Problem 57 - Thinking Mathematically

Question

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. "I do know that this pencil exists; but I could not know this if Hume's principles were true. Therefore, Hume's principles, one or both of them, are false." \- G. E. Moore, Some Main Problems of Philosophy

Step-by-Step Solution

Answer

The argument 'I do know that this pencil exists; but I could not know this if Hume's principles were true. Therefore, Hume's principles, one or both of them, are false' is valid in terms of propositional logic.

Step 1: Translating the Argument

Let's denote the following propositions: P: 'This pencil exists'. Q: 'I know that this pencil exists'. R: 'Hume's principles are true'. S: 'Hume's principles, one or both of them, are false'. The argument in symbolic form would then be: 1) Q (I know that this pencil exists); 2) Q -> ~R (I could not know this if Hume's principles were true); Therefore, 3) S (Hume's principle, one or both of them, are false).

Step 2: Checking the Validity of the Argument

A simple and straight forward way to check the validity of an argument is to create a truth table. After creating a truth table, we can look at the rows where premises are true. If in those rows, the conclusion is also true, your argument is valid. Otherwise, the argument is invalid.

Step 3: Computing the Truth Table

Based on the truth-table analysis mechanism, if the premises are true and the conclusion is also true, then the argument can be considered as valid, otherwise it would be regarded as invalid. Here, apparently, if 'Q -> ~R' is true and 'Q' is also true that implies '~R' must be true. Therefore, 'S' is valid. We don't need to calculate a full truth table here as we can directly infer from logical deduction.

Key Concepts

Validity of Arguments

The validity of arguments is a crucial concept in symbolic logic. When we discuss an argument's validity, we are essentially determining whether the conclusion naturally follows from the premises. In other words, if the premises are true, the conclusion must be true as well. In logical terms, this means that there is no possible scenario where the premises are true and the conclusion is false.

Valid arguments are important in constructing logical proofs. An invalid argument might have a true conclusion, but this truth is not guaranteed by the premises. When analyzing arguments, it's crucial to focus on the structure, not the literal truth of the premises or conclusion. Understanding this helps you recognize that even if the premises and conclusion are factually correct, the argument could still be invalid if the logical connection is not there.

• The premises must logically connect to lead to a conclusion.
• Even one false scenario with true premises and a false conclusion makes the argument invalid.

Truth Tables

Truth tables are a vital tool in evaluating the validity of logical arguments. They systematically show how the truth values of complex statements depend on their components. This can help you visualize whether any combination of true premises could lead to a false conclusion.

A truth table includes rows for every possible truth value combination of the involved propositions. Each proposition can be true or false, so with multiple propositions, the number of rows increases exponentially. You lay out all combinations of truth values and, step by step, calculate the truth values of connected statements and finally the conclusion.

• List all possible truth combinations for the premises.
• Determine the truth value of the conclusion for each combination.
• Check if all true premises lead to a true conclusion.

Truth tables offer a visual tool and can be a step-by-step method, especially helpful when manually validating small to moderately-sized logical arguments.

Logical Deduction

Logical deduction involves deriving a conclusion from given premises using logical reasoning. It is the backbone of sound argumentation and can be thought of as "mathematical" reasoning for logic. Using principles of deduction, we can draw a conclusion guaranteed by the premises, assuming the premises themselves are true.

Logical deduction requires no guessing. Instead, it uses well-established rules and structures, like modus ponens or modus tollens, to ensure the reasoning process is robust. In the provided exercise, the premises lay a straight logical path to the conclusion using deduction. This exercise shows that if "I could not know this if Hume's principles were true," and "I know this pencil exists," the deduction must be that at least one of Hume's principles must be false.

• Deductions are made through chains of reasoning.
• The structure of language plays a critical role in guiding deductions.
• Proper use of logical deduction prevents errors in reasoning and ensures correctness in conclusions.