PHIL222 - Truth Tables L4

Introduction

Course: PHIL222

Lecture: 4

Week: 2

Date: 1/16

Arguments

Joint Truth Tables

• A method used to systematically evaluate the validity of logical arguments, providing a visual representation of how truth values interact within the premises and conclusion.

• Each argument's structure is thoroughly analyzed by allocating one column for each premise alongside one for the conclusion, allowing for an easy tracking of truth values across all possible scenarios.

Criteria for Validity

• An argument is considered valid if there is no scenario (or row) within the truth table where all premises are true and the conclusion is false. This means the logical structure maintains consistency under scrutiny.

• Conversely, the argument is labeled invalid if at least one scenario demonstrates all premises as true while the conclusion is false, indicating a logical inconsistency that highlights flaws within the argument's reasoning. Identifying these flaws is crucial for rigorous logical analysis.

Examples of Arguments

Example 1

Premises and Conclusion: P1) If Ragnar is a husky, then Ragnar is a dog. (H → D) P2) Ragnar is a husky. (H) C) Therefore, Ragnar is a dog. (D)

Truth Table Summary:

• This argument is valid because there is no row that fulfills the condition of having all true premises whilst providing a false conclusion. The structure supports a logically sound inference based on the premises presented, underscoring the role of conditional relationships in argument evaluation.

Example 2

Premises and Conclusion: P1) If Ragnar is a husky, then Ragnar is a dog. (H → D) P2) Ragnar is not a husky. (¬H) C) Therefore, Ragnar is not a dog. (¬D)

Truth Table Summary:

• This argument is deemed invalid, as there exists a row wherein both premises are true but the conclusion is false (i.e., where H is false, making D true). Such an example illustrates the pitfalls of logical reasoning when premises do not correlate correctly with even the simplest forms of conclusions.

Counterexamples

• A crucial aspect of logical reasoning, an argument is invalid if there exists a potential scenario in which all premises are correct to the extent that the conclusion is incorrect. Counterexamples serve as valuable tools for testing arguments against logical scrutiny.

• The counterexample serves as a clear illustration of an invalid argument, specifically the case regarding Ragnar: when H is false (Ragnar is not a husky), the conclusion D (Ragnar is not a dog) becomes true, thereby undermining the argument's credibility.

Soundness

• When analyzing truth tables, each row corresponds to a unique arrangement of truth values across propositions, facilitating a deeper understanding of argument structure and potential outcomes.

Definition:

• An argument qualifies as "sound" if it meets the following criteria:

  1. The argument is valid, as demonstrated through its structure, and

  2. The premises are true in the specific real-world context of the truth row examined. Soundness is a higher standard that encompasses both logical consistency and factual accuracy, which is crucial for establishing reliable arguments.

Example of a Sound Argument:

P1) Snow is white. (S) P2) Grass is green. (G) C) Thus, snow is white and grass is green. (S ∧ G)

Additional Notes on Soundness

• While truth tables efficiently confirm the validity of arguments, they cannot ascertain soundness, as they do not identify the actual row that specifies the truth values. To evaluate soundness, one must verify the premises against real-world facts and consider whether these truths are universally accepted.

• An example that illustrates the principles of soundness might involve exploring the likelihood of life existing in the Andromeda galaxy. Such inquiries not only challenge our logical sensibility but also engage with empirical evidence and scientific reasoning.

Propositions

Single Propositions

• Tautology: A proposition that holds true in every possible row of the truth table and is universally accepted across logic systems, serving as a foundational truth.

• Contradiction: A proposition that is false in every row of the table, representing a logical impossibility. Recognizing contradictions helps in identifying flawed arguments as they pertain to perceived truths.

• Satisfiable Proposition: A statement that is not considered a contradiction and can attain a true value in certain scenarios, highlighting the nuances of truth within varied contexts.

Contingency:

• Contingently True: A proposition that is true in the actual truth row but is not a tautology, indicating conditional truths that depend on specific contexts.

• Contingently False: A proposition that is false in the actual row but is not a contradiction, illustrating that falsehood can co-exist with conditional validity.

Two Propositions

• Logically Equivalent: Two propositions that exhibit the same truth value across all possible rows of the truth table, thus demonstrating complete logical harmony.

• Jointly Unsatisfiable: A pair of propositions that cannot both be true in any arrangement, showcasing mutual exclusivity in logical analysis.

• Contradictory: Two propositions that are jointly unsatisfiable, where there is no row that accommodates both as false, representing stark opposition in their truths.

• Contraries: Two propositions that are jointly unsatisfiable while still allowing for at least one row where both are false, showcasing the complexity of relational truths.

Sets of Propositions

• Definition of Set: In logical terms, a set refers to a collection of objects, which in this context represents a collection of propositions working in tandem to evaluate logical outcomes.

Satisfiability:

• Unsatisfiable Set: A scenario in which there exists no row where all the propositions within that set hold true simultaneously, indicating their inability to coexist under any logical structure.

• Satisfiable Set: A set that possesses at least one row where all propositions can be true concurrently, exemplifying the potential for logical harmony.

Shortcuts in Truth Tables

Steps for More Efficient Evaluation:

1. Begin the truth table evaluation by populating the column corresponding to the simplest proposition in the argument, setting a foundation for thorough assessment.

2. If the evaluated proposition is the conclusion, disregard any rows where that conclusion is shown to be true, streamlining the analysis by negating unhelpful scenarios.

3. Conversely, if it is a premise, eliminate any rows where it is false, thus focusing the evaluation on relevant truth conditions.

4. Progressively work through the next simplest proposition until either a counterexample is established or all rows are exhausted, facilitating a clearer understanding of argument validity while ensuring exhaustiveness in analysis.

Final Examples

Example 3

Premises and Conclusion: ¬P ∨ Q (Q → ¬R) ∧ (¬R → ¬P) ∴ ¬P

• This argument should be examined using a truth table to ascertain its validity, demonstrating a comprehensive step-by-step process across multiple rows to amplify the understanding of logical evaluation and prompt critical thinking.

Conclusion on Example 3

• Following the evaluation, the decision concluded no counterexample was present, thus categorizing the argument as valid according to the compiled findings from the truth table evaluation, reinforcing the importance of detailed analysis in logical discourse.