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Identify complete statements and represent them with a consistent key.
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.
You will distinguish a complete statement from a question or command, assign an atom to the whole statement, and keep its meaning unchanged when it repeats.
You have distinguished a claim from the reason offered for it. Here we isolate the claims before asking how they support one another.
| Term | What it means |
|---|---|
| Statement | A complete claim capable of being true or false. |
| Atomic formula | A single statement letter, with no truth-functional connective. |
| Symbolization key | The declared meaning of each letter. |
| Compound formula | A formula built from one or more formulas with connectives. |
Let G mean 'the signal is green'. G represents that entire claim, not just the signal or the color. A question such as 'Is the signal green?' asks for information; it does not assert G. A command such as 'Turn the signal green' requests an action; it is not assigned a truth value here.
Atomic does not mean short, obvious, or actually true. 'The museum's north door was locked at noon on Monday' can be represented by one atom even though it has many words. 'The door is open and the light is on' combines two claims and is represented by a conjunction once we choose to analyze that structure. Atomicity is relative to the propositional representation we have declared.
Use one letter consistently for one complete claim. If G means the signal is green now, it cannot mean the signal was green yesterday later in the same argument. Different times, places, or people can change the statement. Repetition of a letter repeats its truth value within one assignment. Two occurrences of G do not create two independent facts.
To begin a translation, list the complete simple statements. Give each one a letter and record its meaning. Then identify connective words such as not, and, or, and if. This lesson introduces ~ for not and & for and; the next lesson gives their truth rules in detail. Parentheses group a compound before another connective applies to it. Naming the components does not establish their truth. Evidence about a signal is a separate matter from choosing G as its symbol.
Another way: steps
If you discover that a key omitted a relevant location, revise the key before evaluating the argument. Then check every occurrence against the revised meaning. A correction is useful only when all affected formulas are reviewed together.
A statement is something that can be assessed as true or false, even when you do not know which value it has. 'The archive contains a handwritten letter' is a statement despite your never having visited the archive. Ignorance about its truth is different from the sentence failing to make a claim. A question asks whether a claim holds; a command asks somebody to act. Neither has the same role as asserting the claim.
Some sentences need context before their claim is clear. 'It is open now' leaves the object and time unspecified. Before assigning a letter, resolve those references: perhaps the speaker means the west entrance is open at noon on Tuesday. Record that reading in the key. Another speaker using the same words at a different entrance may be making a different claim. The notation cannot repair an ambiguity you have left unresolved.
An assertion can also be false without ceasing to be a statement. If the west entrance is closed, the assertion that it is open is false, but W still represents that assertion. Do not change W's meaning to 'the entrance is closed' just to make the symbol true. A key fixes what a formula says. A truth assignment later records whether that fixed claim is true in the situation under consideration.
This distinction matters when two reports conflict. Let W mean the west entrance is open at the specified time. One report asserts W and another denies it, written ~W. Giving the denial an unrelated letter without explaining its relation to W hides the conflict. Keeping the positive claim and its negation visible lets a later truth-table analysis detect that they cannot both be true in the same classical assignment.
For now, treat each chosen atomic claim as having exactly one of two values in an assignment. That is the classical propositional framework used in this course. It is a modeling choice, not a claim that every ordinary sentence is already precise or that every investigator knows its value. When a sentence is vague, indexical, or open to several readings, clarify the intended assertion before attempting a two-valued formal analysis.
The word atomic describes a formula's structure in the current representation. It does not describe the number of words in the English sentence. 'The large wooden door beside the upstairs reading room was locked at noon' can be one atom because the propositional analysis does not examine its nouns, adjectives, or relations. Later predicate logic provides tools for some internal structure that this introductory representation leaves intact.
By contrast, the short sentence 'It rains and freezes' can require two atoms once we restore its two complete claims: it rains, and it freezes. The shared subject in ordinary grammar does not prevent the sentence from joining assertions. Write the two meanings explicitly before deciding how to join their letters. This method avoids treating sentence length as a guide to logical complexity.
Not every occurrence of the English word 'and' joins two independent claims. 'Ada and Ben carried the piano together' can describe a joint action, rather than asserting that Ada carried it alone and Ben carried it alone. At this level, one atom for the whole joint-action claim may preserve the intended meaning better. Formalization requires interpretation; replacing every visible word mechanically can change what a sentence says.
The purpose of an argument also helps determine what to expose. If a later inference depends on the claim that a door is open, hiding 'the door is open and the light is on' inside a single unexplained atom would prevent a propositional rule from extracting the door claim. Representing it as D & L makes that structure available. Choose a consistent representation that preserves the connections relevant to the reasoning being tested.
Once the key is fixed for an exercise, follow it. If the key says P means 'the parcel has arrived and has been signed for', then P names that whole assertion in that exercise. You cannot silently redefine P as arrival alone. If a more detailed analysis is needed, declare new atomic meanings explicitly and translate the old claim using them. Changing levels is possible, but the change must be visible so a reader can compare the representations.
Repeated wording can repeat one claim without supplying independent support for it. Suppose a log contains a statement twice because one entry was copied from another. The two occurrences may both be represented by P. The notation records the claim's identity; it does not decide whether the reports come from independent observations. Questions about the quality and independence of evidence require information beyond the statement letters.
Different wording can also express the same chosen assertion. Under an explicitly declared reading, 'the west gate is unlocked' and 'the west gate is not locked' might both be represented as ~L, where L means that gate is locked. But 'the gate is open' need not have that meaning: an unlocked gate can remain physically closed. Do not treat familiar conversational associations as definitions. State the equivalence you intend and check that it preserves the distinction the problem needs.
Time is a particularly common source of accidental changes. Let P mean a pump runs at noon and Q mean it runs at one o'clock. These are distinct claims because a change between the two times is possible. P may be true while Q is false. By contrast, two occurrences of P in one assignment must have the same value, because both occurrences express the identical keyed claim at noon.
When reading back a formula, substitute the complete key meaning for each letter. If P means the parcel arrived on Monday, ~P says it is not the case that the parcel arrived on Monday. It does not by itself say the parcel arrived on Tuesday, was lost, or will never arrive. A denial excludes the specified claim while leaving many alternatives open. Reading back exposes additions that a hurried translation can smuggle in.
Finish a key audit with three checks. Every letter should name a complete claim rather than an object. Every recurrence should keep that claim unchanged, including relevant time and place. Every connective should express an intended relationship among those claims. These checks prepare a representation for later truth tables and proofs. They do not verify its observations, establish causal relations, or turn a repeated assertion into extra evidence. Their contribution is a stable language in which those later questions can be asked precisely.
A fictional maintenance log contains three entries: the pump is running at noon; the warning light is on at noon; the pump is running at noon. Let P represent the first complete claim and W the second. The log becomes P, W, P. There are three entries but only two distinct atoms. The repeated entry does not introduce a third independent condition or provide a second kind of evidence.
Now change the third entry to 'the pump is running at one o'clock'. That is a different claim. Introduce a new atom, such as R, and state its time in the key. P and R may have different values because the pump can stop between the observations. A stable key prevents the reader from mistakenly treating a noon report as proof of the later state.
The instruction 'inspect the pump at one o'clock' is different again. It belongs on a task list, not in this list of asserted observations. Completing a formalization does not mean the inspection was performed.
Finally, if a report says 'the pump is running and the warning light is on at noon', write P & W under the original key. The two claims remain visible rather than being hidden in an unexplained letter. This is useful when the next question asks whether one observation contradicts another or supports a diagnosis. The logical representation organizes the claims that were supplied; checking the actual machinery and the accuracy of the timestamps is a separate job.
If P means 'the parcel arrived', write P for that complete claim. Do not use P to mean 'parcel' and then attempt to attach an English verb to it. Nor should an atom silently include a different date when it reappears.
Identify the sentence's role.
The signal is green.
This asserts a condition; it does not ask a question or issue an instruction.
Find the complete claim in the key.
G = the signal is green
The key includes both the object and what is asserted of it.
Check that the reference is unchanged.
The same signal at the same stated time
A different signal or time would change the claim.
Write its formula.
G
The sentence asserts exactly the atomic claim.
Read back the result.
G: the signal is green
No extra assertion or connective has entered the translation.
Fix the first report.
P = the door is locked at noon
The time belongs to the claim.
Translate the noon entry.
P
It matches the key exactly.
Translate a copied noon entry.
P
Copying the report does not change its claim.
Inspect the later report.
The door is locked at one o'clock.
A later state can differ from the noon state.
Assign the later claim its own atom.
Q = the door is locked at one o'clock
The key must preserve the changed time.
Count and audit.
P, P, Q: three occurrences, two atoms
Only the repeated P must keep the same value within an assignment.
Identify the first positive claim.
P = the pump runs at noon
An atom names a complete assertion.
Identify the second positive claim.
W = the warning light is on at noon
The light is a separate condition.
Read the first clause of the report.
The pump runs: P
This affirms the keyed pump claim.
Read the second clause.
The light is not on: ~W
Negation denies the keyed light claim without changing its meaning.
Join the clauses asserted together.
P & ~W
The report asserts both the pump condition and the light's denial.
Read the full formula back.
The pump runs at noon and the warning light is not on at noon.
The read-back preserves the two claims and their common time.
Check what remains unasserted.
No claim about the pump at one o'clock
The formula cannot extend an observation beyond its keyed time.
Let L mean the light is on at noon.
L
The time belongs in the key.
Represent a report that the light is on at one o'clock.
Use a new atom M with the later time in its key.
The two observations can differ.
Count the distinct claims.
G means the signal is green. Write the formula for the statement that the signal is green.
Answer:
Complete this analysis of the log P, Q, P.
Count all written occurrences.
P, Q, P: occurrences occurrences
Each written letter contributes an occurrence.
Group repeated names.
P and Q: atoms distinct atoms
The repeated P keeps the same meaning.
Record the fixed key.
P has one meaning throughout the log.
A repeated symbol does not name a new statement.
P means the pump is running. Use ~ for not. Write the statement that the pump is not running.
Answer:
L means the light was on at noon; M means it was on at one o'clock. Represent the claim that it was on at one o'clock, without asserting anything about noon.
Answer:
A means the alarm sounds and B means the bell sounds. Express the claim that the bell does not sound; make no claim about the alarm.
Answer:
P means the pump runs and W means the warning light is on. Use & for and. Write the joint report that both statements hold.
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
N means the north gate is open and S means the south gate is open. Write the statement that the south gate is open.
Answer:
You can say what each letter stands for and distinguish an atom from a compound formula.
14. Another time, step 3
2
The time change creates a different statement.