NAND operator alone can form any logical operation exemplifies functional completeness. It highlights the elegance and power of propositional logic in facilitating logical reasoning and computational processes.