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[0] www.yak.net/d

Rainy Day Stories:

This is an introduction to Temporal Logic, based the system called "D",

studied by Arthur Prior and Robert Bull.

I haven't seen Temporal Logic used in Recreational Maths and would like to introduce this system.

It is easy enough that anyone can understand it, but compelling enough to make interesting puzzles.

These slides are on the web at www.yak.net/d

-- Henry Strickland « strick % yak.net »

-- Gathering for Gardner « G4G13 » 2018-04

References:

Copeland, B. Jack. Stanford Encyclopedia of Philosophy » Temporal Logic » Priorâ€™s Basic Tense Logic TL.

https://plato.stanford.edu/entries/logic-temporal/#PriBasTenLogTL

Goldblatt, Robert. Mathematical Modal Logic: a View of its Evolution. Centre for Logic, Language and Computation. Victoria University, P. O. Box 600, Wellington, New Zealand.

http://homepages.mcs.vuw.ac.nz/~rob/papers/modalhist.pdf

Prior, Arthur. Time and Modality. Oxford University Press. 1957.

https://books.google.com/books/about/Time_and_Modality.html?id=K5nymD8qgigC

[1] www.yak.net/d

R = It rains (today).

⚪R = TOMORROW it rains.

◇R = EVENTUALLY it rains.

◻R = FOREVERMORE it rains.

There's only four symbols to learn, so please pay attention, because you're going to need these four.

R means Rain on the day in question, which by default is today.

Circle (⚪) means TOMORROW. Whatever follows the circle is true tomorrow.

Diamond (◇) means EVENTUALLY, at least once. Whatever follows the diamond is true at least once, EITHER today, OR tomorrow, OR some day in the future.

Box (◻) means FOREVERMORE. Whatever follows the box is true today, AND tomorrow, AND every day into the future.

[2] www.yak.net/d

How to remember them:

⚪ = The "point" TOMORROW.

A circle is like a big point, and that point in time is TOMORROW.

◇ = EVENTUALLY at any point

A diamond has many points, and ANY day will do.

◻ = FOREVERMORE

A box looks like a calendar, marking ALL days, starting today.

[3] www.yak.net/d

Four Example Rainy Day Stories (1=Rain, 0=NoRain):

1 0 1 1 1 0 0 0 0 0 0 0 0 0...

0 1 0 1 0 1 0 1 0 1 0 1 0 1...

0 0 0 0 0 0 0 0 0 0 0 0 0 0...

1 1 1 1 1 1 1 1 1 1 1 1 1 1...

This is how I visualize Rainy Day Stories.

Each line of 1s and 0s is a Story.

1 means Rain, 0 means No Rain.

The first digit on each line is whether it rains today.

The second digit is whether it rains tomorrow.

The third digit is the next day, and so forth, into the future.

[4] www.yak.net/d

R = It rains (today).

☑ 1 0 1 1 1 0 0 0 0 0 0 0 0 0...

☐ 0 1 0 1 0 1 0 1 0 1 0 1 0 1...

☐ 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

☑ 1 1 1 1 1 1 1 1 1 1 1 1 1 1...

Raining Today matches the 1st and 4th story, because they start with a 1, meaning Rain Today.

[5] www.yak.net/d

⚪R = TOMORROW it rains.

☐ 1 0 1 1 1 0 0 0 0 0 0 0 0 0...

☑ 0 1 0 1 0 1 0 1 0 1 0 1 0 1...

☐ 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

☑ 1 1 1 1 1 1 1 1 1 1 1 1 1 1...

TOMORROW RAIN matches the 2nd and 4th story, because they have a 1 in the second position, meaning Rain Tomorrow.

[6] www.yak.net/d

◇R = EVENTUALLY it rains.

☑ 1 0 1 1 1 0 0 0 0 0 0 0 0 0...

☑ 0 1 0 1 0 1 0 1 0 1 0 1 0 1...

☐ 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

☑ 1 1 1 1 1 1 1 1 1 1 1 1 1 1...

EVENTUALLY RAIN matches all but the 3rd story, which has no rainy days at all.

[7] www.yak.net/d

◻R = FOREVERMORE it rains.

☐ 1 0 1 1 1 0 0 0 0 0 0 0 0 0...

☐ 0 1 0 1 0 1 0 1 0 1 0 1 0 1...

☐ 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

☑ 1 1 1 1 1 1 1 1 1 1 1 1 1 1...

FOREVERMORE RAIN only matches the last story, in which every day is rainy.

[8] www.yak.net/d

a ∧ b = a AND b

a ∨ b = a OR b

¬ b = NOT b

a → b = a IMPLIES b (IF a THEN b)

All the usual Logic Operators are allowed: And, Or, Not, Implies.

[9] www.yak.net/d

R ∧ ⚪R

☐ 1 0 1 1 1 0 0 0 0 0 0 0 0 0...

☐ 0 1 0 1 0 1 0 1 0 1 0 1 0 1...

☐ 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

☑ 1 1 1 1 1 1 1 1 1 1 1 1 1 1...

It rains TODAY ... AND ... it rains TOMORROW.

That matches only the last story, which starts with "1 1".

[10] www.yak.net/d

R ∧ ⚪¬R

☑ 1 0 1 1 1 0 0 0 0 0 0 0 0 0...

☐ 0 1 0 1 0 1 0 1 0 1 0 1 0 1...

☐ 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

☐ 1 1 1 1 1 1 1 1 1 1 1 1 1 1...

It rains TODAY ... AND ... TOMORROW it does NOT rain.

That matches only the first story, which starts with "1 0".

[11] www.yak.net/d

⚪⚪R ∧ ⚪⚪⚪R

☑ 1 0 1 1 1 0 0 0 0 0 0 0 0 0...

☐ 0 1 0 1 0 1 0 1 0 1 0 1 0 1...

☐ 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

☑ 1 1 1 1 1 1 1 1 1 1 1 1 1 1...

Two circles in a row means "Tomorrow's Tomorrow"; three circles is the day after that.

So, it rains THE DAY AFTER TOMORROW and also THE DAY AFTER THAT.

That matches the first and the last story, which both have "1"s in the 3rd and 4th position.

[12] www.yak.net/d

◇◻R

Eventually, it will rain forevermore.

"It may rain off and on for a while, but some day it will rain and never quit."

The next two get interesting.

Diamond Box R means ... Eventually ... it will rain forevermore.

That is, it may rain off an on for a while, but some day it will rain and never quit.

◻◇R

Forevermore... it will eventually rain.

"No matter how far in the future we go, it will always rain again."

Box Diamond R means Forevermore it will eventually rain.

No matter how far in the future we go, it will always rain again.

If our story is that it only rains on prime-numbered days, that story would match, because no matter how far we go, there's always another prime.

[13] www.yak.net/d

◇ ( R ∧ ⚪R )

"Eventually it will rain (at least) two days in a row."

◻ ( R → ⚪¬R )

"It will never rain two (or more) days in a row."

◻ ( R → ⚪R )

"If it ever rains, it will never quit."

[14] www.yak.net/d

Can you write expressions for these:

(1) "It rains only every other day."

(Your expression must match exactly two stories.)

(2) "Whenever it rains, it rains exactly two days in a row (never one, never three)."

(But it may not rain at all.)

[15] www.yak.net/d

Game of Thrones:

⚪◇W ∧ ¬W

I'll end with some Real World Examples.

From the Game of Thrones:

Starting tomorrow, eventually W; but not W today.

That means: "Winter is coming!"

[16] www.yak.net/d

Game of Thrones:

⚪◇W ∧ ¬W

Morton salt:

◻ ( R → P )

From Morton Salt:

It is always the case that R implies P.

That means: "When it rains, it pours!"

[17] www.yak.net/d

Game of Thrones:

⚪◇W ∧ ¬W

Morton salt:

◻ ( R → P )

Genesis 8 (God's Covenant to Noah):

39 i

¬◇ ∧ ⚪ R

i=0

From Genesis 8, we have God's Covenant to Noah:

It will never happen that,

it rains Today and Tomorrow and the Day after That, etc.,

up to 39 circles and an R.

In other words: "Starting today, it will never rain 40 days in a row!"

[18] www.yak.net/d

Temporal Logic (or Tense Logic) "D" studied by Arthur Prior and Robert Bull.

Search for: Tense Logic, Temporal Logic, Modal Logic.

These slides: http://www.yak.net/d

Source code: https://github.com/strickyak/d-talk

I dedicate these slides to the Public Domain ( Creative Commons CC0 1.0 Universal ).

That's it for now.

I've made postcards with more puzzles.

Some are spread around the tables,

and they will also be in the gift exchange.

[19] www.yak.net/d

Bibliography:

Robert Goldblatt. Mathematical Modal Logic: a View of its Evolution. Centre for Logic, Language and Computation. Victoria University, P. O. Box 600, Wellington, New Zealand. http://homepages.mcs.vuw.ac.nz/~rob/papers/modalhist.pdf

Arthur N Prior. Time and Modality. London : Oxford : At The Clarendon Press, 1968