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Turning a sentence into a formula: which connective the joining word demands, which way the arrow points, and how far a negation reaches.
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.
By the end of this lesson you will be able to write a dictionary of statement letters, symbolise sentences that use and, or, if, only if, unless, neither and not both, decide which way a conditional points, fix the scope of a negation with brackets, and check a symbolisation by asking which situation it rules out.
You can read a formula and find its main connective. Symbolising runs the other way: you are given a sentence and have to produce the formula whose truth table is the sentence's own.
To symbolise a sentence is to write a formula true in exactly the situations the sentence is. A dictionary fixes what each letter stands for, and it must be a whole statement, not a noun. A condition is necessary for another when the second cannot hold without it, and sufficient when it is enough on its own.
Fix a dictionary first: each letter stands for a complete statement, written out. Then the joining word decides the connective, and only the joining word does. And, but, although, yet are all $\wedge$ — the contrast they add is rhetorical and has no truth table. Or is $\vee$ unless the sentence says otherwise. The conditional is where the work is: if $P$ then $Q$ and $Q$ if $P$ are both $P \to Q$, while $P$ only if $Q$ is also $P \to Q$, because only if marks $Q$ as necessary for $P$. $P$ unless $Q$ is $\neg Q \to P$. Negation needs a decision about scope: neither $P$ nor $Q$ is $\neg P \wedge \neg Q$, while not both $P$ and $Q$ is $\neg (P \wedge Q)$, and the two differ on every row where exactly one half holds. The test of a symbolisation is never how similar it looks: it is whether the formula is false in exactly the situations the sentence rules out.
Another way: steps
Another way: example
The door opens only if the card is valid. $P$: the door opens. $Q$: the card is valid. Validity is necessary, so $P \to Q$. The ruled-out situation is an opening door with an invalid card, which is the one row the formula makes false.
The commonest error is reading only if as if and pointing the arrow backwards; P only if Q is $P \to Q$, not $Q \to P$. The second is letting a negation spread further than the English does: not both denies the pair and allows either one alone, while neither denies each. The third is symbolising a letter as something that is not a statement — $P$ has to be the card is valid, never the card.
The alarm sounds unless the door is propped open.
Read unless as if not.
$Q$: the door is propped open. With $\neg Q$, the alarm sounds: $\neg Q \to P$.
The clause after unless is the negated antecedent.
It says nothing about what happens when the door is propped open, which is why it is a conditional and not a biconditional.
Only one direction is claimed.
Not both the gate and the light are on: $\neg (P \wedge Q)$.
The pair is denied.
Neither the gate nor the light is on: $\neg P \wedge \neg Q$.
Each is denied.
They differ on the two rows where exactly one is on: the first is true there, the second false.
A row settles it.
$P$: the report is filed. $Q$: the auditor objects. Unless is if not.
So the formula is $\neg Q \to P$, false only when the auditor does not object and the report is still not filed.
Let P stand for “the file is saved” and Q for “the file is printed”. Write this in symbols: “the file is saved but not printed.”
Answer:
Match each English form to the formula that symbolises it.
| $P \to Q$ | $Q \to P$ | $\neg Q \to P$ | $\neg P \wedge \neg Q$ | $P \wedge Q$ | |
|---|---|---|---|---|---|
| $P$ only if $Q$ | |||||
| $P$ if $Q$ | |||||
| $P$ unless $Q$ | |||||
| neither $P$ nor $Q$ | |||||
| $P$ but $Q$ |
Is $\neg (P \wedge Q)$ a correct symbolisation of 'neither P nor Q'?
Let P stand for “the claim is paid” and Q for “the form is signed”. Write this in symbols: “the claim is not paid if the form is not signed.”
Answer:
Let P be “the parcel arrives” and Q be “the receipt is signed”. Mark every sentence that $P \to Q$ symbolises.
This task has no paper form; do it on a device.
Let P stand for “the lift goes up” and Q for “the lift goes down”. Write this in symbols: “the lift goes up or down, but not both.”
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
'P unless Q' is symbolised ~Q -> P. Give that formula's column, in the usual row order: both true, $P$ only, $Q$ only, neither.
| P | Q | ~Q -> P |
|---|---|---|
You can symbolise a compound English sentence and check the result against the situations the sentence rules out. Say in your own words why 'P only if Q' and 'P if Q' point the arrow in opposite directions.
8. Your turn: *the report is filed unless the auditor objects*, step 2