Deductive Validity and Logical Form

• Question on Deductive Validity

  • Is this argument deductively valid?

  • What is the logical form of this argument?

• Complexity of Argument Forms

  • Each argument possesses a single logical form, which acts as a skeleton for substituting terms to derive a specific argument.

  • This assertion is an oversimplification since a single argument can be derived using multiple logical forms.

• Example Argument:

  • Premise 1: Fred lives in California.

  • Premise 2: If Fred lives in California, then Fred lives in the United States.

  • Conclusion: Fred lives in the United States.

• Derivation from Logical Forms:

  • The argument can fit into at least two forms:

  1. Form (x):

     • Premise: If X, then Y

     • Conclusion: Y

  2. Form (Ss):

     • Premise: X

     • Premise: If X, then Y

     • Conclusion: Y

  • Validity of Forms:

  • Form (a) is valid.

  • Form (b) is invalid.

  • The argument concerning Fred remains valid despite the existence of multiple logical forms.

• Defining Valid and Invalid Arguments:

  • Define argument validity using logical forms while avoiding the misconception that each argument has exclusively one logical form.

Objective Truths in Arguments

• Claim of Objective Truths:

  • The chapter asserts the existence of objective truths.

• Construction of Arguments:

  • Task to construct an argument demonstrating the existence or non-existence of objective truths.

Chapter 3 Overview: Inductive and Abductive Arguments

• Outline of the Chapter:

  • Deductive Validity as a Limitation

  • Nondeductive Inference—A Weaker Guarantee

  • Two Gambling Strategies

  • Universal Laws

  • Detective Work

  • Induction

  • Two Factors Influencing Inductive Strength

  • Abduction

  • Inferring What Isn’t Observed

  • Difference between Abduction and Induction

  • Deducing Observational Predictions from a Theory

  • True and False Predictions Interpretations

  • The Surprise Principle

  • Evidence Discriminating Between Hypotheses

  • True Prediction Isn’t Enough

  • Modest Favoring

  • Summary of the Surprise Principle

  • The Only Game in Town Fallacy

  • Review Questions

  • Problems for Further Thought

  • Suggested Readings, Videos, and Audio

Deductive Validity: Definition and Limitation

• Definition of Deductive Validity:

  • A deductively valid argument guarantees that if premises are true, the conclusion must also be true.

  • An example of deductive reasoning is:

  • If premises P1 = true and P2 = true, then conclusion C must logically be true.

• Limitations of Deductive Arguments:

  • The premises can’t lead to a conclusion that extends beyond their contents.

  • For example, one cannot derive the broader population's percentage composition from a sample without risk.

• Survey Example:

  • Tracking registered voters in a county:

  1. Survey Result: 60 percent of surveyed individuals identify as Democrats.

  2. Invalid Deduction Examples:

     • 1: 60 percent of the people called said they are Democrats.

     • 2: 60 percent of the people from the calls are Democrats.

     • Conclusion: Approximately 60 percent of the county’s voters are Democrats.

  • Justification of Invalidity:

    • Possible lying from respondents or that the actual Democrat population could differ significantly.

Nondeductive Inference: A Weaker Guarantee

• Nature of Nondeductive Inference:

  • Unlike deductive arguments, nondeductive inference provides weaker guarantees.

  • Premises may suggest conclusions that appear probable but do not ensure absolute truth.

• Example of Gambling Analogy:

  • Two gambling types:

  • Extreme Conservative: Only immoral when winning is assured — avoids losses, yet misses potential wins.

  • Thoughtful Risk Taker: Sometimes takes calculated risks hoping to win — may incur losses but could lead to superior understanding.

• Implication for Arguments:

  • Deductive arguments are conservative and limit conclusions to what premises can express.

  • Nondeductive arguments allow for broader conclusions with inherent risks.

Universal Laws and Science

• Nature of Universal Laws:

  • Newton’s universal law of gravitation states:

  • Gravitational attraction $ ext{F} ext{is proportional to} ext{m}1 ext{m}2$ and $ ext{inversely proportional to} ext{d}^2$ ($F = G rac{m1 m2}{d^2}$).

  • Applies regardless of time or place.

• Limitations of Inductive Arguments:

  • Newton couldn't deduce his universal law from limited observations.

  • Inductive conclusions often venture beyond strict observations, creating reliance on probability rather than guarantees.

Detective Work and Nondeductive Reasoning

• Role of Deduction in Investigation:

  • Example: Sherlock Holmes utilizing clues:

  1. Clues: gun with an “M”, cigar butt, fresh footprint.

  2. Conclusion: Moriarty as possible murderer.

• Nondeductive Nature of Holmes’ Inference:

  • Questioning valid conclusions based on incomplete observations where various explanations exist.

Induction and Abduction

Induction

• Definition:

  • Inference that extends a sample description to a broader population.

  • Example: From a sample of 60% Democrats, concluding a similar proportion across a larger group.

• Inductive Strength:

  • Evaluated by sample size and representativeness.

  • Larger and unbiased samples yield stronger arguments.

Abduction

• Definition:

  • Inference to the best explanation based on observations.

• Historical Example:

  • Gregor Mendel’s work on genetics illustrates abduction:

  • Result of controlled experiments leads to the theorization of genes without direct observation of them.

• Difference from Induction:

  • Induction makes claims based purely on observation, while abduction formulates theoretical explanations.

Theory, Predictions, and Conclusion Assessment

• Testing Theories via Observations:

  • Theories deduced from successful predictions do not guarantee the truth of that theory; predictions confirming a theory add support but don’t ensure absolute correctness.

• Deducibility of Truth from Predictions:

  • Correct predictions indicate potential truth.

  • Incorrect predictions invalidate theories linked to those predictions.

The Surprise Principle

• Definition:

  • Observation O strongly supports hypothesis H over H' if:

  1. If H were true, O is expected.

  2. If H' were true, O would be unexpected.

• Application:

  • The principle emphasizes finding elements that differentiate hypothesis merit based on expectations against alternative hypotheses.

  • Example situations illustrating the principle help reveal foundational understanding of predictions versus surprising observations.

Fallacies in Abductive Reasoning

The Only Game in Town Fallacy

• Definition:

  • Accepting a hypothesis simply because no better explanation appears available is an error in reasoning.

• Critique of Reasoning:

  • Failure to provide an alternative does not inherently validate a hypothesis.

Review and Reflection

• Key Questions:

  • Explore conceptual differences, the strength of arguments, surprise factors, and the significance of sound reasoning.

• Practical Applications:

  • Encourage further thought on methodological applications, evaluating perspectives, and grappling with robust argumentation in diverse fields.