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Formal Logic I
Table of Contents
Unit 1 Overview: Logic and Arguments: An Introduction
1.1 Fundamentals of Logic and Reasoning
1.2 Structure and Types of Arguments
1.3 Validity, Soundness, and Cogency
1.4 Inductive vs. Deductive Reasoning

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1.4 Inductive vs. Deductive Reasoning

Citation:

Reasoning comes in two main flavors: deductive and inductive. Deductive reasoning starts with general ideas and leads to specific conclusions. Inductive reasoning does the opposite, using specific observations to draw broader conclusions.

When evaluating arguments, deductive reasoning aims for certainty, while inductive reasoning deals with probability. Deductive arguments are judged on validity and soundness, while inductive arguments are assessed for strength and cogency.

Types of Reasoning

Deductive and Inductive Reasoning

  • Deductive reasoning moves from general premises to a specific conclusion
    • Premises are assumed to be true and the conclusion follows necessarily from them
    • If the premises are true, the conclusion must be true (valid deductive arguments)
    • Deductive arguments are evaluated in terms of validity and soundness
  • Inductive reasoning moves from specific premises to a general conclusion
    • Premises provide evidence that supports the conclusion but do not guarantee its truth
    • Even if the premises are true, the conclusion may still be false (strong or weak inductive arguments)
    • Inductive arguments are evaluated in terms of strength and cogency

General-to-Specific and Specific-to-General Reasoning

  • General-to-specific reasoning starts with a general statement or principle and applies it to a specific case
    • Involves deductive reasoning, drawing a conclusion about a particular instance based on a general rule
    • Example: All humans are mortal. Socrates is human. Therefore, Socrates is mortal.
  • Specific-to-general reasoning starts with specific observations and draws a general conclusion
    • Involves inductive reasoning, inferring a general rule or principle based on particular instances
    • Example: Every swan I have seen is white. Therefore, all swans are probably white.

Evaluating Arguments

Certainty and Probability in Arguments

  • Deductive arguments aim for certainty in their conclusions
    • If the premises are true and the argument is valid, the conclusion must be true with absolute certainty
    • Sound deductive arguments provide conclusive proof of their conclusions
  • Inductive arguments aim for probability in their conclusions
    • The premises provide evidence that makes the conclusion more or less likely to be true
    • The strength of an inductive argument depends on how well the premises support the conclusion
    • Even strong inductive arguments do not provide absolute certainty, only high probability

Strength and Cogency of Inductive Arguments

  • The strength of an inductive argument refers to how well the premises support the conclusion
    • Strong inductive arguments have premises that make the conclusion very likely to be true
    • Weak inductive arguments have premises that provide little support for the conclusion
    • The more relevant and comprehensive the evidence, the stronger the argument
  • The cogency of an inductive argument refers to both its strength and the truth of its premises
    • A cogent inductive argument is strong and has true premises
    • Even if an inductive argument is strong, it may not be cogent if one or more premises are false
    • Cogent arguments provide good reasons to accept the conclusion, but not absolute certainty

Key Terms to Review (16)

Validity: Validity refers to the property of an argument where, if the premises are true, the conclusion must also be true. This concept is essential for evaluating logical arguments, as it helps determine whether the reasoning process used leads to a reliable conclusion based on the given premises.
Soundness: Soundness refers to a property of deductive arguments where the argument is both valid and all of its premises are true, ensuring that the conclusion is necessarily true. This concept is crucial in determining the reliability of an argument, connecting validity to actual truthfulness and making it a cornerstone of logical reasoning.
Premise: A premise is a statement or proposition that provides the foundation for an argument, serving as the evidence or reason that supports the conclusion. Understanding premises is essential for analyzing the structure of arguments, distinguishing between valid and invalid forms, and assessing the overall soundness and cogency of reasoning.
David Hume: David Hume was an 18th-century Scottish philosopher known for his influential work in empiricism and skepticism. His ideas critically examined the principles of human understanding and the nature of knowledge, particularly in relation to inductive and deductive reasoning, where he famously challenged the validity of inductive reasoning as a reliable method for acquiring knowledge.
Syllogism: A syllogism is a form of reasoning in which a conclusion is drawn from two given or assumed propositions (premises). It’s a fundamental tool in logic that helps in constructing valid arguments, and it's crucial for understanding how conclusions can be logically deduced from a set of premises. Syllogisms can illustrate the differences between deductive reasoning, where the conclusion necessarily follows from the premises, and inductive reasoning, which involves generalizing from specific instances. Understanding syllogisms also plays a vital role in analyzing philosophical arguments and the foundational principles of logic and reasoning.
Modus Tollens: Modus Tollens is a valid argument form in deductive reasoning that states if a conditional statement is true, and the consequent is false, then the antecedent must also be false. This logical structure is foundational in understanding validity and soundness, especially in arguments involving implications.
Fallacy of Affirming the Consequent: The fallacy of affirming the consequent is a logical error that occurs when one assumes that if 'P implies Q' is true, then 'Q is true' must mean 'P is true.' This form of reasoning misinterprets the implications of conditional statements. It often confuses necessary and sufficient conditions, leading to invalid conclusions in arguments.
Counterexample: A counterexample is a specific instance or case that demonstrates the falsity of a general statement or argument. It plays a crucial role in distinguishing between inductive and deductive reasoning, as it can show that an inductively derived conclusion is not universally true. Additionally, counterexamples are key in translating quantified statements and applying quantifier rules by providing instances that challenge the validity of the premises or conclusions drawn from them.
Cogency: Cogency refers to the quality of an argument being both strong and having all true premises, making it a compelling basis for conclusion. In the realm of reasoning, especially with inductive arguments, cogency ensures that if the premises are true, the conclusion is likely to be true as well. It plays a crucial role in evaluating the strength of inductive reasoning compared to deductive reasoning, where the focus is more on validity rather than truth of premises.
Hasty Generalization: A hasty generalization is a logical fallacy that occurs when a conclusion is drawn from an insufficient or unrepresentative sample of data. This type of reasoning often leads to stereotypes and inaccurate beliefs about a group based on a limited number of cases, making it important to recognize the difference between inductive reasoning, which can support generalizations when done correctly, and deductive reasoning, which relies on established premises.
Generalization: Generalization is a reasoning process where conclusions are drawn from specific instances to create broader statements or principles. It plays a crucial role in understanding patterns and trends, allowing us to apply knowledge gained from limited observations to wider contexts.
Strength: In the context of reasoning, strength refers to the degree of support that premises provide for a conclusion in an argument. A strong argument is one where the premises, if true, make the conclusion likely to be true, particularly in inductive reasoning, where conclusions are based on observations or patterns. The concept of strength is crucial in evaluating the effectiveness of arguments and distinguishing between types of reasoning.
Deductive Reasoning: Deductive reasoning is a logical process where a conclusion follows necessarily from the premises, leading to a certain outcome if the premises are true. This method emphasizes the relationship between premises and conclusion, establishing validity, soundness, and cogency in arguments.
Modus Ponens: Modus ponens is a fundamental rule of inference in formal logic that allows one to derive a conclusion from a conditional statement and its antecedent. It asserts that if we have a statement in the form of 'If P, then Q' and we know that P is true, then we can conclude that Q must also be true. This logical structure connects to various principles of reasoning and argumentation.
Aristotle: Aristotle was a Greek philosopher and logician who made significant contributions to various fields, including logic, metaphysics, ethics, and natural sciences. His foundational work in formal logic, particularly syllogistic reasoning, set the stage for understanding concepts like validity, soundness, and cogency in arguments.
Inductive Reasoning: Inductive reasoning is a method of reasoning in which a general conclusion is drawn from specific observations or instances. It often involves making predictions or generalizations based on trends or patterns observed in data, which means that while the conclusions can be probable, they are not guaranteed to be true.