Comprehensive Notes on A Brief Guide to Logic and Argumentation
1. WHAT IS AN ARGUMENT?
Purpose in philosophy:
A philosopher aims to provide reasons for believing a conclusion, not just state an answer.
When reading philosophy, identify and assess the author’s arguments; when writing, offer your own arguments.
What is an argument?
A sequence of statements where the last claim is the conclusion to be supported.
Premises: statements asserted without proof in the present argument but may be supported elsewhere.
Example: Argument A (for the existence of God)
(1) The Bible says that God exists.
(2) Whatever the Bible says is true.
(3) Therefore, God exists.
Premises: (1) and (2); Conclusion: (3).
Defense of premises via independent arguments
Argument B (defense of premise (2))
(4) The Bible has predicted many historical events that have come to pass.
(5) Therefore, whatever the Bible says is true.
Combined as Argument C
(6) The Bible has predicted many historical events that have come to pass.
(7) Therefore, whatever the Bible says is true.
(8) The Bible says that God exists.
(9) Therefore, God exists.
Important reading strategy
Distinguish author’s premises (starting points) from conclusions (results to be criticized).
If a statement is a conclusion, it should be supported; premises can be criticized if false or defective, but not simply for lacking proof within the argument.
Clues about function in writing
Clue that a sentence is a conclusion: words like “hence,” “therefore,” “so.”
Clues that a sentence functions as a premise: phrases like “Let us assume that …,” “It seems perfectly obvious that …,” or “Only a fool would deny that … .”
Exercise (practice identifying premises and conclusion)
Passage: "Everyone knows that people are usually responsible for what they do. But you’re only responsible for an action if your choice to perform it was a free choice, and a choice is only free if it was not determined in advance. So we must have free will, and that means that some of our choices are not determined in advance."
Task: Identify the premises and the main conclusion.
2. VALIDITY
Definition
An argument is valid iff it is absolutely impossible for its premises to be true and its conclusion false.
Examples from the text
Argument A (Bible example) is valid: if the premises are true, the conclusion follows.
Argument B is invalid in general: the Bible could be true in some respects and false in others; many worlds exist where the premises hold but the conclusion does not.
Terminology
When premises entail the conclusion, we say the conclusion follows from the premises (entailment).
If an argument is valid, the premises entail the conclusion; equivalently, the form preserves truth from premises to conclusion.
Illustrative oddities
D: All philosophers are criminals. All criminals are short. Therefore, all philosophers are short. (Valid form, though premises are false.)
E: God exists. Therefore, God exists. (Premise and conclusion identical; valid.)
F: The moon is green. The moon is not green. Therefore, God exists. (Shows a contradiction in premises makes the conclusion vacuously true in a way; valid by form since a contradiction in premises makes it impossible for premises to be all true together.)
Turning invalid into valid (adding premises)
G: I can imagine existing without my body. (I can imagine my feet slowly disappearing, etc.) Therefore, I am not my body.
H: I can imagine existing without my body. If I can imagine X existing without Y, then X is not Y. Therefore, I am not my body.
Method: add missing premises that the author might have accepted to make the argument formally valid.
Practice: Spot the valid arguments
(i) If abortion is permissible, infanticide is permissible. Infanticide is not permissible. Therefore, abortion is not permissible. (Valid form; needs premises about permissibility transfer.)
(ii) It is wrong to experiment on a human subject without consent. Dr. X experimented on Mr. Z. Mr. Z consented to this experiment. Therefore, it was not wrong for Dr. X to experiment on Mr. Z. (Valid form; depends on premises about consent sufficiency.)
(iii) I will not survive my death. My body will survive my death. Therefore, I am not my body. (Common argument form; assess validity given tacit premises.)
(iv) Geoffrey is a giraffe. If X is a giraffe, then X’s parents were giraffes. Therefore, all of Geoffrey’s ancestors were giraffes. (Validity varies with premises.)
Exercise: The following arguments are not valid as they stand. Supply missing premises to make them valid.
(v) Every event has a cause. No event causes itself. Therefore, the universe has no beginning in time.
(vi) It is illegal to keep a tiger as a pet in New York City. Jones lives in New York City. Therefore, it would be wrong for Jones to keep a tiger as a pet.
(vii) The sun has risen every day for the past 4 billion years. Therefore, the sun will rise tomorrow.
Necessary truths and validity
Some statements express necessary truths (e.g., mathematical truths like $2+3=5$). If the conclusion is a necessary truth, the argument is automatically valid.
Check your understanding
Prove that if the conclusion is a necessary truth, the argument is valid.
3. SOUNDNESS
Definition
A valid philosophical argument is only as good as its premises; false premises make it a poor argument.
Example: Argument D (invalid premises but valid form)
(1) All philosophers are criminals. (2) All criminals are short. (3) Therefore, all philosophers are short. (Valid form but false premises.)
Soundness
A sound argument = valid argument with true premises.
Important implication
A valid argument with false premises can be persuasive but is not sound.
Check your understanding
Show that if an argument is sound, its conclusion must be true.
4. HOW TO RECONSTRUCT AN ARGUMENT: AN EXAMPLE
Important skill: extracting an explicit argument from dense prose.
Aquinas example (Summa Theologica, Part I, Question 2, Article 3):
Premises and conclusion in the original: Things lacking intelligence act towards an end; hence there exists an intelligent designer who directs natural things to their end.
Step 1: Identify the Conclusion
Main conclusion (explicitly marked by “therefore”): "Some intelligent beings exist by whom all natural things are directed to their end."
Step 2: Interpret the Conclusion
What does it mean for a natural thing to be directed to an end? Function/purpose explanation (e.g., heart’s end is to pump blood).
Reformulation (more familiar): There is an intelligent being that ensures natural objects perform their functions.
Step 3: Reconstruct the Argument
Identify premises and intermediate steps; supply implicit premises to render the argument explicitly valid.
Example of an expanded reconstruction (illustrative):
(1) Unintelligent things always or nearly always act in the same way, so as to achieve the best result.
(2) If a thing always or nearly always acts in a certain way, so as to achieve the best result, then that thing performs a function.
(3) Therefore, unintelligent things perform a function.
(4) If a thing performs a function, it does so by design.
(5) Therefore, unintelligent things perform their functions by design.
(6) If an unintelligent thing performs a function by design, then there exists an intelligent being that ensures that it performs its function.
(7) Therefore, there exists an intelligent being that ensures that natural objects perform their functions.
Step 4: Assess the reconstruction
Even if you grant the premises (1) and (6), the conclusion (7) is not entailed by the premises as given. The reconstruction shows a common problem: the premises may require a single master designer, while the text’s intuition might allow multiple designers.
Verdict in the source: Aquinas’s argument, as reconstructed here, is not valid.
What this teaches about reconstruction
Make tacit assumptions explicit; consider multiple plausible reconstructions; assess validity and premises truth.
Exercise: Provide a reconstruction of Aquinas’s argument that does not commit the logical error noted above in the transition from (6) to (7).
5. FORMAL VALIDITY
Concept
Focus on the form of arguments by abstracting away subject matter to a schematic form.
Example: Argument H and schematic form
Abstract form: Every F is a G. Gs are not H. Therefore, Fs are not H. (Formally valid as a general structure.)
Demonstrations of form validity
J: Every whale is a mammal. Mammals do not lay eggs. So whales do not lay eggs. (Valid form; content of second premise is false in some real-world cases like platypus, but form is valid.)
Formal validity vs validity
An argument can be valid without being formally valid (it may not fit a known valid schematic form).
K: Every crayon in the box is scarlet. So every crayon in the box is red. (Form: Every F is G. So every F is H. This is not necessarily valid without an additional premise like If a thing is scarlet, then it is red.)
What formal validity means for philosophy
Formal logic catalogs valid forms and provides general principles for determining whether a given argument is formally valid.
Common schematic forms and symbols
We use P, Q, R for declarative sentences. Common symbols:
→ (if … then)
~ (not)
∨ (or)
Example: All F are G; All G are H; Therefore All F are H.
More examples of formally valid forms
Modus ponens: P, P → Q ⊢ Q
Syllogistic forms: All F are G; All G are H; Therefore All F are H.
Important distinction
An argument can be valid (the form guarantees the conclusion) without being formally valid (not all instances are captured by a single known valid form).
Reflection on form and content
When analyzing philosophical arguments, prefer identifying a valid form and then check truth of premises.
Check your understanding
Consider a non-formal argument K and discuss why it might be valid but not formally valid; provide a counterexample or add a premise to make it formally valid.
6. A PUZZLE ABOUT FORMAL LOGIC
Sorites paradox setup
A long row of colored squares from red to yellow; rule: (1) Square n and square n+1 are indistinguishable by ordinary means.
(2) If two things are indistinguishable by ordinary means, then if one is red, so is the other.
Derivation to paradox
With premises (1) and (2), modus ponens yields: If square 1 is red, then square 2 is red; then square 3 is red; etc., leading to square 1000 being red, contradicting that square 1000 is yellow.
Dilemma and responses
Options to resolve: (a) reject premise (2) (no sharp cutoff between red and not red), or (b) reject modus ponens as a universally valid rule in this context.
Result
Sorites paradox remains unresolved; illustrates limits of applying simple inductive reasoning to gradual, border-case predicates.
7. WHAT MAKES AN ARGUMENT GOOD?
Beyond validity and soundness
Even a valid or sound argument can be bad if it begs the question or relies on circular premises.
Examples (L, M):
L: God exists. Therefore, God exists. (Formally valid; premise is identical to conclusion; not a contributing argument.)
M: God does not exist. Therefore, God does not exist. (Formally valid; same issue as L.)
Non-begging premises
Argument N (potentially sound) can be valid but still intuitively weak if it presupposes its conclusion rather than supporting it.
Example: God knows when you will die. Therefore, God exists. (May be sound but begs the question because belief in the premise already presumes the conclusion.)
Distinction: begging the question
Begging the question occurs when the premises presuppose the conclusion; the argument as a whole fails to provide independent support.
Example O vs P
O: That thing in the bushes is a platypus. So platypuses exist. (Non-begging; valid if premises are true; can be sound but context-dependent.)
P: This rock in my hand is a material object. So material objects exist. (Similar structure to O but often argued to beg the question when the context assumes a skeptical stance about the external world.)
Takeaway about good arguments
A good argument should not beg the question; its premises should be credible independently of the conclusion.
The notion of a good argument in philosophy is nuanced; not all good arguments must be strictly demonstrative or formally valid.
8. NON-DEMONSTRATIVE ARGUMENTS
What they are
Demonstrative arguments are valid proofs; non-demonstrative are not always valid but can still be powerful.
Examples Q, R, S, T (non-demonstrative)
Q: All who have drunk hemlock die shortly thereafter. Therefore, if I drink hemlock, I will die. (Not valid in all cases; inductive-like reasoning.)
R: No one has seen a unicorn; therefore unicorns do not exist. (Not valid; absence of evidence is not evidence of absence.)
S: The cheese in the cupboard is disappearing; there is a mouse-sized hole; therefore a mouse has moved in. (Provisional conclusion based on circumstantial evidence.)
T: It’s normally wrong to kill a person; bartender is a person; therefore it would be wrong to kill the bartender. (Ethical generalization with exceptions possible.)
Nature of non-demonstrative arguments
They are not formal proofs; they rely on patterns, evidence, or best explanations rather than strict deduction.
Inductive arguments
Premises are observations showing a pattern; conclusion generalizes the pattern to future cases.
Example: Past A events are followed by B events; therefore in the future, A will be followed by B. (Not always reliable; exceptions possible.)
Caution: Even true premises do not guarantee a true universal conclusion (e.g., US presidential elections example with male winners up to 2017).
Inductive validity and its limits
There is no formal test that guarantees a good inductive argument; formal statistics and probability theory (statistical inference) attempt to model good inductive reasoning.
Abductive arguments (inference to the best explanation)
Start from observed facts and reason toward the hypothesis that best explains them.
R and S are abductive: observed facts (e.g., cheese disappearance; no unicorns) are explained by H (mouse in kitchen; no unicorns).
Abduction is central to many scientific explanations (molecules, Darwin’s evolution, etc.).
A form T (non-demonstrative, ethics-focused)
Form: Normally, P. Therefore, P. (e.g., normally cats have four legs; Felix is a cat; therefore Felix has four legs—unless there is information to the contrary.)
Such arguments are common in ethics where general principles allow exceptions.
General remark on non-demonstrative arguments
There is no universally accepted formal account of when non-demonstrative arguments are good; it remains an active research area.
9. SOME GENERAL REMARKS ON ARGUMENTATION IN PHILOSOPHY
The standard in mathematics vs philosophy
In mathematics, the ideal is that all good arguments are valid (formally valid proofs).
In philosophy, non-demonstrative arguments often play a crucial role, and not all good philosophical reasoning is strictly demonstrative.
The balance philosophers strike
While the formal standard (valid and non-begging premises) is highly valued, philosophy also accepts robust non-demonstrative arguments (induction, abduction, and principled generalizations).
Caution about standards
Avoid overgeneralizing the demand for formal validity; non-demonstrative arguments can still provide genuine, rational support if premises are credible and not begging the question.
Practical guidance for philosophers
When reconstructing arguments, try to identify the best charitable version of the argument.
Seek tacit assumptions and test their reasonableness.
Where possible, present arguments in valid form and evaluate the truth of premises and the independence of premises from the conclusion.
Final reflection
Philosophy rests on a mixture of methods: formal validity, soundness, non-demonstrative reasoning, and careful analysis of premises and assumptions.
Check your understanding (recap prompts you may attempt)
What makes an argument valid? What makes an argument sound? Why can a valid argument have false premises?
How do you reconstruct an argument from a dense text? What counts as a tacit assumption?
Why is it not enough for philosophy to rely solely on demonstrative proofs? When might non-demonstrative arguments be acceptable?
Notes: The above captures the major and minor points, examples, and exercises from the transcript. LaTeX is used for key logical forms and equations where appropriate, including formal validity forms, entailment notation, and common inference rules. Throughout, the emphasis is on understanding what makes arguments valid, sound, and good, and on practical methods for reconstructing and evaluating philosophical arguments.