Question

Consider the following argument : "If it is a good watch then it is a Titan watch. It is a Titan watch therefore it is a good watch". This argument is (1) Valid (2) invalid (3) may be valid or invalid (4) invalid if conditional connective is replaced by biconditional connective

Short answer

The argument is invalid.

Step-by-step solution

Identify premises and conclusion

The premises are: 'If it is a good watch then it is a Titan watch' and 'It is a Titan watch'. The conclusion is: 'Therefore, it is a good watch.'

Translate premises into logical statements

Let G represent 'It is a good watch' and T represent 'It is a Titan watch'. The first premise can be written as: \[ G \rightarrow T \]. The second premise can be written as: \[ T \].

Analyze the argument structure

The argument concludes that \( G \) (It is a good watch) based on \( T \) (It is a Titan watch) and \( G \rightarrow T \) (If it is a good watch, then it is a Titan watch).

Determine logical validity

An argument is valid if the conclusion necessarily follows from the premises. In this case, knowing that \( T \) is true does not necessarily make \( G \) true from \( G \rightarrow T \). Therefore, the argument is invalid.

Evaluate alternative options

Option (4) suggests the argument is invalid if the conditional connective is replaced by a biconditional connective. Biconditional means \( G \leftrightarrow T \), which would imply that \( G \) is true if and only if \( T \). However, this change makes the argument structure different and does not affect the given conditional argument.

Key concepts

Logical Statements

Logical statements are sentences that can be classified as true or false. Think of them as building blocks for more complex logical reasoning. In the given exercise, we use logical statements to transform everyday language into precise logical expressions.

For example:
- 'If it is a good watch, then it is a Titan watch'
- 'It is a Titan watch'
Each of these statements can be symbolized in a formal way for clarity and analysis. Representing them with symbols like \( G \rightarrow T \) (for 'If it is a good watch, then it is a Titan watch') and \( T \) (for 'It is a Titan watch') helps to eliminate ambiguity.

Here, \( G \) stands for 'It is a good watch' and \( T \) stands for 'It is a Titan watch'. This symbolic representation is a key part of understanding and analyzing logical statements.

Conditional Logic

Conditional logic, also known as 'if-then' logic, involves statements that assert a condition and its result. A conditional statement has the form \( p \rightarrow q \), which reads as 'If p, then q'. It claims that if the first statement (the antecedent) is true, then the second statement (the consequent) must be true as well.

In the exercise, the conditional statement is: 'If it is a good watch, then it is a Titan watch', which is written as \( G \rightarrow T \). Conditional logic can help evaluate the relationships between different statements by setting clear conditions.

It's important to note that just because the consequent (\( T \)) is true, it doesn't necessarily mean the antecedent (\( G \)) must be true. For instance, knowing 'It is a Titan watch' does not logically conclude 'It is a good watch', which highlights a common mistake in interpreting conditional statements.

Argument Validity

An argument is valid if the conclusion logically follows from the premises. Validity doesn't concern itself with the truth of the premises, only with the logical structure. If it's impossible for the premises to be true while the conclusion is false, then the argument is valid.

In our example, the premises are:
- \( G \rightarrow T \): 'If it is a good watch, then it is a Titan watch.'
- \( T \): 'It is a Titan watch.'
The conclusion is: \( G \): 'Therefore, it is a good watch.'

The argument claims that because 'It is a Titan watch' (\( T \)), it must be a good watch (\( G \)). This line of reasoning is flawed because \( G \rightarrow T \) does not logically infer \( G \) when \( T \) is true.

Thus, the argument is invalid, as the conclusion doesn't follow necessarily from the premises. Understanding the validity of arguments is crucial in logical reasoning to ensure that our conclusions are always backed by solid premises.