Connectives · printable packet

Atoms, the five connectives, the formation rules, and finding the main connective of a formula.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

By the end of this lesson you will be able to say which strings the formation rules build, find the main connective of a formula and the subformulas it joins, read a formula written with precedence rather than brackets, and give the column a connective has over the rows of a truth table.

2. What you already have

You have written mathematical statements with and, or, not and if ... then, and you have used a truth table to settle whether an implication holds. This lesson makes the language itself the object of study: which strings count as statements, and how one is put together.

3. Terms to use precisely

4. Formulas and connectives

Term	What it means
Negation	~P is true exactly when P is false.
Conjunction	P & Q requires both parts to be true.
Inclusive disjunction	P	Q requires at least one true part and permits both.
Material conditional	P -> Q excludes just P true with Q false.
Biconditional	P <-> Q requires the same truth value on both sides.
Scope	The whole formula to which a connective applies.

The language has atoms $P, Q, R, \ldots$ and five connectives: $\neg$, $\wedge$, $\vee$, $\to$, $\leftrightarrow$. The formation rules say what a formula is, and they are the whole definition: every atom is a formula; if $\phi$ is a formula then so is $\neg \phi$; if $\phi$ and $\psi$ are formulas then so is $(\phi \wedge \psi)$, and likewise for $\vee$, $\to$ and $\leftrightarrow$; and nothing else is a formula. Because every formula is built by those rules, every formula has exactly one main connective, the one applied last, and the pieces it joins are its immediate subformulas. Outer brackets are usually dropped and precedence used instead: $\neg$ binds tightest, then $\wedge$ and $\vee$, then $\to$ and $\leftrightarrow$. A valuation gives each atom a value, and the truth tables of the connectives then fix the value of every formula built from them: $\phi \to \psi$ is false on one row only, the row where $\phi$ is true and $\psi$ false.

Another way: steps

Strip any bracket that encloses the whole string.
Find the connective with the widest scope; that is the main connective.
Split the formula into the pieces it joins, and repeat on each piece.
Stop at atoms. The tree you have drawn is the order a truth table fills its columns.

Another way: example

$\neg P \to Q \wedge R$ has main connective $\to$: negation reaches only $P$ and conjunction binds tighter than the arrow, so the two pieces are $\neg P$ and $Q \wedge R$.

5. What a conditional leaves open

A material conditional is a rule for combining truth values. It does not by itself describe a cause, a promise, a probability, or a time sequence. If P means that the alarm sounds and Q that the door is open, P -> Q forbids an alarm while the door is closed. It permits an open door with a silent alarm. Reversing the arrow adds a claim the first sentence never made.

Likewise, an inclusive disjunction permits both alternatives. Everyday speakers sometimes mean exactly one, but the formula P | Q does not express that restriction. Write (P | Q) & ~(P & Q) when the intended reading explicitly excludes both. Parentheses tell you which denial is intended: ~(P & Q) rules out the pair, whereas ~P & ~Q rules out each member. Try P true and Q false to see the difference.

The keyboard forms used here are ~, &, |, -> and <->. Unicode forms are accepted too. Evaluate parentheses from the inside outward rather than relying on English word order. A letter keeps the same value every time it occurs in one row; it may change on the next row.

6. Read structure before calculating values

A connective combines formulas according to a fixed truth rule. Negation takes one input. Conjunction, disjunction, conditional, and biconditional take two. The inputs can themselves be compound formulas. When you read (P & Q) -> R, the conditional takes the whole P & Q as its first input and R as its second. The conjunction is an operation inside the first input, not a rival description of the whole expression.

The main connective is the outermost operation. Find it by asking which operation produces the value of the entire formula after the smaller pieces have been evaluated. In ~(P | Q), disjunction is evaluated first and negation last. In ~P | Q, negation applies only to P, and disjunction is last. These formulas can differ: with P false and Q true, the first is false and the second true. Parentheses are therefore information, not optional decoration.

A useful working method is to name the immediate parts without assigning their values yet. For (P & Q) -> ~R, write antecedent P & Q and consequent ~R. Next identify the inputs inside each part. Only then use a stated row to calculate. This structural pass prevents applying a truth rule to the wrong pair of atoms, especially when an expression contains several occurrences of the same connective.

Repeated letters keep the same value within a row. If P is true, both occurrences in P & ~P begin from true P; the negation reverses only its own occurrence's value. You cannot make the first P true and the second P false to satisfy the conjunction. The repeated name indicates the same claim. It is the connective around one occurrence that changes the value of the compound part.

After finding the whole value, check it using the main connective's decisive condition. A conjunction marked true should have two true inputs. A disjunction marked false should have two false inputs. A conditional marked false should have true antecedent and false consequent. These checks use the immediate compound parts where necessary, rather than jumping back to unrelated individual letters.

7. Distinguish the connectives by their decisive cases

Conjunction is demanding: both inputs must be true. Inclusive disjunction is permissive: at least one input must be true. If both inputs are true, both conjunction and inclusive disjunction are true. Their difference appears on mixed rows. A true first input and false second input make conjunction false but disjunction true. Finding a distinguishing row is often clearer than memorizing a verbal contrast.

Negation reverses a whole input value. It does not mean 'probably false', 'unknown', or 'not yet checked'. If an exercise gives no evidence about P, that lack of evidence does not license ~P. A truth-table row, however, stipulates a value for the calculation. You may evaluate ~P on that stipulated row without asserting that the row describes the actual world. The exercise is conditional reasoning about a possible assignment.

A biconditional requires matching values, so it is true when both sides are false as well as when both are true. This sometimes surprises readers who interpret it as asserting both claims. P <-> Q connects the values of P and Q; it does not independently assert P or Q. To assert both, write P & Q. On the false-false row the conjunction is false and the biconditional true, which supplies a direct test of the difference.

The material conditional P -> Q says that P does not occur without Q. On a row where P is false, that prohibited combination is absent whatever value Q has. This explains the two true rows with false antecedent. The truth rule does not claim that P caused Q, that an actual promise was made, or that the speaker has evidence for a causal law. Those are richer questions than the truth-functional operation represents.

The reversed conditional Q -> P prohibits a different combination. A row with P false and Q true satisfies P -> Q but violates Q -> P. If you require both directions, use a biconditional or conjoin the two conditionals. Before choosing notation for an everyday rule, ask whether the intended requirement is one-way or two-way. A small record that one wording permits and another forbids can reveal the intended distinction.

8. Use a truth table without turning rows into statistics

For two distinct atoms, list four assignments: true-true, true-false, false-true, and false-false. These exhaust the combinations of two values for each atom. They do not claim that real observations occur equally often. If a conditional column contains three true entries, that is a fact about the formula's truth function. It is not evidence that a proposed policy succeeds seventy-five percent of the time.

Write intermediate columns whenever a formula contains several operations. For ~(P & Q), first calculate P & Q across the four rows, obtaining true, false, false, false. Then reverse each entry, obtaining false, true, true, true. Negating the whole conjunction is different from negating both atoms and then conjoining them. On a mixed row, the former is true while the latter is false.

You can also audit a single row before committing to a whole column. Suppose P is true, Q false, and R true in (P | Q) & ~R. The disjunction is true, the negation false, and the conjunction false. Giving the intermediate results shows where an error occurred if your final answer differs. A bare final value hides whether you misunderstood inclusive or, negation, or conjunction.

When explaining a correction, identify the operation that failed. 'The answer should be false' is less useful than 'the right conjunct is false because R is true, so the conjunction fails despite its true left side'. That explanation connects the result to a reusable truth rule. Once you can do this reliably, longer expressions require more bookkeeping but no new truth-functional principles.

9. Checking a workshop access rule

A fictional workshop tests this proposed rule: if a person enters, that person has completed training. Let P mean the person enters and Q mean the person has completed training. The rule is P -> Q. List four possible records. A trained entrant gives P true and Q true, so the rule is satisfied. An untrained entrant gives P true and Q false, so the rule is violated. A trained person who stays outside gives P false and Q true, which does not violate the rule. An untrained person who stays outside gives two false atoms and also does not violate it.

The resulting column is T, F, T, T in that order. There is one violating combination out of four logical combinations. This is not a claim that one quarter of actual visitors violate the rule: the four rows have not been assigned frequencies. A truth table enumerates possibilities, not a visitor survey.

If the manager instead writes Q -> P, every trained person must enter, including people who choose to go home. That is a different policy. If the manager writes P <-> Q, both directions are required. A useful review therefore includes a concrete record that distinguishes the proposed wording from the reversed wording. The trained non-entrant is such a record. State which rule the organization actually means before anyone implements a door controller from it.

10. Where this goes wrong

Two mistakes are worth naming. The first is reading precedence as word order: in $P \vee Q \to R$ the arrow is the main connective, even though it is written last, because it has the widest scope. The second is treating $\neg$ as reaching to the end of the formula; $\neg P \wedge Q$ is a conjunction whose left half is negated, not the negation of a conjunction, and the two differ on the row where $P$ and $Q$ are both false.

11. A denied conjunction on a mixed row

12. Compound inputs to a conditional

13. Combine a biconditional and a denial

14. Calculate before reversing

15. Guided practice

16. Guided practice

Complete the worked count for the column P & Q across all assignments of two atoms.

17. Guided practice

18. Practice

19. Practice

A display is lit only if its power supply is on. P: display lit; Q: power supply on. Express this necessary condition; do not assert that power guarantees light.

20. Somewhere new

21. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

22. Test question

23. What you can do now

You can decide whether a string is a formula, find its main connective, and write the column of each connective. Say in your own words why every formula has exactly one main connective.

Working for the steps left to you

The outer negation reverses the whole input.