Question

Is the following argument valid? "If it is raining, then the streets are wet. Either it is snowing, or the streets are not wet. It is not snowing. Therefore, it is not raining."

Short answer

The argument is valid, as all premises logically lead to the given conclusion. Using logical rules and representation, we deduced that it is not raining (¬R) from given premises: If it is raining (R), then the streets are wet (W), either it is snowing (S) or the streets are not wet (¬W), and it is not snowing (¬S).

Step-by-step solution

Represent each premise and the conclusion with logical symbols.

Represent the given argument using the symbols: 1. R → W (If it is raining, then the streets are wet) 2. S ∨ ¬W (Either it is snowing or the streets are not wet) 3. ¬S (It is not snowing) Conclusion: ¬R (It is not raining)

Analyze each premise and the conclusion through logical rules.

Use logical rules to determine the validity of the argument: From premise 1 and 3, we know: - R → W (If it is raining, then the streets are wet) - ¬S (It is not snowing) From premise 2, we have: - S ∨ ¬W (Either it is snowing or the streets are not wet) - Since ¬S (It is not snowing), we can conclude ¬W (The streets are not wet), using the disjunction elimination rule. Now we have: - R → W (If it is raining, then the streets are wet) - ¬W (The streets are not wet) From these two statements, we can apply the contrapositive rule to the first statement. The contrapositive of R → W is ¬W → ¬R. Therefore, we have: - ¬W → ¬R (If the streets are not wet, then it is not raining) Since we also know ¬W (The streets are not wet), we can now logically deduce the conclusion ¬R (It is not raining), using the modus ponens rule.

Validate the conclusion.

The conclusion can now be validated using the logical deduction derived in Step 2: Given everything that follows logically from the premises, we deduced ¬R (It is not raining). Therefore, the argument is valid, as all premises logically lead to the given conclusion.

Key concepts

Logical Symbols

Logical symbols are fundamental tools in formal logic that help us represent complex arguments clearly and succinctly. These symbols are like a shorthand language that everyone in logic can understand, making it easier to convey arguments without ambiguity.

Here’s a quick rundown:

• R stands for "It is raining."
• W means "The streets are wet."
• S represents "It is snowing."
• ¬ is the symbol for "not." So, ¬S means "It is not snowing."
• → is used for "implies." Hence, R → W means "If it is raining, then the streets are wet."
• ∨ stands for "or." Therefore, S ∨ ¬W means "Either it is snowing, or the streets are not wet."

By using these symbols, complex logical arguments can be broken down into manageable and understandable parts. Understanding how to translate sentences into logical symbols is the first step in evaluating an argument’s validity.

Disjunction Elimination

Disjunction elimination is a logical rule that helps us simplify statements involving the "or" symbol (∨). When faced with a disjunction like S ∨ ¬W, disjunction elimination allows us to make deductions by eliminating one possibility if we know the other to be false.

In our exercise, we have the statement \( S \lor eg W \) which translates to "Either it is snowing, or the streets are not wet." Given we have ¬S (It is not snowing), we can eliminate "it is snowing."

Through disjunction elimination, we confidently conclude the remaining option, ¬W: "The streets are not wet."

This rule effectively narrows down the possible outcomes to the one that must hold true based on the established premises. It's a crucial step in moving forward with our logical proof.

Contrapositive Rule

The contrapositive rule is an important concept in logic that allows us to infer a conclusion by rewriting an implication statement in a reverse manner. The contrapositive of an implication \( R \to W \) is \( eg W \to eg R \). Simply put, if the conclusion is not true, then the premise can't be true either.

Why does this help? Often, it transforms a complex logical statement into something easier to evaluate.

In our exercise, we use this rule by taking the first premise "If it is raining, then the streets are wet" (\( R \to W \)) and forming its contrapositive: "If the streets are not wet, then it is not raining" (\( eg W \to eg R \)). This conversion aids us in linking ¬W ("The streets are not wet") directly to our conclusion ¬R ("It is not raining").

Recognizing and applying contrapositives can simplify proofs and reinforce the consistency and validity of an argument.

Modus Ponens

Modus ponens is a fundamental rule of inference that enables us to deduce a conclusion when we have an implication statement and its antecedent. This principle operates by stating that if we know \( R \to W \) ("If it is raining, the streets are wet") and R is true, then W must also be true.

In our particular exercise, after applying the contrapositive, we have \( eg W \to eg R \), which means "If the streets are not wet, then it is not raining."

Given that we know ¬W ("The streets are not wet"), modus ponens allows us to affirm ¬R ("It is not raining").

By using this logical rule, we can efficiently arrive at the conclusion simply and directly from our modified premises. Modus ponens is a powerful tool for logical reasoning, helping us draw concrete conclusions from established premises.