Math Problem Statement
Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If he tries hard, he will do poorly. He will do well. He does not try hard. Let p be the statement "he tries hard," and q be the statement "he will do well." Select the correct answer below and fill in the answer box to complete your choice. (Type the terms of your expression in the same order as they appear in the original expression.) A. The argument is valid. In symbolic form, the argument is enter your response here. B. The argument is invalid. In symbolic form, the argument is enter your response here. Valid Arguments Direct Reasoning Contrapositive Reasoning Disjunctive Reasoning Transitive Reasoning pright arrowq pright arrowq plogical orq plogical orq pright arrowq p ~q ~p ~q qright arrowr q ~p q p pright arrowr ~rright arrow~p
Invalid Arguments Fallacy of the Converse Fallacy of the Inverse Misuse of Disjunctive Reasoning Misuse of Transitive Reasoning pright arrowq pright arrowq plogical orq plogical orq pright arrowq q ~p p q qright arrowr p ~q ~q ~p rright arrowp ~pright arrow~r
Solution
Let’s analyze the argument step by step to determine its symbolic form and validity.
Step 1: Translating statements into symbolic form
-
If he tries hard, he will do poorly.
- This can be interpreted as:
("If (he tries hard), then not (he will do poorly).")
- This can be interpreted as:
-
He will do well.
- This is:
-
He does not try hard.
- This is:
Step 2: Writing the argument in symbolic form
The argument becomes:
1. & \ p \rightarrow \sim q \\ 2. & \ q \\ 3. & \ \sim p \\ \end{aligned}$$ ### Step 3: Testing validity We need to see if this argument is valid. One way to do this is to check if it matches any standard valid or invalid forms, or we can use a **truth table**. #### Standard Form Comparison - **Premise 1** ($$p \rightarrow \sim q$$) states that if $$p$$, then $$\sim q$$. - **Premise 2** ($$q$$) directly contradicts $$\sim q$$, implying $$p$$ must be false ($$\sim p$$). - **Premise 3** ($$\sim p$$) supports this. This aligns with **contrapositive reasoning**: $$(p \rightarrow \sim q) \wedge q \implies \sim p$$ Thus, the argument is **valid**. ### Final Answer: #### The symbolic form of the argument is: $$(p \rightarrow \sim q), q \vdash \sim p$$ ### The argument is **valid**. --- ### Follow-Up Questions: 1. What is contrapositive reasoning, and how does it apply to this argument? 2. Can you explain why the truth table confirms this argument's validity? 3. How does this argument differ from the Fallacy of the Converse? 4. What other standard valid forms might help confirm this argument's validity? 5. How would the argument change if $$q$$ were false instead? ### Tip: When evaluating validity, always check if the argument’s premises lead logically to the conclusion using standard forms or a truth table.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Propositional Logic
Logical Arguments
Validity Testing
Formulas
Implication (p → q)
Negation (~p)
Contrapositive Reasoning
Theorems
Contrapositive Reasoning
Standard Forms of Logical Validity
Suitable Grade Level
Undergraduate - Logic/Mathematics