What is being asserted?
Make the conclusion explicit.
Many disagreements become clearer once the actual proposition under dispute is stated precisely.
Side 11
Logic
A study of what follows from what. Logic separates the structure of an argument from the truth of its premises, then gives us tools for testing deduction, induction, explanation, contradiction and proof.
claim→premises→inference→conclusion
05inference modes
06formal operators
08argument traps
11Side
Arguments are structures.
Before deciding whether an argument is persuasive, identify exactly which statements are premises, which statement is the conclusion and what inferential bridge connects them.
What is offered in support?
List each supporting proposition.
Do not smuggle the conclusion into a premise or treat background assumptions as invisible.
How is support supposed to travel?
Deductive, probabilistic or explanatory?
The standard of evaluation depends on the type of inference being attempted.
Could the premises be true and the conclusion false?
Test structure independently of content.
If no such case is possible, the deduction is valid.
Are the premises actually true?
Validity is not enough.
A sound deductive argument requires both valid form and true premises.
Core separationValidity concerns form. Truth concerns propositions. Soundness requires both.
Deduction preserves necessity.
A deductively valid argument is one in which true premises cannot lead to a false conclusion.
Valid forms
MP
Modus ponens
If P, then Q. P. Therefore Q.
MT
Modus tollens
If P, then Q. Not Q. Therefore not P.
HS
Hypothetical syllogism
If P then Q; if Q then R; therefore if P then R.
DS
Disjunctive syllogism
P or Q. Not P. Therefore Q.
Invalid lookalikes
AA
Affirming the consequent
If P then Q. Q. Therefore P. Q may have another cause.
DA
Denying the antecedent
If P then Q. Not P. Therefore not Q. Q may occur without P.
EQ
Confusing implication and equivalence
P → Q does not by itself establish Q → P.
Countermodels are powerful.
To show a deductive form invalid, it is enough to construct one possible case where the premises are true and the conclusion false.
Most real-world reasoning is not deductive.
Induction extends beyond observed cases; abduction selects an explanation. Their conclusions can be well supported without being guaranteed.
Observed cases → broader pattern
Strength depends on sample quality, coverage and the stability of the process being generalized.
Similar in some respects → perhaps similar in another
The relevant similarities matter more than the number of superficial similarities.
Difference in outcome → possible difference in cause
Confounding, selection and alternative pathways must be ruled out or bounded.
Evidence → best current explanation
Competing explanations are compared by fit, simplicity, background knowledge and predictive reach.
Model + conditions → expected observation
A successful prediction can support a model without proving it uniquely true.
Prior plausibility + evidence → revised plausibility
The evidential impact depends on how expected the observation was under competing hypotheses.
deductionasksmust it follow?·inductionaskshow strongly?·abductionaskswhat explains it?
Make the structure explicit.
Formal notation removes some ambiguity by representing logical relationships independently of ordinary-language content.
| Operator | Symbol | Meaning | Key question |
|---|---|---|---|
| Negation | ¬P | Not P | What follows when the proposition is false? |
| Conjunction | P ∧ Q | P and Q | Are both required? |
| Disjunction | P ∨ Q | P or Q | Inclusive or exclusive? |
| Conditional | P → Q | If P, then Q | Is P sufficient for Q? |
| Biconditional | P ↔ Q | P iff Q | Are both directions established? |
| Quantification | ∀ / ∃ | For all / there exists | Universal claim or existence claim? |
Quantifier caution¬∀x P(x) means “not everything is P,” which is equivalent to “there exists at least one thing that is not P.”
∀x ¬P(x) means “nothing is P.”
∀x ¬P(x) means “nothing is P.”
Bad arguments fail in different ways.
A fallacy name is not a rebuttal. The useful task is to identify precisely where support breaks down.
Attack a weaker position.
Misrepresent the claim, then refute the substitute.
Collapse the option space.
Present two alternatives as exhaustive when other possibilities exist.
Shift a word’s meaning mid-argument.
The same term silently carries different senses in different premises.
Conclusion hides in the premises.
The support presupposes the very claim it is meant to establish.
Infer too much from too little.
The sample is too small, biased or unrepresentative for the conclusion.
Sequence becomes causation.
B happened after A, therefore A caused B.
Replace the argument with the person.
A trait of the speaker is treated as if it settled the proposition.
Retreat to the easy claim.
A strong controversial position is defended by shifting to a weaker defensible one, then later restoring the stronger claim.
Proof is disciplined compression.
A proof shows why a conclusion follows, not merely that it is true. Different strategies expose different structures.
Direct proof
Assume the stated premises or definition and derive the conclusion through valid steps. Best when the structure already points toward the target.
Proof by contradiction
Assume the negation of the conclusion, derive an impossibility, and conclude that the negation cannot hold.
Contrapositive
To prove P → Q, prove the logically equivalent ¬Q → ¬P. Often simpler when negating Q exposes useful structure.
Mathematical induction
Establish a base case, then prove that truth at one stage carries to the next. The chain then covers the intended natural-number domain.
Counterexample
To refute a universal claim, one valid counterexample is enough.
Introduction to LogicIrving Copi et al. · classical argument analysis
forall x: CalgaryOpen formal logic textbook
How to Prove ItDaniel Velleman · proof technique
The Logic BookBergmann, Moor & Nelson · formal systems