Rules, forms & definitions reference

An argument is invalid exactly when there is at least one possible situation in which every premise is true and the conclusion is false. Such a situation is called a counterexample to the argument. An argument is valid exactly when no counterexample exists.

What is this saying? To show an argument is invalid, you don't need to show its conclusion is false. You only need to describe one situation, real or imagined, where the premises all hold and the conclusion still fails. One such situation settles it.

Why should I care? "It doesn't follow" is a claim people make and dispute all the time. This test turns it into something you can demonstrate: produce the counterexample and the argument is finished, however true its conclusion happens to be.

Intuition. Validity is a promise: whenever the premises are true, the conclusion is true too. A promise about every situation is broken by a single situation where it fails, the same way one black swan breaks "all swans are white."

Ingredients.

• Possible situation: any consistent way the world could be, not only the way it happens to be.
• Every premise true: all of them at once, in that same situation.
• Conclusion false: in that same situation, not in some other one.

Example. "Every spider has eight legs. Charlotte has eight legs. So Charlotte is a spider." Counterexample: Charlotte is a scorpion. Both premises are true in that situation and the conclusion is false, so the argument is invalid, even if the real Charlotte turns out to be a spider.

What goes wrong without "every premise true." A situation where the conclusion is false but some premise is false too proves nothing. A valid argument can have a false conclusion when a premise is false as well.

Call a form valid when none of its substitution instances can have all true premises and a false conclusion. Then:

1. Every substitution instance of a valid form is a valid argument.
2. If even one substitution instance of a form has all true premises and a false conclusion, the form is invalid.

What is this saying? The letters in a form are blanks, and an argument you get by filling the blanks is an instance of the form. If a pattern of reasoning is good, it is good whatever you fill it with. If one filling of the pattern leads from truths to a falsehood, the pattern itself is broken.

Why should I care? This is why logic can study patterns instead of topics. Show a form is valid once, and every argument with that form is valid, about weather, medicine, or anything else. The reverse doesn't hold: an argument with an invalid form can still be valid because of what its words mean. It also gives a practical way to attack a bad argument: keep its form, swap in content where the premises are plainly true and the conclusion plainly false.

Intuition. A valid form gets its guarantee from its connecting words, such as if … then, or, and, and not, and those are the same in every instance. The topic words only fill the blanks, so they can't switch the guarantee on or off.

Ingredients.

• Form: an argument with each distinct claim replaced by a letter, with the same letter used every time the same claim appears.
• Substitution instance: what you get by putting a claim in for each letter, the same claim for every copy of that letter.
• All true premises, false conclusion: the counterexample combination, here found by choosing the fillings.

Example. "If Paris is in Germany, then Paris is in Europe. Paris is in Europe. So Paris is in Germany." The premises are true and the conclusion is false. By part 2, the form behind it (if P then Q; Q; so P) is invalid, and so no argument is made valid just by having that form.

Where the rule stops. Part 2 condemns the form, not every argument that has it. "If Ann is taller than Ben, then Ben is shorter than Ann. Ben is shorter than Ann. So Ann is taller than Ben." This has the invalid form above, yet it is valid, because of what taller and shorter mean. An invalid form can't prove an argument valid, but some other feature of the argument might.

The form

1. If P, then Q.
2. Q.

\therefore P.

is invalid. It has instances with true premises and a false conclusion.

What is this saying? Knowing that P leads to Q, and that Q happened, does not tell you P happened. Something else may have caused Q.

Why should I care? This is one of the most common reasoning mistakes there is, because it looks almost exactly like modus ponens. "If the engine is flooded, the car won't start. The car won't start. So the engine is flooded." A dead battery also stops the car.

Intuition. "If P, then Q" says P is one way to get Q. It never says P is the only way. The second premise lands on the result, and a result can have many causes.

Ingredients.

• Antecedent: the "if" part, P.
• Consequent: the "then" part, Q.
• Affirming the consequent: the second premise asserts the consequent and the conclusion asserts the antecedent. Modus ponens runs the other way: it asserts the antecedent and concludes the consequent.

Example. "If Paris is in Germany, then Paris is in Europe. Paris is in Europe. So Paris is in Germany." Both premises are true. The conclusion is false.

Non-example. "If Paris is in Germany, then Paris is in Europe. Paris is in Germany. So Paris is in Europe." This is modus ponens, and it is valid. Its first premise is still true, its second is false, and validity only promises a true conclusion when every premise is true.

The form

1. If P, then Q.
2. Not P.

\therefore Not Q.

is invalid. It has instances with true premises and a false conclusion.

What is this saying? Knowing that P leads to Q, and that P did not happen, does not tell you Q didn't happen. Q could still come about some other way.

Why should I care? It sounds like careful reasoning and often slips past unnoticed. "If you study, you'll pass. You didn't study. So you won't pass." Some people pass without studying, and the premises never ruled that out.

Intuition. "If P, then Q" tells you what happens when P is true. It says nothing about what happens when P is false. Denying P walks into territory the conditional never covered.

Ingredients.

• Antecedent: the "if" part, P.
• Consequent: the "then" part, Q.
• Denying the antecedent: the second premise denies the antecedent and the conclusion denies the consequent. Modus tollens runs the other way: it denies the consequent and concludes that the antecedent is false.

Example. "If Paris is in Germany, then Paris is in Europe. Paris is not in Germany. So Paris is not in Europe." Both premises are true. The conclusion is false.

Non-example. "If Paris is in Germany, then Paris is in Europe. Paris is not in Europe. So Paris is not in Germany." This is modus tollens, and it is valid.

The form

1. If P, then Q.
2. If Q, then R.

\therefore If P, then R.

is valid: no instance can have both premises true and the conclusion false.

What is this saying? If one thing leads to a second, and the second leads to a third, then the first leads to the third. You can link conditionals into a chain and skip the middle.

Why should I care? Much everyday reasoning is a chain of "if … then" steps: if the flight is late, I miss the connection; if I miss the connection, I miss the meeting. Hypothetical syllogism is what lets you jump from the first link to the last: if the flight is late, I miss the meeting.

Intuition. Picture falling dominoes. Knocking over P knocks over Q, and knocking over Q knocks over R, so knocking over P knocks over R.

Ingredients.

• P, Q, R: any three claims. Q must be the same claim in both premises; it is the link that joins them.
• If … then: a conditional. The only way "if P, then R" is false is when P is true and R is false.

Example. "If the reservoir drops below the line, the town restricts watering. If the town restricts watering, lawns turn brown. So if the reservoir drops below the line, lawns turn brown."

Look-alike to avoid. "If P, then Q. If Q, then R. So if R, then P" reverses the chain. Brown lawns could come from a heat wave with the reservoir full.

The form

1. P or Q.
2. Not P.

\therefore Q.

is valid: no instance can have both premises true and the conclusion false.

What is this saying? If at least one of two options holds and you've ruled one out, the other must hold.

Why should I care? This is the process of elimination, used in diagnosis, repair, and detective work. It is also the valid form hiding behind the fake either-or mistake of chapter 6: the reasoning is fine, but the first premise wrongly claims there are only two options.

Intuition. "P or Q" says the answer is somewhere in two boxes. Opening one box and finding it empty tells you where the answer is.

Ingredients.

• P or Q: a disjunction, true when at least one part is true (both is allowed). It is false only when both parts are false.
• Not P: the second premise rules out one part. It must deny a part. Asserting a part tells you nothing about the other.

Example. "Either the fuse blew or the bulb burned out. The fuse didn't blow. So the bulb burned out."

Look-alike to avoid. "Either the fuse blew or the bulb burned out. The fuse blew. So the bulb didn't burn out." This is invalid: "or" allows both, so both might have failed. The mistake is called affirming a disjunct.

What breaks without a true first premise. If the options weren't really the only two (maybe the switch is broken), the argument stays valid but becomes unsound. That is exactly how a fake either-or works.

The form

1. P.
2. Q.

\therefore P and Q.

is valid: no instance can have both premises true and the conclusion false.

What is this saying? If two claims are each true, the claim that joins them with "and" is true.

Why should I care? It looks too obvious to name, but longer arguments use it constantly. Before you can use a premise that needs two facts together, you have to be allowed to put them together.

Intuition. "P and Q" asks for nothing beyond P being true and Q being true. The premises supply exactly that.

Ingredients.

• P and Q: a conjunction, true when both parts are true and false as soon as either part is false.

Example. "The contract is signed. The deposit is paid. So the contract is signed and the deposit is paid."

Look-alike to avoid. "P or Q; so P and Q" is invalid. An "or" claim can be true with only one part true.

The form

1. P and Q.

\therefore P.

is valid, and so is the version that concludes Q instead.

What is this saying? If a claim of the form "P and Q" is true, each part of it is true on its own.

Why should I care? Premises often bundle several facts into one sentence. Simplification lets you pull out just the fact you need.

Intuition. An "and" claim is a package deal. If the package is true, everything in it is true.

Ingredients.

• P and Q: a conjunction, true only when both parts are true.
• The main "and": the rule applies when "and" joins the two halves of the whole premise. From "either P and Q, or R," you can't take out P, because that "and" sits inside an "or."

Example. "The store is open late and parking is free. So parking is free."

Look-alike to avoid. "P or Q; so P" is invalid. The "or" claim could be true because of Q alone.

The form

1. P.

\therefore P or Q.

is valid for any claim Q whatsoever.

What is this saying? If a claim is true, then the claim "that, or anything else" is true too.

Why should I care? Addition feels like cheating, because Q can be anything, even something false or off-topic. Seeing why it is still valid is a good test of whether you've understood that logic's "or" is inclusive. Addition also has real uses: it builds the "or" claim a later step needs, such as the "either this or that" premise of an argument that works through both cases.

Intuition. "P or Q" is a weaker claim than P. It only says at least one of the two is true. Once you know P, that is already guaranteed, so saying less can't make you wrong.

Ingredients.

• P: the premise, any claim.
• Q: any claim at all. It does not need to be true, related, or mentioned anywhere else.
• P or Q: a disjunction, true when at least one part is true.

Example. "The meeting is on Monday. So the meeting is on Monday or the moon is made of cheese." The conclusion is true, because its first part is true. It is an odd thing to say, but it is not false.

Look-alike to avoid. "P or Q; so P" runs addition backward and is invalid.

The form

1. If P, then Q.
2. If R, then S.
3. P or R.

\therefore Q or S.

is valid: no instance can have all three premises true and the conclusion false.

What is this saying? You have two conditionals and know that at least one of their "if" parts holds. Then at least one of their "then" parts holds too.

Why should I care? It is the shape of every "either way" argument. "If I take the bus, I'll be late. If I drive, I'll pay for parking. I'll either take the bus or drive. So either I'll be late or I'll pay for parking." It is also modus ponens run on two tracks at once.

Intuition. Whichever of P or R is true, its own conditional fires. If P, you get Q. If R, you get S. Either way you end up with Q or S.

Ingredients.

• P, Q, R, S: four claims. Each conditional pairs one condition with one result.
• P or R: the third premise asserts the two conditions, the "if" parts, not the results.
• Or: inclusive. The conclusion is true when at least one of Q and S is true.

Example. "If the package went by air, it arrives Tuesday. If it went by ground, it arrives Friday. It went by air or by ground. So it arrives Tuesday or Friday."

Look-alike to avoid. If the third premise asserts the results ("Q or S") and the conclusion claims the conditions ("P or R"), the argument is invalid. That is affirming the consequent on two tracks.

The form

1. If P, then Q.
2. If R, then S.
3. Not Q or not S.

\therefore Not P or not R.

is valid: no instance can have all three premises true and the conclusion false.

What is this saying? You have two conditionals and know that at least one of their "then" parts fails. Then at least one of their "if" parts fails too.

Why should I care? It is modus tollens run on two tracks. "If the suspect was at the party, the guests saw him. If he was at the office, the cameras recorded him. Either the guests didn't see him or the cameras didn't record him. So either he wasn't at the party or he wasn't at the office."

Intuition. Whichever result fails, its own conditional runs backward. If Q fails, P must fail. If S fails, R must fail. Either way, one of the conditions fails.

Ingredients.

• P, Q, R, S: four claims. Each conditional pairs one condition with one result.
• Not Q or not S: the third premise denies the results, the "then" parts.
• Or: inclusive. The conclusion is true when at least one of "not P" and "not R" is true.

Example. "If the recipe is right, the bread rises. If the oven is hot enough, the crust browns. Either the bread didn't rise or the crust didn't brown. So either the recipe is wrong or the oven isn't hot enough."

Look-alike to avoid. Denying the conditions ("not P or not R") and concluding that a result fails ("not Q or not S") is invalid. That is denying the antecedent on two tracks.

Take the four categorical statements with the same subject term S and predicate term P:

• A: All S are P.
• E: No S are P.
• I: Some S are P.
• O: Some S are not P.

On the Boolean reading, A and O are contradictories, and E and I are contradictories: in each pair exactly one statement is true. No other pair of the four has a fixed logical relation.

What is this saying? "All S are P" and "Some S are not P" can never both be true or both be false; the same goes for "No S are P" and "Some S are P." Every other link you might expect, such as getting "some" from "all," can fail.

Why should I care? The square tells you what you may infer from one categorical statement about the other three. Knowing that only the diagonals are safe keeps you from drawing conclusions about a class that might have no members.

Intuition. Each diagonal pair talks about the same region of a Venn diagram. The A statement says the region "S but not P" is empty; the O statement says that same region has something in it. A region is either empty or not, so one of the two is right and the other is wrong.

Ingredients.

• S, P: the subject and predicate classes. They may have members or may be empty.
• Contradictories: two statements that always have opposite truth values.
• Boolean reading: "all" and "no" statements make no claim that anything exists; "some" means at least one thing exists.

Example. "All phones in the box are charged" and "Some phones in the box are not charged": whatever is in the box, one is true and the other false.

What breaks without the Boolean reading's care about empty classes. Let S be the class of unicorns. "All unicorns are white" and "No unicorns are white" are both true (there are no unicorns to break either), so A and E are not contraries. "Some unicorns are white" and "Some unicorns are not white" are both false, so I and O are not subcontraries. And A is true while I is false, so the step from "all" to "some" (subalternation) fails. The traditional square assumed every subject class has members; the Boolean square does not.

On the Boolean reading:

For example, "No S are P" says the same as "No P are S." "All S are P" says the same as "No S are non-P." And "All S are P" says the same as "All non-P are non-S."

What is this saying? Each of the three ways of rewriting a categorical statement is safe for some statement types and not others. Obversion is always safe. Conversion is safe only for "no" and "some … are" statements. Contraposition is safe only for "all" and "some … are not" statements.

Why should I care? Real arguments rarely state their premises in the shape a syllogism needs. These operations let you rewrite a statement into the shape you need, as long as you use one that keeps its meaning.

Intuition. A categorical statement is a claim that one region of the Venn diagram is empty or occupied. An operation is safe exactly when the new statement is a claim about the same region.

Ingredients.

• S, P: the subject and predicate terms.
• Complement (non-S, non-P): everything not in that class.
• Quality: affirmative (A, I) or negative (E, O).
• Equivalent: true in the same situations, and false in the same ones.

Example. "No reptiles are mammals" converts to "No mammals are reptiles." But "All cats are mammals" does not convert: "All mammals are cats" is false.

What breaks. Converting an A statement or an O statement, or contraposing an E or an I statement, can take a true statement to a false one.

A standard-form categorical syllogism is valid on the Boolean reading if and only if it satisfies all five rules:

1. The middle term is distributed in at least one premise.
2. Any term distributed in the conclusion is distributed in the premise that contains it.
3. The premises are not both negative.
4. A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.
5. If both premises are universal, the conclusion is not particular.

Breaking rule 1 is the undistributed middle; breaking rule 2 is illicit major or illicit minor; rule 3 is exclusive premises; rule 5 is the existential fallacy. Breaking rule 4 is an affirmative conclusion from a negative premise (or a negative conclusion from affirmative premises).

What is this saying? You can check a syllogism without drawing anything: look at which terms each statement is about "all of," count the negatives, and check whether the conclusion claims existence that the premises don't. Pass all five checks and the syllogism is valid; fail any one and it is invalid.

Why should I care? The rules are faster than a Venn diagram and they name exactly what is wrong with a bad syllogism, which is what you need to explain or repair it.

Intuition. The middle term is the bridge between the other two terms; if neither premise talks about the whole middle class, the two premises may be talking about different parts of it and never meet. And a conclusion can't say more about a class, or claim more existence, than the premises gave it.

Ingredients.

• Distributed term: one the statement says something about every member of. A distributes its subject; E distributes both terms; I distributes neither; O distributes its predicate.
• Middle term: in both premises, not in the conclusion.
• Negative: an E or O statement. Universal: A or E. Particular: I or O.

Example. "All dogs are mammals. All cats are mammals. So all cats are dogs." The middle term "mammals" is the predicate of two A statements, so it is never distributed. Rule 1 fails: undistributed middle.

What breaks without rule 5. "All unicorns are horses. All horses are animals. So some unicorns are animals." Both premises are true (there are no unicorns to break the first), but the conclusion claims a unicorn exists.

On the Boolean reading, exactly fifteen of the 256 standard-form categorical syllogism forms are valid:

What is this saying? A syllogism's form is fixed by its mood (the types of its three statements) and its figure (where the middle term sits). Of all possible forms, only these fifteen guarantee their conclusion, and every one of them passes all five syllogism rules.

Why should I care? If you can name a syllogism's mood and figure, you can look up whether it is valid. The list is also a cross-check: a Venn diagram test or a rules test that disagrees with the table has a mistake in it.

Intuition. Most combinations break a rule, often by leaving the middle term undistributed or by drawing a "some" conclusion from "all" premises. The fifteen survivors are the combinations that break none.

Ingredients.

• Mood: three letters for the major premise, minor premise, and conclusion, in that order.
• Figure: 1, middle term is the subject of the major premise and the predicate of the minor; 2, predicate of both; 3, subject of both; 4, predicate of the major and subject of the minor.
• Boolean reading: no "all" or "no" statement asserts that its subject class has members.

Example. "All poets are dreamers. Some teachers are poets. So some teachers are dreamers" is AII in figure 1, in the table, so it is valid.

What breaks under the Boolean reading. The traditional list has nine more forms (such as AAI in figure 1) that are valid only if some class is assumed to have members. On the Boolean reading each of them commits the existential fallacy.

What is this saying? These four rows are the complete definition of the five connectives. "Not" flips a value. "And" is true only when both parts are. "Or" is false only when both parts are. "If … then" is false only when the first part is true and the second false. "If and only if" is true when the two parts match.

Why should I care? Every truth-table calculation in the course, and every rule in the proof system, comes back to these rows. If you know them cold, you can work out the value of any sentence on any row.

Intuition. Read each column as the one situation that breaks the claim. A conjunction breaks when any part fails; a disjunction breaks only when every part fails; a conditional breaks only when its promise is tested and not kept.

Ingredients.

• A, B: any sentences, atomic or compound.
• T, F: true and false.
• \neg (not), \wedge (and), \vee (inclusive or), \to (if … then), \leftrightarrow (if and only if).

Example. With A false and B true (row 3): \neg A is T, A \wedge B is F, A \vee B is T, A \to B is T, and A \leftrightarrow B is F.

What to watch. \vee is inclusive: row 1 is T. \to is not symmetric: rows 2 and 3 differ.

Suppose "if A then B" is truth-functional, is false when A is true and B false, and is true when both are true. Then its only reasonable table is the material conditional:

Every other way of filling the last two rows turns "if A then B" into a sentence that says something different.

What is this saying? Once you agree on the first two rows, which nearly everyone does, the rows where A is false have to be true. Any other choice makes "if" mean "and," or "if and only if," or makes the "if" part irrelevant.

Why should I care? The rows where a false antecedent makes the conditional true are the part of the table students find strangest. This argument shows they aren't arbitrary: they are what is left when the alternatives are ruled out.

Intuition. A conditional is a promise about what happens when A is true. When A is false the promise was never tested, so it hasn't been broken.

Ingredients.

• Truth-functional: the value of "if A then B" depends only on the values of A and B.
• Rows 1 and 2: the promise kept (T) and the promise broken (F).
• Rows 3 and 4: the rows with a false antecedent, the ones in question.

Example. "If you score over 90, you get an A." A student who scores 85 and gets an A hasn't shown the promise false, and neither has a student who scores 85 and gets a B.

Where the fit is poor. Because the table depends only on truth values, "If the moon is cheese, then 2 + 2 = 4" comes out true. English "if" often suggests a connection between the parts that the table doesn't record.

A complete truth table for sentences built from n distinct sentence letters has 2^n rows, one for each valuation.

What is this saying? Each new letter doubles the number of rows: 1 letter gives 2 rows, 2 letters give 4, 3 give 8, 4 give 16, and so on.

Why should I care? It tells you how big a table you're about to build, and it's why tables stop being practical: 6 letters already need 64 rows. That growth is the reason the course develops faster tests and formal proofs.

Intuition. Each letter can be T or F independently of the others, so every choice for one letter pairs with every combination for the rest.

Ingredients.

• n: the number of different sentence letters, counting each letter once however many times it appears.
• Valuation: one assignment of T or F to every letter; one row.

Example. (P \wedge Q) \to (R \vee P) has three distinct letters, P, Q, and R, so its table has 2^3 = 8 rows.

Standard order. List the rows so the rightmost letter alternates every row, the next letter every 2 rows, the next every 4, and the leftmost letter is T for the top half and F for the bottom half. Then no row is missed or repeated.

Every sentence of sentential logic is exactly one of:

• a tautology: true on every row of its truth table;
• a contradiction: false on every row;
• a contingency: true on at least one row and false on at least one.

Moreover, A is a tautology if and only if \neg A is a contradiction, and A is contingent if and only if \neg A is contingent.

What is this saying? Look down the main connective's column. All T, all F, or a mix: those are the only possibilities, and they don't overlap. Negating a sentence swaps "always true" and "always false" and leaves "it depends" alone.

Why should I care? A tautology is true because of its form alone, so it tells you nothing about the world. A contradiction can't be true however the world is. A contingency is the kind of sentence whose truth you have to go and check.

Intuition. A column of T's and F's has only three shapes: no F's, no T's, or some of each.

Ingredients.

• Row: one valuation of the sentence letters.
• \neg: negation, which flips every entry in the column.

Example. P \vee \neg P is a tautology, P \wedge \neg P is a contradiction, and P \to Q is a contingency (false only when P is T and Q is F).

What breaks if you check only some rows. A sentence that is true on every row you happened to check might still be false on another, so it may be contingent rather than a tautology.

For every sentence A, A \vee \neg A is a tautology.

What is this saying? Any sentence is either true or false; there is no third, middle option. So "A or not A" can't come out false.

Why should I care? It is the simplest tautology and a building block for longer arguments: reasoning by cases often starts by saying "either this is true or it isn't." In chapter 17 you'll also prove it from no premises at all.

Intuition. Whichever value A has, one of the two disjuncts is true, and a disjunction needs only one.

Ingredients.

• A: any sentence, atomic or compound.
• \vee, \neg: inclusive or, and not.

Example. "The store is open or it isn't open" is true whatever the store's hours are.

Not to be confused with. Excluded middle says a sentence OR its negation is true. It does not say you know which. "Either the meeting is on Tuesday or it isn't" is true even if you have no idea when the meeting is.

For every sentence A, \neg(A \wedge \neg A) is a tautology; equivalently, A \wedge \neg A is a contradiction.

What is this saying? A sentence and its negation are never true together, so "A and not A" is false on every row, and its negation is true on every row.

Why should I care? Much of logic runs on this. A set of sentences that contains both A and \neg A can't all be true, and deriving such a pair in a proof shows that an assumption has to go.

Intuition. \neg A has the opposite value of A on every row, so the two are never both T.

Ingredients.

• A: any sentence.
• \wedge, \neg: and, and not.

Example. "The light is on and the light is not on" is false on every reading, whatever the light is doing.

Not to be confused with. Non-contradiction is about a sentence and its exact negation. "The room is warm" and "the room is cold" can both be false (a mild room is neither), because neither sentence is the negation of the other.

Sentences A_1, A_2, \dots, A_n are jointly consistent if and only if the conjunction A_1 \wedge A_2 \wedge \dots \wedge A_n is not a contradiction. They are inconsistent if and only if that conjunction is a contradiction.

What is this saying? Asking whether several sentences can be true together is the same as asking whether "all of them" is ever true. If some row makes the big conjunction true, that row makes every sentence true.

Why should I care? It turns a question about a set of sentences into a question about one sentence, which a truth table answers directly. It also explains why inconsistent premises behave oddly in arguments (chapter 15).

Intuition. A conjunction is true exactly on the rows where every conjunct is true.

Ingredients.

• A_1, \dots, A_n: any sentences; the dots mean "and so on up to A_n."
• Consistent: some one row makes every A_i true.
• Contradiction: false on every row.

Example. P \vee Q, \neg P, and \neg Q: their conjunction is false on every row, so the three are inconsistent, even though every pair of them is consistent.

What breaks if you check pairs. Consistency is about one row that works for all the sentences at once. Showing that each pair can be true together is not enough, as the example shows.

Two sentences A and B are logically equivalent, written A \equiv B, exactly when they have the same truth value on every row. This holds if and only if A \leftrightarrow B is a tautology.

What is this saying? Equivalent sentences are two ways of saying the same thing: no possible situation makes one true and the other false. Since \leftrightarrow is true just when its two sides match, "they always match" is the same as "their biconditional is always true."

Why should I care? It lets you check an equivalence with the tools you already have: build one table and look at one column. It also explains why equivalent sentences can replace each other in reasoning.

Intuition. A \leftrightarrow B is false only on rows where A and B disagree. No such rows means no F in the biconditional's column.

Ingredients.

• \equiv: "is logically equivalent to." It is a claim ABOUT two sentences, not a connective inside a sentence.
• \leftrightarrow: the biconditional connective, which builds one sentence from two.
• Tautology: true on every row.

Example. P \to Q and \neg P \vee Q have the columns T, F, T, T and T, F, T, T. They match on every row, so P \to Q \equiv \neg P \vee Q.

What breaks if you check one row. Two sentences can agree on one row and disagree on another: P \wedge Q and P \vee Q agree when both letters are T but not when only P is. Equivalence needs every row.

For every sentence A, \neg\neg A \equiv A.

What is this saying? Denying a denial gets you back where you started: "it is not the case that it is not raining" says it is raining.

Why should I care? Double negations turn up constantly when you push a negation inward with De Morgan's laws or rewrite a conditional. This law is what lets you clean them up.

Intuition. Each \neg flips the truth value. Two flips return the original value.

Ingredients.

• A: any sentence.
• \neg\neg A: the negation of the negation of A.
• \equiv: same truth value on every row.

Example. "It's not true that the store isn't open" is equivalent to "the store is open."

What doesn't follow. Double negation cancels two negations applied to the same sentence. In \neg(\neg P \wedge Q) the two \neg signs apply to different sentences, and they don't cancel: that sentence is equivalent to P \vee \neg Q, not to P \wedge Q.

For any propositions p and q: \neg(p \wedge q) \equiv \neg p \vee \neg q \neg(p \vee q) \equiv \neg p \wedge \neg q

What is this saying? To deny an "and," deny at least one of its parts: "not both" means "one or the other fails." To deny an "or," deny both parts: "not either" means "neither." In each law the \neg moves onto each part, and the connective swaps.

Why should I care? These two laws are how you push a "not" inside a bracket. You need them to simplify a sentence, to put it in a normal form, and to state plainly what a denial claims.

Intuition. "Not both a towel and a mat" allows a towel alone. So it can't mean "no towel and no mat." It means "no towel, or no mat." The "and" had to turn into "or" once the "not" moved in.

Ingredients.

• p, q: any two propositions. Capital letters such as A and B work the same way.
• \neg: "not." \wedge: "and." \vee: "or."
• \equiv: the two sides have the same truth value on every row.

Example. "It's not true that the café is open and serving lunch" is equivalent to "the café is closed, or it isn't serving lunch."

What doesn't follow. The \neg does not just copy onto each part. \neg(p \wedge q) is not equivalent to \neg p \wedge \neg q: on the row p = T, q = F, the first is true and the second false.

For all sentences A, B, C:

What is this saying? You can swap the two sides of an "and" or an "or." In a chain of "and"s only, or of "or"s only, the brackets can go anywhere. And an "and" spreads over an "or" (and the other way round) the way multiplication spreads over addition.

Why should I care? These are the laws you use most often in a chain of equivalences, and knowing which regroupings are NOT allowed is just as important.

Intuition. A conjunction is true when all its parts are true, however they are ordered or bracketed; a disjunction is true when any part is. The distributive laws say that "A, and also B or C" is true in exactly the cases "A and B, or A and C" covers.

Ingredients.

• A, B, C: any sentences.
• \equiv: same truth value on every row.

Example. "Soup, and salad or bread" is equivalent to "soup and salad, or soup and bread."

What does not hold.

• \to is not commutative: A \to B and B \to A differ when A is T and B is F.
• Brackets can't be moved across two different connectives: (A \wedge B) \vee C is not equivalent to A \wedge (B \vee C) (make A F and C T).

For every sentence A, with P \vee \neg P standing for any tautology and P \wedge \neg P for any contradiction:

What is this saying? Saying the same thing twice adds nothing. Adding a tautology with "and," or a contradiction with "or," changes nothing. A tautology swamps any "or" it joins, and a contradiction swamps any "and." And A with its own negation gives a contradiction ("and") or a tautology ("or").

Why should I care? These laws are how a step-by-step rewrite of a sentence gets shorter: they remove repetition and throw away parts that can't affect the truth value.

Intuition. A tautology is T on every row, so in an "and" it never changes the result, and in an "or" it makes the result T on every row. A contradiction does the mirror image.

Ingredients.

• A: any sentence.
• P \vee \neg P: stands for any tautology; any other tautology works the same way.
• P \wedge \neg P: stands for any contradiction.

Example. "The shop is open, and it's raining or it isn't" is equivalent to "the shop is open."

Watch the pairing. Identity and domination swap roles between \wedge and \vee: a tautology is harmless in an "and" but takes over an "or."

For all sentences A and B:

1. Material implication: A \to B \equiv \neg A \vee B
2. Contraposition: A \to B \equiv \neg B \to \neg A
3. Negated conditional: \neg(A \to B) \equiv A \wedge \neg B
4. Biconditional as two conditionals: A \leftrightarrow B \equiv (A \to B) \wedge (B \to A)
5. Biconditional as matching: A \leftrightarrow B \equiv (A \wedge B) \vee (\neg A \wedge \neg B)

What is this saying? A conditional says "either the condition fails or the result holds." It says the same as its contrapositive ("if not the result, then not the condition"). It is false only when the condition holds and the result doesn't. A biconditional is two conditionals, one each way, or equivalently "both true or both false."

Why should I care? These let you remove \to and \leftrightarrow from any sentence, which is the first step toward a normal form, and they explain which rewrites of a conditional are safe.

Intuition. Each law restates the conditional's one false row (A T, B F) or the biconditional's two false rows (the rows where A and B differ).

Ingredients.

• A, B: any sentences.
• Contrapositive: \neg B \to \neg A. The converse is B \to A and the inverse is \neg A \to \neg B; neither is equivalent to A \to B.

Example. "If it rains, the game is postponed" ≡ "It doesn't rain, or the game is postponed" ≡ "If the game isn't postponed, it didn't rain."

What does not hold. \neg(A \to B) is not \neg A \to \neg B. And A \to B is not equivalent to its converse B \to A or its inverse.

If B \equiv C, and A' is the result of replacing one occurrence of B inside the sentence A with C, then A \equiv A'.

What is this saying? If two sentences always have the same truth value, you can swap one for the other inside a bigger sentence without changing the bigger sentence's truth value on any row.

Why should I care? This is what makes a chain of equivalences work. The laws are stated for whole sentences, but you usually apply them to a part: rewriting \neg\neg Q as Q inside P \wedge \neg\neg Q, for example.

Intuition. Every connective is truth-functional: the value of a compound depends only on the values of its parts. If a part keeps its value on every row, the whole does too.

Ingredients.

• A: the larger sentence. B: a part of it. C: a sentence equivalent to B.
• A': A with that occurrence of B replaced by C.
• Truth-functional: depending only on the truth values of the parts.

Example. Since \neg\neg Q \equiv Q, the sentence P \wedge \neg\neg Q is equivalent to P \wedge Q.

What breaks if the replaced part doesn't match. The part you replace must be a whole subsentence exactly matching B. In \neg P \wedge Q there is no subsentence \neg(P \wedge Q), so De Morgan's law can't be applied there.

Every sentence of sentential logic is logically equivalent to a sentence in disjunctive normal form (an "or" of "and"s of literals) and to a sentence in conjunctive normal form (an "and" of "or"s of literals).

What is this saying? However tangled a sentence is, you can rewrite it as a list of alternatives, each a plain "these letters true, those false" combination. You can also rewrite it as a list of requirements, each a plain "at least one of these" condition.

Why should I care? Normal forms give every sentence a standard shape, so two sentences can be compared by putting both into it. The DNF construction also shows that the five connectives can express any truth table at all.

Intuition. A sentence is true on certain rows. Describe each of those rows with an "and" of literals, then join the descriptions with "or." The result is true on exactly those rows.

Ingredients.

• Literal: a sentence letter or its negation, such as P or \neg Q.
• DNF: a disjunction of one or more conjunctions of literals.
• CNF: a conjunction of one or more disjunctions of literals.

Example. P \leftrightarrow Q is true on the rows TT and FF, so its DNF is (P \wedge Q) \vee (\neg P \wedge \neg Q). Its CNF is (\neg P \vee Q) \wedge (P \vee \neg Q).

Not unique. A sentence has many equivalent DNFs and CNFs; the theorem says at least one exists, not that it is the only one.

An argument in sentential logic with premises A_1, \dots, A_n and conclusion C is valid if and only if no row of their joint truth table makes every premise true and C false. When this holds we write A_1, \dots, A_n \models C. Because a table has finitely many rows, this test always gives an answer.

What is this saying? To test an argument, build one table with a column for each premise and for the conclusion, and look only at the rows where every premise is T. If the conclusion is T on all of those rows, the argument is valid. If it is F on even one, the argument is invalid, and that row is a countermodel.

Why should I care? Chapter 3 defined validity in terms of possible situations, which you had to imagine. For sentential arguments this turns the definition into a mechanical check that is guaranteed to finish.

Intuition. In sentential logic, all that matters about a situation is which sentence letters it makes true. So "every possible situation" comes down to "every row."

Ingredients.

• A_1, \dots, A_n: the premises. C: the conclusion.
• \models: "entails." A_1, \dots, A_n \models C says every row making all the premises true makes C true.
• Countermodel: a row with every premise T and C F.

Example. For P \to Q, Q, therefore P: the row P F, Q T makes both premises T and the conclusion F. So the argument (affirming the consequent) is invalid.

What does not count. A row where every premise AND the conclusion is T shows nothing. Valid and invalid arguments both usually have such rows.

For any sentences A_1, \dots, A_n and C, A_1, \dots, A_n \models C \quad\text{if and only if}\quad (A_1 \wedge \dots \wedge A_n) \to C \text{ is a tautology.}

What is this saying? An argument is valid exactly when the single "if all the premises, then the conclusion" sentence can never be false.

Why should I care? It links two ideas you have met separately: validity of arguments and tautologies of single sentences. Any question about one can be turned into a question about the other, and it explains why a valid argument form gives a tautology when you write it as one conditional.

Intuition. The conditional is false only on rows where its antecedent, the conjunction of the premises, is T and its consequent, the conclusion, is F. Those are exactly the countermodel rows.

Ingredients.

• \models: entails.
• A_1 \wedge \dots \wedge A_n: the conjunction of all the premises.
• Tautology: true on every row.

Example. Modus ponens is valid, and ((P \to Q) \wedge P) \to Q is a tautology.

What does not follow. A tautologous conditional says the argument is valid. It does not say the premises are true, so it says nothing about soundness.

1. If the premises A_1, \dots, A_n are inconsistent, then A_1, \dots, A_n \models C for every sentence C.
2. If C is a tautology, then A_1, \dots, A_n \models C for any premises at all, including none.

What is this saying? An argument whose premises can never all be true is valid whatever its conclusion. An argument whose conclusion can never be false is valid whatever its premises. In both cases the premises and conclusion may have nothing to do with each other.

Why should I care? These cases look like mistakes, and a common first reaction is to call them invalid. They follow directly from the definition, and seeing why makes the definition clearer. They are also why validity alone isn't enough: an argument with inconsistent premises is valid but can never be sound.

Intuition. Validity only forbids one kind of row: premises all T, conclusion F. If the premises are never all T, or the conclusion is never F, that kind of row can't exist.

Ingredients.

• Inconsistent: no row makes all the premises true together.
• Tautology: true on every row.
• \models: entails.

Example. P, \neg P, therefore Q is valid. So is R, therefore Q \vee \neg Q.

What this does not mean. Valid is not the same as good. Neither kind of argument could persuade anyone: one has premises that can't all be true, and the other proves something that needed no premises.

Given p \implies q and p, conclude q. \big[(p \implies q) \wedge p\big] \implies q

What is this saying? If one thing guarantees another, and the first thing happens, then the second happens too. "If it rains, the street gets wet. It is raining. So the street is wet."

Why should I care? This is the most common step in everyday reasoning, and the first rule most proof systems give you. Its look-alike, affirming the consequent ("the street is wet, so it rained"), is a fallacy, and telling the two apart is a basic skill.

Intuition. The if–then sentence is a promise: whenever p is true, q is true. Knowing p is true puts you in the only case the promise covers, so q follows.

Ingredients.

• p, q: any statements, simple or compound.
• p \implies q: the conditional "if p then q" (also written p \to q).
• p: the antecedent, asserted on its own as a second premise.

Given p \implies q and \neg q, conclude \neg p. \big[(p \implies q) \wedge \neg q\big] \implies \neg p

What is this saying? If one thing guarantees another, and the second thing did not happen, then the first did not happen either. "If it rained, the street is wet. The street is dry. So it didn't rain."

Why should I care? This is how a failed prediction counts against a claim: if the claim were true, you would see q, and you don't. Its look-alike, denying the antecedent ("it didn't rain, so the street is dry"), is a fallacy.

Intuition. Suppose p were true. Then the promise p \implies q would make q true, but you know q is false. So p can't be true.

Ingredients.

• p, q: any statements, simple or compound.
• p \implies q: the conditional "if p then q" (also written p \to q).
• \neg q: "not q," the denial of the consequent, as a second premise.

For any sentences A_1, \dots, A_n and C, A_1, \dots, A_n \vdash C \quad\text{if and only if}\quad A_1, \dots, A_n \models C. The left-to-right direction is soundness; the right-to-left direction is completeness.

What is this saying? Every argument you can prove with the course's proof rules is valid by truth table, and every argument that is valid by truth table can be proved with the rules. The two methods never disagree.

Why should I care? It means the choice between a table and a proof is a practical one. A proof never proves an invalid argument, and when an argument is valid, a proof exists, even when you haven't found it yet. To show an argument INVALID, you still need a countermodel.

Intuition. Each rule was chosen so that it can't lead from true lines to a false line; that gives soundness. Completeness says the rules are strong enough to reach every valid conclusion, which is harder to show.

Ingredients.

• \vdash: "proves." A_1, \dots, A_n \vdash C says there is a formal proof that ends with C and uses only A_1, \dots, A_n as premises.
• \models: "entails," as defined by the truth-table test.

Example. P \to Q, \neg Q \vdash \neg P (by a short proof), and the same argument passes the truth-table test.

What this doesn't give you. Failing to find a proof doesn't show an argument is invalid; only a countermodel does.

Conjunction introduction (\wedgeI). From A on line i and B on line j, write A \wedge B, citing "\wedgeI i, j."

Conjunction elimination (\wedgeE). From A \wedge B on line i, write A, or write B, citing "\wedgeE i."

What is this saying? If you have both parts, you may put them together; if you have them together, you may take either one out.

Why should I care? These are the simplest rules in the system and the pattern every other pair follows: an introduction rule to build a connective, an elimination rule to use it.

Intuition. A conjunction is true exactly when both conjuncts are, so building one from both parts, or taking a part out, can never lead from truth to falsehood.

Ingredients.

• A, B: any sentences.
• Line numbers i, j: earlier lines that are still available (not inside a closed subproof).
• \wedgeE applies only when \wedge is the main connective of line i.

Example.

What breaks. From (P \wedge Q) \vee R you cannot write P by \wedgeE: the main connective is \vee, and the whole line can be true with P false.

Conditional elimination (\toE). From A \to B on line i and A on line j, write B, citing "\toE i, j."

Reiteration (R). From A on an available line i, write A again, citing "R i."

What is this saying? If you have a conditional and its antecedent, you may write its consequent. And you may always repeat a line you already have.

Why should I care? \toE is modus ponens from chapter 4, now as a rule you can cite in a proof. Reiteration looks trivial, but you need it to bring a line into a subproof or to end a subproof on a line you already have.

Intuition. The conditional's only false row has A T and B F. If both A \to B and A are true, that row is ruled out, so B is true.

Ingredients.

• A \to B: must be the whole line, with \to as its main connective.
• A: must match the antecedent exactly.
• Available: not inside a subproof that has already closed.

Example.

What breaks. From A \to B and B you cannot write A: that is affirming the consequent, and the row A F, B T is a countermodel.

Conditional introduction (\toI). If a subproof begins with the assumption A on line i and ends with B on line j, then after the subproof closes you may write A \to B, citing "\toI i–j." The assumption A is discharged.

What is this saying? To prove "if A then B," suppose A for the sake of argument, reason your way to B, then close off the supposition and write the conditional.

Why should I care? It is the standard way to prove a conditional, in logic and in mathematics, and it introduces subproofs, which the rules for \vee, \neg, and \leftrightarrow also use.

Intuition. The subproof shows that whenever A is true (together with the lines available above), B is true too. That is exactly what A \to B claims.

Ingredients.

• Assumption (AS): a line supposed without justification, starting a subproof.
• Subproof: the indented lines from the assumption to its last line.
• Discharge: once the subproof closes, its lines may not be cited individually; only the whole subproof i–j may be cited.

Example.

What breaks. After line 6 you may not cite line 4 or 5: they depended on the assumption P, which is no longer in force. A proof may not end inside an open subproof.

Disjunction introduction (\veeI). From A on line i, write A \vee B or B \vee A, where B is any sentence, citing "\veeI i."

Disjunction elimination (\veeE). From A \vee B on line i, a subproof assuming A and ending with C (lines j–k), and a subproof assuming B and ending with the same C (lines l–m), write C, citing "\veeE i, j–k, l–m."

What is this saying? From one part you may conclude any "or" that contains it. To use an "or," show that the same conclusion follows whichever part is true.

Why should I care? \veeI is addition from chapter 4. \veeE is proof by cases, one of the most common patterns in all reasoning: "either way, C."

Intuition. An inclusive "or" is true whenever one part is. And if C follows from each part, and at least one part is true, C is true.

Ingredients.

• B in \veeI: any sentence at all, even one unrelated to the proof.
• C in \veeE: must be the SAME sentence at the end of both subproofs.
• Both subproofs close, discharging A and B.

Example.

(Lines 2–3 and 4–5 are two separate subproofs.)

What breaks. Reaching C from only one disjunct proves nothing: the other disjunct may be the true one.

Biconditional introduction (\leftrightarrowI). From a subproof assuming A and ending with B (lines i–j) and a subproof assuming B and ending with A (lines k–l), write A \leftrightarrow B, citing "\leftrightarrowI i–j, k–l."

Biconditional elimination (\leftrightarrowE). From A \leftrightarrow B on line i and A on line j, write B; or from A \leftrightarrow B and B, write A. Cite "\leftrightarrowE i, j."

What is this saying? To prove "A if and only if B," prove each direction. To use it, knowing either side gives you the other.

Why should I care? Definitions and "exactly when" claims are biconditionals, so these rules are how you prove and use them.

Intuition. A \leftrightarrow B says the same as (A \to B) \wedge (B \to A), so its rules are the conditional rules done in both directions.

Ingredients.

• Two subproofs for \leftrightarrowI, one for each direction.
• \leftrightarrowE works from either side, unlike \toE.

Example.

What breaks. One subproof gives only one direction, which proves a conditional, not a biconditional.

Negation elimination (\negE). From A on line i and \neg A on line j, write \bot, citing "\negE i, j."

Negation introduction (\negI). From a subproof assuming A and ending with \bot (lines i–j), write \neg A, citing "\negI i–j."

Explosion (X). From \bot on line i, write any sentence at all, citing "X i."

What is this saying? A sentence together with its negation is a contradiction, recorded as \bot. If an assumption leads to a contradiction, the assumption is false. And from a contradiction, anything may be written.

Why should I care? These rules give proofs a way to reason about "not." \negI is how you show something is false, and explosion is what lets a proof finish off a case that can't happen.

Intuition. A and \neg A are never true together, so any line of reasoning that reaches both has started from something false. Explosion is the proof-rule version of the fact that inconsistent premises entail every sentence.

Ingredients.

• \bot: the contradiction symbol, "falsum." It stands for a sentence false on every row.
• \negE needs a sentence and its exact negation, not merely two sentences that clash in meaning.

Example.

What breaks. X may be cited only from a line that is \bot. Two sentences that merely seem to disagree, like "the room is warm" and "the room is cold," are not A and \neg A.

Indirect proof (IP). From a subproof assuming \neg A and ending with \bot (lines i–j), write A, citing "IP i–j."

What is this saying? To prove A, suppose it is false, derive a contradiction, and conclude that A is true after all.

Why should I care? Some sentences, such as P \vee \neg P, have no easy direct proof. IP is the last-resort strategy that works when nothing else does.

Intuition. If assuming "not A" leads to something impossible, then "not A" is false, so A is true.

Ingredients.

• The assumption is the NEGATION of the goal.
• The subproof must end with \bot.

How it differs from \negI. \negI assumes A and concludes \neg A. IP assumes \neg A and concludes A. Check which one the assumption is.

Example.

What breaks. Concluding \neg A from a subproof that assumed \neg A is not a rule; IP delivers A itself.

Each of these may be cited as a single step, because each can be replaced by a proof that uses only the basic rules.

What is this saying? Patterns you would otherwise prove again and again, some of them the named forms from chapter 4, can be used in one line.

Why should I care? Derived rules make proofs shorter and easier to read, without letting you prove anything the basic rules can't.

Intuition. A derived rule is a saved proof: each time you cite it, imagine the justifying proof pasted in its place.

Ingredients.

• Basic rules: R and the introduction and elimination rules for each connective, plus X and IP.
• Derived rule: a rule whose every use can be replaced by basic-rule steps.

Example. From P \to Q and \neg Q, MT gives \neg P in one line; the negation-rules card shows the five basic-rule lines it replaces.

What breaks. A derived rule has the same fixed pattern as a basic rule. DS needs the NEGATION of a disjunct; from A \vee B and A nothing follows about B.

Each of these may be cited as a single step:

What is this saying? The three chaining and dilemma forms you met in chapter 4 in English are now rules you may cite in a formal proof, each justified by a proof in the basic rules.

Why should I care? Long arguments often chain conditionals or reason by dilemma. Citing the rule saves a subproof each time.

Intuition. HS is two uses of \toE inside a conditional proof. CD is proof by cases with \toE in each case. DD is CD with modus tollens in place of modus ponens.

Ingredients.

• A, B, C, D: any sentences. In HS, the B in both premises must be the same sentence.
• Each rule's premises must match its pattern exactly, including which parts are negated.

Example. From P \to Q and Q \to R, HS gives P \to R; the conditional-introduction card shows the four lines it replaces.

What breaks. HS does not run backwards: from A \to B and B \to C you may not write C \to A. DD denies the consequents, not the antecedents.

The form

1. If P, then Q.
2. If Q, then R.

\therefore If P, then R.

is valid: no instance can have both premises true and the conclusion false.

What is this saying? If one thing leads to a second, and the second leads to a third, then the first leads to the third. You can link conditionals into a chain and skip the middle.

Why should I care? Much everyday reasoning is a chain of "if … then" steps: if the flight is late, I miss the connection; if I miss the connection, I miss the meeting. Hypothetical syllogism is what lets you jump from the first link to the last: if the flight is late, I miss the meeting.

Intuition. Picture falling dominoes. Knocking over P knocks over Q, and knocking over Q knocks over R, so knocking over P knocks over R.

Ingredients.

• P, Q, R: any three claims. Q must be the same claim in both premises; it is the link that joins them.
• If … then: a conditional. The only way "if P, then R" is false is when P is true and R is false.

Example. "If the reservoir drops below the line, the town restricts watering. If the town restricts watering, lawns turn brown. So if the reservoir drops below the line, lawns turn brown."

Look-alike to avoid. "If P, then Q. If Q, then R. So if R, then P" reverses the chain. Brown lawns could come from a heat wave with the reservoir full.

The form

1. If P, then Q.
2. If R, then S.
3. P or R.

\therefore Q or S.

is valid: no instance can have all three premises true and the conclusion false.

What is this saying? You have two conditionals and know that at least one of their "if" parts holds. Then at least one of their "then" parts holds too.

Why should I care? It is the shape of every "either way" argument. "If I take the bus, I'll be late. If I drive, I'll pay for parking. I'll either take the bus or drive. So either I'll be late or I'll pay for parking." It is also modus ponens run on two tracks at once.

Intuition. Whichever of P or R is true, its own conditional fires. If P, you get Q. If R, you get S. Either way you end up with Q or S.

Ingredients.

• P, Q, R, S: four claims. Each conditional pairs one condition with one result.
• P or R: the third premise asserts the two conditions, the "if" parts, not the results.
• Or: inclusive. The conclusion is true when at least one of Q and S is true.

Example. "If the package went by air, it arrives Tuesday. If it went by ground, it arrives Friday. It went by air or by ground. So it arrives Tuesday or Friday."

Look-alike to avoid. If the third premise asserts the results ("Q or S") and the conclusion claims the conditions ("P or R"), the argument is invalid. That is affirming the consequent on two tracks.

The form

1. If P, then Q.
2. If R, then S.
3. Not Q or not S.

\therefore Not P or not R.

is valid: no instance can have all three premises true and the conclusion false.

What is this saying? You have two conditionals and know that at least one of their "then" parts fails. Then at least one of their "if" parts fails too.

Why should I care? It is modus tollens run on two tracks. "If the suspect was at the party, the guests saw him. If he was at the office, the cameras recorded him. Either the guests didn't see him or the cameras didn't record him. So either he wasn't at the party or he wasn't at the office."

Intuition. Whichever result fails, its own conditional runs backward. If Q fails, P must fail. If S fails, R must fail. Either way, one of the conditions fails.

Ingredients.

• P, Q, R, S: four claims. Each conditional pairs one condition with one result.
• Not Q or not S: the third premise denies the results, the "then" parts.
• Or: inclusive. The conclusion is true when at least one of "not P" and "not R" is true.

Example. "If the recipe is right, the bread rises. If the oven is hot enough, the crust browns. Either the bread didn't rise or the crust didn't brown. So either the recipe is wrong or the oven isn't hot enough."

Look-alike to avoid. Denying the conditions ("not P or not R") and concluding that a result fails ("not Q or not S") is invalid. That is denying the antecedent on two tracks.

For any sentences A_1, \dots, A_n and C, A_1, \dots, A_n \vdash C \quad\text{if and only if}\quad A_1, \dots, A_n \models C. The left-to-right direction is soundness; the right-to-left direction is completeness.

What is this saying? Every argument you can prove with the course's proof rules is valid by truth table, and every argument that is valid by truth table can be proved with the rules. The two methods never disagree.

Why should I care? It means the choice between a table and a proof is a practical one. A proof never proves an invalid argument, and when an argument is valid, a proof exists, even when you haven't found it yet. To show an argument INVALID, you still need a countermodel.

Intuition. Each rule was chosen so that it can't lead from true lines to a false line; that gives soundness. Completeness says the rules are strong enough to reach every valid conclusion, which is harder to show.

Ingredients.

• \vdash: "proves." A_1, \dots, A_n \vdash C says there is a formal proof that ends with C and uses only A_1, \dots, A_n as premises.
• \models: "entails," as defined by the truth-table test.

Example. P \to Q, \neg Q \vdash \neg P (by a short proof), and the same argument passes the truth-table test.

What this doesn't give you. Failing to find a proof doesn't show an argument is invalid; only a countermodel does.