Notes on Validity, Circular Reasoning, Contradictions, Augmentation, Induction, and Learning Strategies
2.1 First Weird Case of Validity: Circular Reasoning
Circular reasoning definition: a form of reasoning that assumes what you are trying to prove; the premise and the conclusion are the same thing, at least in effect.
Example given: “The Chief is a Martian. Therefore: The Chief is a Martian.”
Answer: Valid. It is a circular argument because the premise and conclusion are effectively identical.
Takeaway: Circular reasoning is an extreme case that reveals something odd about validity, but it is still a valid form of argument when the premise is true.
The broader point about circularity:
Circularity is any case of assuming what you are trying to prove, even if the premise and conclusion are stated in slightly different terms.
Why study circular reasoning: understanding its structure helps when proving when arguments are valid or invalid in difficult cases.
Practical exercise: drag words to form circular arguments (interactive activity).
Important nuance: circular reasoning is labeled a fallacy in everyday reasoning because it is tempting and often flawed in persuasion, but explicitly the book notes that validity can make circular arguments valid in a formal sense if the premise holds.
Practical examples to illustrate the idea:
- George example: 1) George’s reasoning was tempting but flawed. 2) Therefore, George committed a fallacy. (This shows that restating the conclusion in slightly different terms can illuminate the flaw without endorsing it as a model of good reasoning.)
2.2 Second Weird Case of Validity: Contradictory Premises
Definition recap:
Contradiction: a logical falsehood; a sentence that is necessarily false; e.g., "Quinn is telling the truth" and "Quinn is not telling the truth" cannot both be true.
Contradictory set: a group of sentences that cannot all be true at the same time.
Key claim: Any argument with contradictory premises is valid.
Rationale: If the premises cannot all be true together, the condition for validity (that it is impossible for all premises to be true and the conclusion false) is vacuously satisfied.
This is an application of the principle that from a contradiction, anything follows (explosion). In logic: from a contradictory set, any conclusion can be derived.
Examples from the text:
Contradictions in the “bin”: e.g.,
Some police dogs are not dogs, Pia is not Pia. (an illustrative contradictory set piece)
Mixed with extra premises:
1. If Quinn is innocent, then someone doctored the file.
2. No one doctored the file.
→ Quinn is innocent.
Takeaway: When premises form a contradictory set, every conclusion follows; hence the argument is valid, even if the conclusion is ridiculous or irrelevant.
- Important caveat: The fact that an argument is valid due to contradictory premises does not endorse the truth of the premises themselves; it is a feature of formal validity, not a guarantee about real-world truth.
2.3 Third Weird Case of Validity: Impossible Premises
Idea: If it is impossible for the premises to be true, then it is also impossible for the premises to be true and the conclusion false.
Example construction (to illustrate a valid form):
Premises: 1) If all police dogs are smart, then Rufus is smart. 2) Rufus is smart.
Conclusion: All police dogs are smart.
Assessment: Invalid. This is a familiar fallacy (affirming the consequent) dressed as a validity exercise.
Takeaway: The impossibility of all premises being true turns on a particular form of reasoning; not every argument with a seemingly reasonable structure under these conditions is actually valid.
- Note: This section motivates recognizing tempting but flawed patterns in reasoning (to be discussed later in the book).
2.4 Necessary vs. Contingent (and empty premises)
Clarifying terms:
Necessary truth (logical truth): true in all possible worlds by virtue of logical laws. E.g., a tautology under the laws of logic.
Logical falsity (contradiction): false in all possible worlds by virtue of logical laws.
Contingent truth/falsehood: true or false depending on the way the world is, not determined by logic alone.
Examples of logical status:
All police dogs are dogs. (Logical truth)
Rufus is Rufus. (Logical truth)
Some police dogs are not dogs. (Logical falsity)
If Rufus is a dog, then Rufus is a dog. (Logical truth)
Rufus isn’t Rufus. (Logical falsity)
Contingent example: Pia is innocent. This is neither logically true nor logically false; its truth value depends on the real world, not on logic alone.
Important notion: Necessary vs contingent applies to logical laws; physical or moral necessities are different kinds of necessity and are not what is meant by logical necessity here.
The status of certain sentences when framed as tests:
1) Some dogs are not police dogs. → Contingent truth
2) Some police dogs are not dogs. → Logical falsity
3) All dogs are police dogs. → Contingent falsity
4) All police dogs are dogs. → Logical truth
Augmentation vs logical necessity:
Augmentation: adding premises to an argument to form a new one.
Logical necessity is distinct from augmentation; augmented arguments can remain logically valid if the original argument was valid.
Clarification about necessity types:
- “Necessary” refers to logical necessity for this logic course, not physical or other kinds of necessity.
2.5 Augmentation
Definition: To augment an argument is to add one or more premises to make a new argument.
Key point: If you start with a valid argument and augment it, you cannot make it invalid in a deductive sense.
Example of a valid argument:
Base: 1. Stan and Uma are innocent. 2. Uma is innocent. 3. Therefore, Uma is innocent. (The structure is already valid.)
Augmentation variants:
2: Add a premise to line 2 (the exact form of line 2 can be chosen). Result: a new argument with the same conclusion.
2**: Add an irrelevant premise (e.g., Tamar is innocent) and still arrive at the same conclusion.
2***: Add a contradictory premise (e.g., Uma is guilty). The conclusion remains entailed (since the original premise set is unchanged in its entailment relation to the conclusion).
Three cases illustrating why augmentation does not destroy validity:
Relevant augmentation: the added premise supports the conclusion directly or indirectly; validity preserved.
Irrelevant augmentation: the added premise does not affect the link between the original premises and the conclusion; validity preserved.
Contradictory augmentation: the added premise contradicts other premises or the conclusion; validity preserved because the original premises still entail the conclusion.
Important contrast with inductive reasoning: augmentation cannot destroy deductive validity, but it can destroy inductive support (see below).
Special note: It is possible to move from an invalid argument to a valid one through augmentation (e.g., by adding the conclusion as a premise).
- Inductive note: Induction is probabilistic; premises may support a conclusion to some degree (>50%), not guarantee it.
2.6 Learning About Learning (Desirable Difficulties and Active Learning)
Desirable difficulty: A problem that requires effort but is surmountable; designed to create puzzlement and curiosity, leading to better long-term learning.
Common misconception: learners learn best when everything is easy; the science of learning argues that a certain level of challenge enhances retention and understanding.
Example question: True or False: Generating an incorrect answer impedes learning because it teaches the wrong approach. Answer: False (errors can be productive learning experiences when engaged properly).
Critique of passive learning:
The “ideal of the passive teacher” and the “ideal of the passive student” are illusions; effective learning requires active engagement and effort.
People often rely on passive strategies (re-reading, highlighting) that are less effective for long-term retention than active problem solving and retrieval practices.
The science of memory and learning:
Learning strengthens synaptic connections in long-term memory; active practice leads to better retention than passive exposure.
The brain reorganizes itself through practice and prior knowledge integration; the more you practice retrieval, the stronger the memory traces become.
Practical study guidance:
If time is limited, solving problems (exam-like conditions) with closed book and no notes is more effective than extensive rereading.
Retrieval practice (actively recalling information) yields greater learning than passive review; this is a form of deliberate practice.
A “snowball effect” can occur: the more you engage in active learning, the easier it becomes to integrate new information and build upon existing knowledge.
Bonus lesson on retrieval practice:
Generating answers from memory strengthens learning more than recognizing correct answers from options; it is harder but more beneficial for long-term retention.
Actionable takeaway:
If you want to master the material, list all skills learned and self-check against that list; this active self-assessment often reveals gaps and new connections.
Gentle reminder: the feeling of mastery from passive rereading is a cognitive illusion; true mastery requires effortful practice.
Closing note: the text emphasizes active learning and the practical benefits of deliberate, effortful study strategies for deep understanding and long-term retention.
"This notes set is designed to mirror the structure and content of the transcript, capturing major and minor points, examples, and the practical implications of validity, circular reasoning, contradictions, augmentation, inductive vs deductive reasoning, and learning strategies. The included LaTeX snippets provide formal representations where appropriate to support a rigorous study aid."