Question

Suppose that in a formal logic, green \((x)\) means that \(x\) has the attribute of being green, \(\operatorname{frog}(x)\) means that \(x\) has the attribute of being a bullfrog, and \(J\) stands for the specific entity Jeremiah. Translate the following formal statements into English: a. \(\operatorname{frog}(J)\) b. \((\forall x)(\) frog \((x) \rightarrow \operatorname{green}(x))\)

Short answer

a. Jeremiah is a bullfrog. b. All bullfrogs are green.

Step-by-step solution

Understanding the symbols

In this problem, we need to translate statements from symbolic logic into plain English. We have two primary predicates: \( \operatorname{frog}(x) \) and \( \operatorname{green}(x) \). Additionally, \( J \) stands for Jeremiah.

Statement (a) Translation

The statement \( \operatorname{frog}(J) \) uses the predicate \( \operatorname{frog}(x) \) applied to the specific entity \( J \). Thus, it translates to "Jeremiah is a bullfrog."

Statement (b) Translation

The statement \( (\forall x)(\operatorname{frog}(x) \rightarrow \operatorname{green}(x)) \) is a universally quantified statement. It reads as "For all entities \( x \), if \( x \) is a bullfrog, then \( x \) is green." So, it means "All bullfrogs are green."

Key concepts

Predicate Logic

Predicate Logic is a framework used to express propositions and quantify variables. It extends propositional logic by incorporating quantifiers and predicates which provide more nuances.
A **predicate** is a statement that may be true or false depending on the values of its variables. In this exercise, \(\operatorname{frog}(x)\) suggests that \(x\) is a bullfrog, and \(\operatorname{green}(x)\) suggests that \(x\) is green.
**Quantifiers** in predicate logic, like \(\forall\) ("for all") and \(\exists\) ("there exists"), are used to specify the quantity of specimens in the domain of discourse that satisfy a given predicate. Here, the statement \( (\forall x) \, (\operatorname{frog}(x) \rightarrow \operatorname{green}(x))\) uses the universal quantifier \(\forall\) to assert that every instance of \(x\) that is a bullfrog is also green.

Symbolic Logic

Symbolic Logic, also known as formal logic, uses symbols to articulate logical ideas and complexities. This concise representation aids in the analysis and understanding of logical assertions.
In our scenario, **symbols** like \(\operatorname{frog}(x)\) and \(\operatorname{green}(x)\) represent specific conditions or properties. By associating Jeremiah with \(J\), we simplify logical statements.
For instance, \(\operatorname{frog}(J)\) denotes that Jeremiah has the property of being a bullfrog. The use of these symbols helps readers and scholars effortlessly parse complex relations and verify logical constructs. The abstraction using symbols is fundamental in eliminating ambiguity and improving clarity.

Translation of Logic Statements

The translation of logic statements involves converting expressions from symbolic form into natural language, making them easily comprehensible. Understanding the rules and syntax used in symbolic logic is essential for accurate translation.
When dealing with statements like \(\operatorname{frog}(J)\), the translation is straightforward: "Jeremiah is a bullfrog." This is because \(J\) directly represents Jeremiah, and \(\operatorname{frog}\) indicates that bullfrog is his attribute.
More complex translations often involve logical connectors and quantifiers. The expression \( (\forall x) \, (\operatorname{frog}(x) \rightarrow \operatorname{green}(x))\) transforms into "All bullfrogs are green". This captures the meaning that the property of being green follows from the property of being a bullfrog for every entity in the domain.

• The predicate \(\rightarrow\) ("implies") indicates conditional relationships.
• Quantifiers guide the interpretation of the scope of the statement.

These translations enable us to reveal the logical structure of statements in the natural language, facilitating better understanding and discussion.