Notes on Informal Logic: Arguments, Deduction, and Evaluation
Informal Logic: Key Concepts from the Lecture
Purpose of the lecture sequence
- Part 1: informal logic — introduce correct reasoning, identify arguments, and explain how to evaluate arguments.
- Part 2 (next time): moral reasoning.
Working definition of logic
- Logic is the study of correct reasoning, including the methods for evaluating arguments.
- Three guiding questions:
- What is correct reasoning?
- What is an argument?
- How do we evaluate arguments?
What is correct reasoning?
- Correct reasoning adheres to the laws of logic.
- Three laws of logic to be aware of:
- Law of noncontradiction: two opposite statements cannot both be true.
- Example: it cannot be true that this is a computer on my desk and this is not a computer on my desk.
- Symbolically: \neg (P \land \neg P) is true for any statement P.
- Law of the excluded middle: a statement cannot be both true and false; every statement is either true or false.
- Example: this is a computer or it is not a computer. There is no middle ground.
- Symbolically: P \lor \neg P is true.
- Law of identity: something is what it is; a statement is true if it is true and not true if it isn’t.
- Simple articulation: P \rightarrow P (a statement implies itself).
- Violating these laws means you’re not engaging in correct reasoning.
What is an argument? (Philosophical sense)
- An argument is a group of statements; one or more claims provide support for another statement.
- Example layout:
- Statement 1 (premise)
- Statement 2 (premise)
- Statement 3 (conclusion)
- Premises (1 and 2) infer or give reasons for the truth of the conclusion (3).
- Propositions
- The statements in an argument are propositions — they are either true or false.
- Functions of propositions in an argument
- Premises: propositions that give support/reasons.
- Conclusion: the proposition that is supported.
- How premises support a conclusion
- Premises give good reasons to believe that the conclusion is true.
- A well-constructed argument has premises that provide reasons for the conclusion to be true.
- Structure of an argument
- There can be many premises, potentially a whole book of premises, but there is typically one conclusion.
- Premises provide reasons to believe the conclusion.
- Markers and indicators
- Conclusion markers indicate the conclusion (e.g., "therefore").
- There are premise markers as well; a rough list will be provided later in the lecture.
- Important takeaway
- An argument is about the logical relationship: premises support the conclusion.
Example 1 (deductive conclusion): all crimes are violations of the law; theft is a crime; therefore, theft is a violation of the law.
- Premises: 1) All crimes are violations of the law. 2) Theft is a crime.
- Conclusion: 3) Therefore, theft is a violation of the law.
- Connection: the marker word here is "therefore" indicating the conclusion.
- Note: the conclusion can appear anywhere in an argument; it is not necessarily the last sentence.
Two kinds of arguments (as introduced in the lecture)
- 1) Deductive argument: structured so that if the premises are true, they force the conclusion to be true (necessity).
- The deductive structure is evaluated for validity and soundness:
- Validity: the premises imply the conclusion with necessity (proper form).
- Soundness: valid form and all premises are true.
- If both hold, the argument is a sound deductive argument.
- Example focus: several standard deductive forms are named and analyzed for validity.
Deductive argument forms and examples
- Modus ponens (affirming the antecedent)
- Form: If $p$, then $q$. $p$. Therefore, $q$.
- Symbolic form: (p \rightarrow q),\ p \vdash q
- Example: If it rains, then the sidewalk will be wet. It rains. Therefore, the sidewalk is wet.
- This is a valid deductive argument (premises force the conclusion).
- Modus tollens (denying the consequent)
- Form: If $p$, then $q$. Not $q$. Therefore, not $p$.
- Symbolic form: (p \rightarrow q),\ \neg q \vdash \neg p
- Example: If it rains, the sidewalk will be wet. The sidewalk is not wet. Therefore, it did not rain.
- This is a valid deductive argument.
- Disjunctive syllogism (disjunctive, either/or)
- Form: $P$ or $Q$. Not $P$. Therefore, $Q$.
- Symbolic form: (P \lor Q),\ \neg P \vdash Q
- Example: Either God exists or everything is meaningless. God does not exist. Therefore, everything is meaningless.
- This is a valid deductive argument structurally, regardless of whether you agree with the conclusion.
- Pure hypothetical syllogism (chain of conditionals)
- Form: If $P$ then $Q$, and If $Q$ then $R$. Therefore, If $P$ then $R$.
- Symbolic form: (P \rightarrow Q),\ (Q \rightarrow R) \vdash (P \rightarrow R)
- Example: If all actions are determined, then no actions are free. If no actions are free, then no one is responsible for anything they do. Therefore, if all actions are determined, then no one is responsible for anything they do.
- This is a valid deductive argument.
- Note on the two-kinds distinction
- The lecture emphasizes these deductive forms and their validity; more forms exist, and the discussion could continue to a broader treatment of logic.
Invalid deductive forms (denying the antecedent and affirming the consequent)
- Denying the antecedent (invalid)
- Form: If $p$, then $q$. Not $p$. Therefore, not $q$.
- Example: If you are a lawyer, then you have a job. You are not a lawyer. Therefore, you do not have a job.
- Observation: all premises could be true, but the conclusion does not necessarily follow; the argument is invalid.
- Affirming the consequent (invalid)
- Form: If $p$, then $q$. $q$ is the case. Therefore, $p$.
- Example: If you’re a lawyer, you have a job. You have a job. Therefore, you are a lawyer.
- This is invalid because the truth of the consequent does not guarantee the antecedent.
Additional notes on deductive validity and soundness
- An argument can have true premises and a true conclusion but still be invalid if the logical connection does not entail the conclusion.
- A valid deductive argument with true premises is sound; otherwise, it may be valid but unsound (if any premise is false).
- The distinction between form (valid/invalid) and content (true/false premises) is crucial for evaluating arguments.
Practical takeaways for evaluating arguments
- Identify premises and conclusion.
- Check for conclusion markers (e.g., "therefore").
- Determine if the form is valid (premises force the conclusion) or invalid.
- If valid, assess whether premises are true to determine soundness.
- Be aware that conclusions can appear in different positions within an argument.
- Recognize that some arguments can be strongly reasoned but still be invalid if the form is not correct.
Context and scope
- The lecturer notes that logic is a vast field; the two-week focus here is to give a practical grasp of how arguments work and how to evaluate them, with deeper exploration possible in a longer course or book.
Key terms to remember
- Premises: supporting propositions.
- Conclusion: proposition being supported.
- Propositions: statements that are true or false.
- Deductive argument: aims at necessity (if premises true, conclusion must be true).
- Validity: proper form where premises entail the conclusion.
- Soundness: validity plus true premises.
- Conclusion marker: e.g., "therefore" indicating the conclusion.
- Premise marker: signals the premises within an argument.
- Law of noncontradiction, law of the excluded middle, law of identity: the foundational laws of logic.
Quick reference formulas
- Modus ponens: (p \rightarrow q),\ p \vdash q
- Modus tollens: (p \rightarrow q),\ \neg q \vdash \neg p
- Disjunctive syllogism: (P \lor Q),\ \neg P \vdash Q
- Pure hypothetical syllogism: (P \rightarrow Q),\ (Q \rightarrow R) \vdash (P \rightarrow R)
- Denying the antecedent (invalid): (P \rightarrow Q),\ \neg P \nvdash \neg Q
- Affirming the consequent (invalid): (P \rightarrow Q),\ Q \vdash P
Note on markers
- Conclusion markers (e.g., "therefore") help identify the conclusion.
- Premise markers help identify premises; a list will be provided later in the course.