Epistemic Logic Kevin: I know. Tom: I know you know. Kevin: I know you know I know.

What is Epistemic Logic? Epistemic Logic is the logic of knowledge It formalizes our intuitions about who knows what about what It matters in Magic because we are constantly trying to figure out who knows what

Red Sligh 4 cards in hand Green Beatdown 2 cards in hand Hand You’re green. Do you block? what do you know? what do you know he knows? what do he think you know?

How is it modeled? Epistemic Logic is a sub-field of Modal Logic Modal Logic deals with different modes – It qualifies the truth of a statement For example – “It is possible that” – “I know that” – “It will happen that” Just like other logics, we have a formal language for it

Review of Propositional Logic A ∧ B (and) – is true when both A is true and B is true A ∨ B (or) – is true when either A is true or B is true ¬ A (not) – is true when A is false A → B (implication) – is true when either A is false or B is true

First-Order Logic Extends prop logic with quantifiers that specify objects in a domain ∃ x A(x) (existential) – Is true when there exists an x such that A(x) is true – For example A(x) is true when x is a girl ∃ x A(x) is probably not true for the domain of this class ∀ x A(x) (universal) – Is true when for all objects in the domain, A(x) is true

So why do we need it? Began because C I Lewis didn’t like → Problem of monotocity with implication – If we know that something is true, we can add assumptions, and it’ll still be true For example, given A → B, it is necessarily true that (A ∧ C) → B – why? – why would this be a problem?

How to justify partying via logic Let A be “I will study tonight” Let B be “I will do well on my midterm tomorrow” A → B is reasonable – “If I will study tonight, I will do well on my midterm tomorrow” Let C be “I will drink tonight” (A ∧ C) → B ? – “If I will study tonight and drink tonight, I will do well on my midterm tomorrow.”

Modal operators ☐ T (box) means that Tom knows φ – Box is borrowed from some Japanese script  T (diamond) means that it is consistent with what Tom knows that φ – Diamond is borrowed from wingdings and is usually not filled in

Kripke Structures! Composed of a set of worlds or states – Representing different modes of possibility And relations that connect them – Representing related worlds

A B A,B B,C C w1w1 w2w2 w3w3 w4w4 w5w5 w 4 Rw 2 Each circle represents a possible world Each connection represents 2 worlds that are indistinguishable from each other by someone

A B A,B B,C C w1w1 w2w2 w3w3 w4w4 w5w5 w 4 Rw 2 Things that are true in this world: w 1 |= A means that A is true in world w 1 w 1 |=  B because w 1 can see a world where B is true w 2 |= ☐ B because in all worlds w 2 can see, B is true

Kripke Structures in action

Card on the left is what they play Card on the right is what they have in hand How are these linked? Which of these are distinguishable for your opponent?

So what else can it do for us? Now I cheat and use someone else’s slides

3Zo 3Zo

B/G Elves – Charles Gindy 1 Urborg, Tomb of Yawgmoth 4 Treetop Village 2 Swamp 1 Pendelhaven 4 Mutavault 4 Llanowar Wastes 4 Gilt-Leaf Palace 3 Forest 1 Boreal Druid 4 Llanowar Elves 4 Wren's Run Vanquisher 4 Tarmogoyf 4 Imperious Perfect 4 Civic Wayfinder 3 Chameleon Colossus 4 Thoughtseize 4 Terror 3 Profane Command 2 Garruk Wildspeaker 1 st place, PT Hollywood 2008, TS/LOR Standard

U/W Merfolk – Jan Ruess 4 Wanderwine Hub 11 Snow-Covered Island 4 Mutavault 1 Faerie Conclave 4 Adarkar Wastes 2 Venser, Shaper Savant 2 Tideshaper Mystic 2 Sygg, River Guide 3 Stonybrook Banneret 3 Sower of Temptation 4 Silvergill Adept 4 Merrow Reejerey 4 Lord of Atlantis 4 Cursecatcher 3 Sage's Dousing 3 Cryptic Command 2 Ancestral Vision 2 st place, PT Hollywood 2008, TS/LOR Standard

20 Life Llanowar Elves in graveyard 20 Life 3 cards in hand Banneret in graveyard Attack step. Do you attack? What do you do?

Relevant facts You did not have a turn 2 play You did not have a turn 3 play Jan did not counter the first Colossus Jan has mana to double-protect

You can play 1 removal He has a counter You can play 1 removal He has a counter You can play >2 removal He doesn’t have a counter You can play >2 removal He doesn’t have a counter You can play 2 removal He has a counter You can play 2 removal He has a counter You can play a creature He has a counter You can play a creature He has a counter You can play 1 removal He doesn’t have a counter You can play 1 removal He doesn’t have a counter You can play 2 removal He doesn’t have a counter You can play 2 removal He doesn’t have a counter You can play >2 removal He has a counter You can play >2 removal He has a counter You can play a creature He doesn’t have a counter You can play a creature He doesn’t have a counter

You can play 1 removal He has a counter You can play 1 removal He has a counter You can play >2 removal He doesn’t have a counter You can play >2 removal He doesn’t have a counter You can play 2 removal He has a counter You can play 2 removal He has a counter You can play a better creature He has a counter You can play a better creature He has a counter You can play 1 removal He doesn’t have a counter You can play 1 removal He doesn’t have a counter You can play 2 removal He doesn’t have a counter You can play 2 removal He doesn’t have a counter You can play >2 removal He has a counter You can play >2 removal He has a counter You can play a better creature He doesn’t have a counter You can play a better creature He doesn’t have a counter

You can play 1 removal He has a counter You can play 1 removal He has a counter You can play 2 removal He has a counter You can play 2 removal He has a counter You can play 1 removal He doesn’t have a counter You can play 1 removal He doesn’t have a counter You can play 2 removal He doesn’t have a counter You can play 2 removal He doesn’t have a counter

You can play 1 removal He doesn’t have a counter You can play 1 removal He doesn’t have a counter You can play 2 removal He doesn’t have a counter You can play 2 removal He doesn’t have a counter

20 Life 3 cards in hand Banneret in graveyard 20 Life Llanowar Elves in graveyard He double-blocks. What do you do?

5 Cards in hand 20 Life Elves in graveyard 20 Life Banneret in graveyard No pump again What do you do?

Drafting We talked about signals in drafting Basically, let your neighbors know what you’re doing Let’s take Martin Juza from GP Tampa

Take Rampaging Baloths In green Take Rampaging Baloths In green Take Hideous End In black Take Hideous End In black Take Hideous End In red Take Hideous End In red Take Rampaging Baloths In red Take Rampaging Baloths In red Take Rampaging Baloths In black Take Rampaging Baloths In black Take Rampaging Baloths In white Take Rampaging Baloths In white Take Rampaging Baloths In blue Take Rampaging Baloths In blue Take Hideous End In green Take Hideous End In green Take Hideous End In white Take Hideous End In white Take Hideous End In blue Take Hideous End In blue

Uncommon

How are you a better player? Understand what information you are and aren’t revealing Guess what your opponent has based on how they play Exploit what your opponent doesn’t know Send signals Think about you games in terms of who knows what

Want more? Opportunities to play such that you aren’t revealing information – rolling a dice to block Cards and plays that develop strong ambiguity

Where Can You Learn More? Phil 154, potentially Phil 151 What They Think You’re Thinking – Steve Sadin Telepathy – Josh Utter-Leyton

∧∨ ¬ →↔ ∃∀ ☐  φ