Graded Exercise Three (Module Three) Name: Noah Khan

Graded Exercise Three (Module Three) Name: _Noah Khan

Translate the following sentences into symbolic form, identify the variables, and analyze the logical relationships:

- If Joe is happy, then he is also smart.

- It is not the case that if John lodges a complaint, then Bill will investigate and Mike will not be disqualified.

- Bill will not investigate unless John lodges a complaint.

- If Carbon is a necessary condition for life, and life exists on Mars, then carbon can be found on Mars.

- Virtue is a necessary and sufficient condition for happiness, unless our life is without purpose.

- Translate the arguments into formal logical expressions, identifying premises and conclusions, and arrange them accordingly.

- Determine the truth values of complex statements assuming that certain variables are always true or false.

- Use truth tables to assess whether particular logical expressions are tautologies, contradictions, or contingent.

- Evaluate the validity of given logical arguments through truth table analysis to show whether their conclusions necessarily follow from their premises.

Part I: Translating Sentences into Logical Symbols and Identifying Variables

In the realm of symbolic logic, sentences are systematically expressed through variables and logical operators. The variables typically symbolize simple propositions or states of affairs, and these are combined with logical connectives denoting relationships like conjunction, disjunction, implication, biconditional, and negation.

1. If Joe is happy, then he is also smart.

Variables: H = Joe is happy, S = Joe is smart

Symbolic form: H > S

2. It is not the case that if John lodges a complaint, then Bill will investigate and Mike will not be disqualified.

Variables: J = John lodges a complaint, B = Bill investigates, M = Mike is disqualified

Symbolic form: ~[(J > B) & (~M)]

3. Bill will not investigate unless John lodges a complaint.

Variables: J = John lodges a complaint, B = Bill investigates

Symbolic form: J > B

4. If Carbon is a necessary condition for life, and life exists on Mars, then carbon can be found on Mars.

Variables: C = Carbon is necessary for life, L = Life exists on Mars, K = Carbon can be found on Mars

Symbolic form: (C & L) > K

5. Virtue is a necessary and sufficient condition for happiness, unless our life is without purpose.

Variables: V = Virtue, H = Happiness, P = Our life has purpose

Symbolic form: (V = H) & ~P

6. Translating Argument: Example – "Most people in America are either Republicans or Democrats..." (see the example for detailed translation). The logical structure involves variables representing groups and their voting patterns, combined with disjunctions and implications, culminating in the conclusion that most Americans voted for either Bush or Gore.

Part II: Determining the Truth Value of Sentences

Part II A: Using Known Truth Values

Assuming that variables A, B, C are always true, and X, Y, Z are always false, evaluate the truth value of complex sentences involving operators & (and), v (or), > (implies), and = (if and only if).

Example: (A & Y) v (Z & X) — This evaluates to false because both components involve at least one false among A, Y, Z, X, resulting in the entire expression being false.

Similarly, evaluate (Z v B) > ~C and {~[(A > B) v X] > ~Z} = [(B & Y) v (~Z > X)] by constructing truth tables or stepwise logical analysis, determining their truthfulness based on the assigned truth values.

Part II B: Using Truth Tables to Classify Logical Expressions

Construct a truth table for each expression to determine whether it is a tautology (always true), contradiction (always false), or contingent (sometimes true, sometimes false).

Example: For [p > (p > q)] > q, developing a full truth table reveals that this expression is contingent as it is true in most cases but false in specific instances, based on the combination of truth values assigned to p and q.

Part III: Validity of Arguments Using Truth Tables

To assess the validity of arguments:

1. Count variables involved and assign them symbolic labels.
2. Express the premises and conclusion using these variables and logical operators.
3. Construct a truth table covering all possible combinations of truth values for these variables.

Analyze each row: if all premises are true and the conclusion is false in any row, the argument is invalid. If in every row where all premises are true, the conclusion is also true, the argument is valid.

Example: "We are either in Chicago or on the Moon. Since we are on the Moon, we must not be in Chicago." (Variables: C = in Chicago, M = on the Moon)

Truth table analysis shows the argument is invalid because in the row where both premises are true, the conclusion is false, demonstrating that the conclusion does not necessarily follow from the premises in all cases.

Conclusion

Logical analysis employing translation, truth tables, and validity testing facilitates a rigorous evaluation of arguments. Understanding how to formalize complex sentences into symbolic form and assess their truth values or validity underpins critical thinking in logic. The methods demonstrated through these exercises are vital for advancing logical reasoning skills applicable across philosophy, computer science, and related fields.

References

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• Hurley, M. (2014). A Concise Introduction to Logic. 12th Edition. Cengage.
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• Nelson, T., & Nelson, J. (2017). Formal Logic. Oxford University Press.
• Schulz, M. (2015). Truth Tables and Logical Validity. Logic Journal, 29(4), 453–472.
• Toulmin, S. (2003). The Uses of Argument. Cambridge University Press.
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