Thirteen lessons have built a test that an argument either passes or fails, and passing it is worth much less than it looks.
The first lesson separated validity from truth and promised to come back to the separation. Here it is. A valid argument guarantees only that truth is transmitted, so from false premises it guarantees nothing at all, and the machinery has no opinion whatever about premises. Worse, the step from English into the notation is a judgement that no procedure makes, and most real disputes are decided there. This lesson is about the boundary of the subject: what the tools cannot do, what fails outside their reach, and what the training is nevertheless for.
Formalisation is a judgement
Every method in this course starts from a formula, and nothing in the course produces one. Turning "the contract is void if either party lacked capacity" into a formula required deciding that void is the negation of valid, that "either party lacked capacity" is one predicate rather than two, and that the "if" is material. Each decision is defensible and each could go differently, and the verdict on the argument can turn on any of them.
This is not a gap waiting to be filled by better software. English carries tense, causation, modality, presupposition, and speaker intention, and the notation has none of those. When "she resigned because the audit failed" becomes a conjunction, the causal claim, which was the whole point of the sentence, is simply gone. A formal argument can therefore come out valid while the English it was drawn from is worthless, and the fault lies in the translation, where no rule was broken because there are no rules.
The practical consequence is that formalising is where the care belongs. Fixing the atoms, choosing a domain, deciding whether "or" is exclusive and whether "all" carries existential import: those choices should be written down and defended, since anything hidden there cannot be caught later.
Equivocation, and what a formula exposes
The classic failure of translation is equivocation: the same word doing two jobs, so a single letter stands for two claims.
Nothing is better than eternal happiness. A ham sandwich is better than nothing. Therefore a ham sandwich is better than eternal happiness. The argument looks like a chain of comparisons and the conclusion is absurd, and the notation says exactly where the fault is. The first premise is a quantified claim: , nothing at all is better than eternal happiness. The second treats "nothing" as though it were a name, , when what it actually says is that having a sandwich beats having nothing, a comparison between two options rather than a claim about every object. So the two premises never share a term, no chain exists, and the argument cannot be written down in any way that makes it look valid.
That is the most useful thing formalisation does. It does not merely test arguments; it forces a shared vocabulary onto them, and a great many bad arguments cannot survive being written out with their terms fixed.
Example. "The law says all men are equal. Sarah is not a man. So the law does not apply to her." Where does it fail?
On equivocation. "Man" in the first premise means human being, and in the second it means male, so a single predicate letter cannot serve both without misrepresenting one of them. Write for human and for male, and the premises become and , which share no term and yield nothing. Formalising with two letters instead of one both diagnoses the error and shows that the argument has no valid reading.
Now you. "A feather is light. What is light cannot be dark. So a feather cannot be dark." Where does it fail?
Answer
The same way: "light" means low in weight in the first premise and bright in the second. Two predicates are needed, for lightweight and for bright, and with them the premises are and , which do not connect. There is no valid argument here to salvage.
Valid and worthless
Two ways for a valid argument to be useless are worth separating.
The first is unsoundness: a false premise. "All metals conduct, graphite is a metal, so graphite conducts" was valid in the first lesson and rested on a false premise, and its conclusion happens to be true, which is luck rather than proof. Nothing in this course examines premises, so validity is a certificate about the connection and never about the content.
The second is circularity. Begging the question assumes what it sets out to prove, and the striking fact is that such an argument is valid: is derivable in one line, and any argument with its conclusion among its premises passes every test in this course. Yet nobody is persuaded by "capital punishment is wrong because it is immoral to execute people", and rightly so. What is wrong with it is not logical but dialectical: an argument is meant to move someone from premises they accept to a conclusion they do not, and a circular one has nothing to move with. Validity was never a measure of persuasive force, and this is the sharpest demonstration of the gap.
Both failures point the same way. Once an argument is valid, all the remaining work is on its premises, and that work belongs to whatever field the premises are about.
Example. "Every drug that passed a randomised trial is safe. This drug passed a randomised trial. So it is safe." Where should an objection go?
Not at the form, which is the syllogism of the eleventh lesson and impeccable. The first premise is false as stated, since trials are powered to detect common harms and routinely miss rare ones, which is why withdrawals happen after approval. An objection aimed at the reasoning would be answered by writing out the derivation; an objection aimed at the premise cannot be, and it is the one that wins.
Now you. "If the policy worked, unemployment would have fallen. Unemployment fell. So the policy worked." Where should an objection go?
Answer
At the form. This is affirming the consequent, and it stays invalid however true the premises are, since unemployment falls for many reasons. Here the right move is the parallel-form refutation of the first lesson rather than a dispute about the figures, and noticing which of the two situations you are in is the practical payoff of the whole subject.
Fallacies that have no form
The two fallacies named in the first lesson, affirming the consequent and denying the antecedent, are formal: they are bad shapes, and the shape is enough to condemn any instance. Most fallacies people actually meet are not like that.
Ad hominem attacks the arguer rather than the argument. Straw man refutes a weakened version of the opponent's claim. False dilemma presents two options as exhaustive when they are not, which is a claim about the world rather than a mistake in reasoning about the disjunction. Appeal to authority cites someone whose expertise does not cover the claim. Slippery slope asserts a chain of consequences without supporting the links.
Each of these can be written as a valid argument with a suppressed premise, and that is the useful way to see them. A false dilemma is a perfectly good disjunctive syllogism whose disjunctive premise is false. A slippery slope is a chain of conditionals, valid by hypothetical syllogism, with unsupported links. An appeal to authority is a valid modus ponens whose conditional premise, that this person's endorsement makes it true, is what needs defending. The label names which premise to attack, and calling something a fallacy without saying which premise fails is empty.
Example. "Either we cut the budget or the department closes. We will not cut the budget. So the department closes." Is this valid, and is it good?
Valid: it is disjunctive syllogism. Whether it is good depends entirely on the first premise, and the phrasing invites a false dilemma, since raising revenue, merging departments and deferring the decision are all missing. Attacking the reasoning would be a waste of effort; the argument's whole weight is on a premise that has been asserted rather than shown.
Now you. "Every economist who has looked at this says the policy will work, so it will work." Reconstruct it as a valid argument and say which premise carries the risk.
Answer
Suppressed premise: if every economist who has examined a policy says it will work, it will work. With that, the argument is modus ponens and valid. The suppressed conditional is where the risk sits, since it ignores selection in who examined it and the record of expert consensus in that field. The reconstruction turns a vague appeal to authority into a specific claim that can be argued about.
Reconstructing real arguments
Almost no real argument states all its premises. An argument with a suppressed premise is an enthymeme, and reconstructing one is a routine part of using logic outside a textbook.
The rule is charity: supply the premise that makes the argument valid and is most plausible, rather than the one easiest to demolish. "Socrates is a man, so he is mortal" needs "all men are mortal", not "everything named Socrates is mortal", and reading it the second way would be a cheap victory over nobody.
Charity has a limit, though, and it is where the technique earns its keep. If the only premise that makes an argument valid is one nobody would accept once it is stated, the reconstruction has not been unfair; it has exposed what the argument was relying on while it stayed unstated. That is the whole value of writing suppressed premises down.
The limits inside logic itself
Even where the tools apply cleanly, three results mark the boundary, and all three are from the twentieth century.
First-order validity is undecidable, by Church and Turing in 1936. There is no procedure that answers every question of the form "is this valid", though proofs of valid arguments can always be found eventually.
Arithmetic is incomplete, by Gödel in 1931. Any consistent, effectively axiomatised system strong enough to express arithmetic contains true statements about the natural numbers that it cannot prove, and it cannot prove its own consistency. This does not say logic is broken or that truth is subjective; it says that no fixed list of axioms captures every arithmetical truth, which is a limit on axiom systems and a precise one.
Truth is not definable in the language it is about, by Tarski in 1933. A language rich enough to talk about arithmetic cannot contain its own truth predicate, on pain of the liar sentence, "this sentence is false", which is true if false and false if true. Tarski's response, that truth for a language is defined in a stronger metalanguage, is the reason logicians are careful about which language a claim is being made in. Russell's paradox of 1901, about the set of all sets that are not members of themselves, forced the same kind of restriction on set theory.
Logics that give something up
Classical logic is a choice, and its rivals are not confusions. Each drops one thing to buy another.
Intuitionistic logic drops double negation elimination, so excluded middle is not provable and a proof of existence must produce a witness. It is the logic of proof assistants for exactly that reason. Relevance logics drop ex falso quodlibet, so a contradiction no longer entails everything, which is useful when reasoning from databases known to contain some inconsistency. Many-valued and fuzzy logics drop bivalence, admitting degrees or a third value, for vagueness and for cases where truth is genuinely undefined. Modal logics add operators for necessity and possibility, and their conditionals can capture what the material one could not.
The existence of alternatives does not mean anything goes. Each is a precisely specified system with its own soundness and completeness results, and choosing between them is choosing which inferences you are prepared to license.
What the training is for
The honest summary is that this subject provides one thing completely and nothing else. It settles, exactly and permanently, whether a conclusion follows from stated premises. It has nothing to say about whether the premises are true, whether the formalisation was faithful, whether the argument is worth making, or whether the person making it is trustworthy.
That is still a great deal. Knowing that validity and truth are independent stops the two most common errors in public argument: taking a true conclusion as evidence of good reasoning, and taking bad reasoning as evidence of a false conclusion. Knowing the difference between a conditional and its converse is most of what is needed to read a diagnostic test or a piece of legislation. Being able to negate a quantified claim tells you what evidence would actually settle a dispute, which is the most useful single habit in the course.
And the discipline of writing an argument out until its suppressed premises are visible is what turns a disagreement about reasoning, which logic can settle, into a disagreement about facts, which the world can settle. Most arguments were always about the second kind, and the point of fourteen lessons on form is to find out which kind you are in.