Arguments & Validity

This is Unit One of Logic: Arguments and Validity. Logic asks one question about a piece of reasoning, and it is narrower than people expect. Not whether the claims are true. Whether the conclusion actually follows from what came before it. That narrowness is the whole discipline. It means an argument built entirely from falsehoods can be flawless, and an argument in which every sentence is true can be worthless. Those two facts are where most people get lost, and this unit spends real time on both. The unit builds four moves: recognising an argument when you see one, laying it out so its parts are visible, mapping how the support is arranged, and naming what kind of guarantee is being offered. Everything later in the course is machinery for testing those claims. By the end you will be able to do five things. One: tell an argument from a report, from an explanation, and from an opinion with nothing behind it. Two: put a passage into standard form and say what each line is doing there. Three: say what validity is, and why every premise being false does not touch it. Four: keep truth, validity, and soundness apart, and say which of them attaches to what. Five: prove an argument invalid by building a counterexample that shares its form.

Start with what the subject is for, and then with the only material it can operate on. Logic works on sentences that can be true or false. A great deal of ordinary language cannot be, and the first job is telling the two apart.

What the subject is.

Logic 001 What Logic Is

Take the classic three lines about Socrates being mortal. The interesting question is not whether he existed — that belongs to history. It is why the third line cannot escape the first two.

The material it works on.

Logic 002 Statements

Name Illinois's capital correctly and you have said something true. Name it wrongly and you have said something false. Both count, because being wrong is not the same as saying nothing.

And what it cannot work on.

Logic 003 Non-Statements

Shut the door is an instruction, not a claim, so nothing about it is true or false. Watch the rhetorical question, though — it usually has a claim hidden inside it.

Now the object the whole course is about. An argument is not a quarrel. It is a set of statements where some are being offered to make you accept another, and it has exactly two roles inside it — plus a small vocabulary that flags which is which.

The object itself.

Logic 004 Arguments

Icy roads, therefore a delayed opening. One reason, one claim, and that is already a complete argument.

The first role.

Logic 005 Premises

In the argument about trout and gills, the opening two lines are the support. Note that support does not have to be correct to count as support — bad evidence is still evidence offered.

The second role.

Logic 006 Conclusions

It is the line the whole thing exists to get you to. And it can sit at the start of a passage, at the end, or buried in the middle — where it sits tells you nothing.

The words that flag the roles.

Logic 007 Indicator Words

Some words announce a conclusion and others announce support. But the same words also mark time and cause, so the test is what the word is actually doing in that sentence.

Most prose is not argument, and two kinds of passage fool people especially reliably. One states things without offering any reason to believe them. The other uses the exact same connecting words for a completely different job.

The first look-alike.

Logic 008 Arguments vs. Non-Arguments

A market report telling you the index dropped three hundred points is not arguing for anything. Nor is a firmly held view, until reasons get attached to it.

The second is subtler.

Logic 009 Explanations vs. Arguments

Corroded bolts brought the bridge down. Nobody is disputing that the bridge came down, so nothing is being argued — the sentence accounts for a fact rather than establishing one.

One small piece of formatting, and it earns its place because of what it exposes. Prose lets an argument hide its gaps behind flow. Stacking the lines up strips the flow away.

The layout.

Logic 010 Standard Form

Each supporting line gets its own numbered row, and the claim sits underneath a rule. Arguments that read smoothly often stop looking convincing once they are stacked up like this.

Here is the idea the rest of the course rests on, and it is worth slowing down for. Validity is a claim about impossibility — about what the premises rule out — and it says nothing whatsoever about whether those premises are actually the case.

The definition.

Logic 011 Validity

Rex the dog is a mammal because every dog is. There is no way to arrange the world so that both opening lines hold and the closing one fails.

Its negation.

Logic 012 Invalidity

Turn it around: Rex is a mammal, and every dog is a mammal, so Rex is a dog. Rex could be a cat. One available scenario is all it takes.

And the point most people lose.

Logic 013 Validity Is About Form, Not Truth

Flying fish, whales among them: not one sentence there is true, and the reasoning is perfect. This is the confusion that costs students the most, so sit with it.

Two combinations follow from that, and both feel wrong the first time. An argument can be flawless and ridiculous. An argument can be composed entirely of true sentences and still be broken. Neither is a paradox — they follow directly from what validity was defined to mean.

The first combination.

Logic 014 Valid with False Premises

Cube-shaped planets, Mars among them, therefore a cube-shaped Mars. Nothing is wrong with the reasoning; the promise was only ever conditional on the premises holding.

The second is the mirror.

Logic 015 Invalid with All True Statements

Paris sits in France, grass is green, and water is wet. Every sentence checks out and none of them has anything to do with the others. A right answer is not evidence of good reasoning.

Validity was only ever half the job, because it never asked whether the premises were true. The word that asks for both is soundness, and the topic closes by sorting out which of these three words attaches to which kind of thing.

The property that adds the missing half.

Logic 016 Soundness

The Socrates argument has the form and the facts together, which is why the conclusion is not merely guaranteed conditionally but actually delivered.

The relation between the two.

Logic 017 Valid vs. Sound

Every argument with the full property has the partial one; the reverse fails constantly, as the cube-shaped Mars showed. And "valid point" in ordinary speech means something else entirely — never import it here.

Which word goes with what.

Logic 028 Truth, Validity, Soundness: The Map

Sentences get called true. Forms get called valid. Whole arguments get called sound. Policing that in your own speech is the quickest route to actually holding the distinctions.

Not every argument claims its conclusion is guaranteed. Some only claim it is likely, and those are not failed attempts at guarantees — they are a different enterprise, judged by a measure that comes in degrees rather than in a yes or a no.

The two claims.

Logic 018 Deductive vs. Inductive

Every raven anyone has looked at was black, so expect the next one to be. That is a good argument that could still turn out wrong, and it is not trying to be anything else.

The measure for the second kind.

Logic 019 Inductive Strength

Ten thousand observations behind a prediction, or three: identical shape, entirely different force. That difference is a matter of degree.

Its version of the full property.

Logic 020 Cogency

A poll drawn from a rigged sample can be impeccably structured and still tell you nothing, because the structure was never the part that was in doubt.

And how to tell which you have.

Logic 029 Deciding Deductive vs. Inductive

Words like "must" and "certainly" announce one kind; "likely" and "probably" announce the other. Where nothing announces anything, read it in its strongest form.

The unit's first genuine technique, and it comes with an asymmetry worth naming up front: it can settle the question one way and never the other.

The idea.

Logic 021 The Counterexample Method

Keep the skeleton, swap in dogs and cats, and the conclusion comes out plainly wrong while the openings stay plainly right. That settles it. Finding nothing, though, settles nothing — you may only have run out of ideas.

Watch it run, and watch the trap.

Logic 022 Testing Validity by Counterexample

Strip the argument to letters, then rebuild it with content that makes the openings true and the ending false. The trap is quietly changing the shape while you change the subject — then you have tested a different argument.

Almost no real argument states all of itself. Something is always left for the listener to fill in, and there is a discipline about how to fill it in that separates genuine criticism from the appearance of it.

The gap.

Logic 023 Enthymemes

He is a politician, so expect disappointment — which only works if all of them disappoint. That unsaid line is usually the shakiest part, which is precisely why it went unsaid.

The discipline for filling it.

Logic 024 The Principle of Charity

If the gap could be filled with "all of them" or with "most of them," take the second. Demolishing the weaker reading demolishes nothing at all, however satisfying it feels.

Real arguments are rarely one premise and one conclusion. Support arranges itself in three different ways, and knowing which one you are looking at changes what happens when you attack a single line.

The three arrangements.

Logic 025 Argument Structure: Serial, Convergent, Linked

A fever and a rash each point at illness on their own. The trout premises do not — either one alone is useless. Remove a line from the first kind and the rest holds; remove one from the second and it all goes.

Watch a passage get mapped.

Logic 026 Diagramming an Argument

Number every sentence, then draw the arrows showing which supports which. The care goes into telling the separate kind of support from the joint kind, because on the page they look identical.

The statement that does two jobs.

Logic 027 Complex Passages: Sub-Conclusions

Rates went up, so borrowing fell; borrowing fell, so growth will too. The middle claim gets established and then immediately put to work. Miss that and the map turns to mush.

The unit closes on one real editorial, run through every tool at once — and then a single entry naming what you can now do, which is more than it may feel like.

The capstone. A library, and its shortening hours.

Logic 030 Capstone: Full Passage Analysis

Number the sentences, strip the decoration, find the line nobody said, read it charitably, draw the arrows, name what kind of argument it is, and deliver a verdict. Watch the sentence that sounds like a reason and is only feeling.

And what you have now.

Logic 031 Unit 1 Recap: The Skeleton

Four moves: find the argument, lay it out, map it, and name what it is claiming. Everything ahead is apparatus for testing those claims.

Five things you can now do. You can tell an argument from a report, from an explanation, and from an opinion with nothing attached — and the explanation test is the useful one: ask whether anyone is actually disputing the point. You can put a passage into standard form, name each line's role, and read the indicator words for what they are doing rather than for what they usually mean. You can say what validity is: the premises leave no room for the conclusion to fail. Every sentence in the argument can be false and that stays true. You can keep the three words apart — sentences are true, forms are valid, whole arguments are sound — and you know that valid is only half of sound. And you can prove an argument invalid by building one with its exact shape, true throughout, that lands on something false. Failing to build one proves nothing.