key elements of the propositional calculuskryten.mm.rpi.edu/courses/ahr/prop_calc.pdf · key...

Key Elements of the Propositional Calculus

(with Help From PBS Masterpiece SHERLOCK)

Selmer BringsjordAre Humans Rational?

9/21/15 (updates 9/24/15 & 9/28/15)

[email protected]

http://www.pbs.org/wgbh/masterpiece/sherlock/

mailto:[email protected]

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Success v Failure

√

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So what’s got to be in that mind to make it rational?!?

Success v Failure

√

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?

?

So what’s got to be in that mind to make it rational?!?

Success v Failure

√

A1: Deductive ToolsA2: Inductive ToolsA3: Analysis

2015

Syntax

S ::=Object | Agent | Self @ Agent | ActionType | Action v Event |

Moment | Boolean | Fluent | Numeric

f ::=

action : Agent⇥ActionType ! Action

initially : Fluent ! Boolean

holds : Fluent⇥Moment ! Boolean

happens : Event⇥Moment ! Boolean

clipped : Moment⇥Fluent⇥Moment ! Boolean

initiates : Event⇥Fluent⇥Moment ! Boolean

terminates : Event⇥Fluent⇥Moment ! Boolean

prior : Moment⇥Moment ! Boolean

interval : Moment⇥Boolean

⇤ : Agent ! Self

payoff : Agent⇥ActionType⇥Moment ! Numeric

t ::= x : S | c : S | f (t1 , . . . , tn)

f ::=

t : Boolean | ¬f | f^y | f_y |

P(a, t,f) | K(a, t,f) | C(t,f) | S(a,b, t,f) | S(a, t,f)

B(a, t,f) | D(a, t,holds( f , t0)) | I(a, t,happens(action(a⇤ ,a), t0))

O(a, t,f,happens(action(a⇤ ,a), t0))

Rules of Inference

C(t,P(a, t,f)! K(a, t,f))[R1 ] C(t,K(a, t,f)! B(a, t,f))

[R2 ]

C(t,f) t t1 . . . t t

n

K(a1 , t1 , . . .K(an

, tn

,f) . . .)[R3 ]

K(a, t,f)

f[R4 ]

C(t,K(a, t1 ,f1 ! f2))! K(a, t2 ,f1)! K(a, t3 ,f2)[R5 ]

C(t,B(a, t1 ,f1 ! f2))! B(a, t2 ,f1)! B(a, t3 ,f2)[R6 ]

C(t,C(t1 ,f1 ! f2))! C(t2 ,f1)! C(t3 ,f2)[R7 ]

C(t,8x. f ! f[x 7! t])[R8 ] C(t,f1 $ f2 ! ¬f2 ! ¬f1)

[R9 ]

C(t, [f1 ^ . . .^fn

! f]! [f1 ! . . .! fn

! y])[R10 ]

B(a, t,f) f ! y

B(a, t,y)[R11a

]B(a, t,f) B(a, t,y)

B(a, t,y^f)[R11b

]

S(s,h, t,f)

B(h, t,B(s, t,f))[R12 ]

I(a, t,happens(action(a⇤ ,a), t0))

P(a, t,happens(action(a⇤ ,a), t))[R13 ]

B(a, t,f) B(a, t,O(a⇤ , t,f,happens(action(a⇤ ,a), t0)))


K(a, t,I(a⇤ , t,happens(action(a⇤ ,a), t0)))[R14 ]

f $ y

O(a, t,f,g)$ O(a, t,y,g)[R15 ]

1

3

always position some particular work he and likeminded collaboratorsare undertaking within a view of logic that allows a particularlogical system to be positioned relative to three dimensions, whichcorrespond to the three arrows shown in Figure 2. We have positionedDCEC ⇤within Figure 2; it’s location is indicated by the black dottherein, which the reader will note is quite far down the dimensionof increasing expressivity that ranges from expressive extensionallogics (e.g., FOL and SOL), to logics with intensional operators forknowledge, belief, and obligation (so-called philosophical logics; foran overview, see Goble 2001). Intensional operators like these arefirst-class elements of the language for DCEC ⇤. This language isshown in Figure 1.

Syntax

S ::=Object | Agent | Self � Agent | ActionType | Action � Event |


f ::=










⇤ : Agent ! Self


t ::= x : S | c : S | f (t1 , . . . , tn)

f ::=

t : Boolean | ¬f | f^� | f�� | 8x : S. f | �x : S. f


B(a, t,f) | D(a, t,holds( f , t�)) | I(a, t,happens(action(a⇤ ,a), t�))

O(a, t,f,happens(action(a⇤ ,a), t�))

Rules of Inference

C(t,P(a, t,f) ! K(a, t,f))[R1 ]

C(t,K(a, t,f) ! B(a, t,f))[R2 ]

C(t,f) t t1 . . . t tn

K(a1 , t1 , . . .K(an , tn ,f) . . .)[R3 ]

K(a, t,f)

f[R4 ]

C(t,K(a, t1 ,f1 ! f2) ! K(a, t2 ,f1) ! K(a, t3 ,f3))[R5 ]

C(t,B(a, t1 ,f1 ! f2) ! B(a, t2 ,f1) ! B(a, t3 ,f3))[R6 ]

C(t,C(t1 ,f1 ! f2) ! C(t2 ,f1) ! C(t3 ,f3))[R7 ]

C(t,8x. f ! f[x �! t])[R8 ]

C(t,f1 � f2 ! ¬f2 ! ¬f1)[R9 ]

C(t, [f1 ^ . . .^fn ! f] ! [f1 ! . . . ! fn ! �])[R10 ]

B(a, t,f) B(a, t,f ! �)

B(a, t,�)[R11a ]

B(a, t,f) B(a, t,�)

B(a, t,�^f)[R11b ]

S(s,h, t,f)

B(h, t,B(s, t,f))[R12 ]

I(a, t,happens(action(a⇤ ,a), t�))


B(a, t,f) B(a, t,O(a⇤ , t,f,happens(action(a⇤ ,a), t�)))


K(a, t,I(a⇤ , t,happens(action(a⇤ ,a), t�)))[R14 ]

f � �

O(a, t,f,�) � O(a, t,�,�)[R15 ]

1

Fig. 1. DCEC ⇤Syntax and Rules of Inference

Fig. 2. Locating DCEC ⇤in “Three-Ray” Leibnizian Universe

The final layer in our hierarchy is built upon an even more expres-sive logic: DCEC ⇤

CL. The subscript here indicates that distinctiveelements of the branch of logic known as conditional logic are

U

ADR M

DCEC ⇤

DCEC ⇤CL

Moral/Ethical Stack

Robotic Stack

Fig. 3. Pictorial Overview of the Situation Now The first layer, U, is, assaid in the main text, inspired by UIMA; the second layer is based on whatwe call analogico-deductive reasoning for ethics; the third on the “deonticcognitive event calculus” with a indirect indexical; and the fourth like thethird except that the logic in question includes aspects of conditional logic.(Robot schematic from Aldebaran Robotics’ user manual for Nao. The RAIRLab has a number of Aldebaran’s impressive Nao robots.)

included.8 Without these elements, the only form of a conditionalused in our hierarchy is the material conditional; but the materialconditional is notoriously inexpressive, as it cannot represent coun-terfactuals like:

If the robot had been more empathetic, Officer Smith would have thrived.

While elaborating on this architecture or any of the four layersis beyond the scope of the paper, we do note that DCEC ⇤(and afortiori DCEC ⇤

CL) has facilities for representing and reasoning overmodalities and self-referential statements that no other computationallogic enjoys; see (Bringsjord & Govindarajulu 2013) for a more in-depth treatment.

B. Augustinian Definition, Formal VersionWe view a robot abstractly as a robotic substrate rs on which wecan install modules {m1,m2, . . . ,mn}. The robotic substrate rs wouldform an immutable part of the robot and could neither be removednor modified. We can think of rs as akin to an “operating system”for the robot. Modules correspond to functionality that can be addedto robots or removed from them. Associated with each module miis a knowledge-base KBmi that represents the module. The substratealso has an associated knowledge-base KBrs. Perhaps surprisingly,we don’t stipulate that the modules are logic-based; the modulescould internally be implemented using computational formalisms (e.g.neural networks, statistical AI) that at the surface level seem far awayfrom formal logic. No matter what the underlying implementation ofa module is, if we so wished we could always talk about modulesin formal-logic terms.9 This abstract view lets us model robots that

8Though written rather long ago, (Nute 1984) is still a wonderful intro-duction to the sub-field in formal logic of conditional logic. In the finalanalysis, sophisticated moral reasoning can only be accurately modeled forformal logics that include conditionals much more expressive and nuancedthan the material conditional. (Reliance on conditional branching in standarprogramming languages is nothing more than reliance upon the materialconditional.) For example, even the well-known trolley-problem cases (inwhich, to save multiple lives, one can either redirect a train, killing oneperson in the process, or directly stop the train by throwing someone in frontof it), which are not exactly complicated formally speaking, require, whenanalyzed informally but systematically, as indicated e.g. by Mikhail (2011),counterfactuals.

9This stems from the fact that theorem proving in just first-order logic isenough to simulate any Turing-level computation; see e.g. (Boolos, Burgess& Jeffrey 2007, Chapter 11).

1666

Leibniz

“Universal Computational Logic”

The Future?

20201956

Logic Theorist(birth of modern

logicist AI)(birth of agent-based/

behavioral econ)

Simon

1.5 centuries < Boole

CC

1854

2015

Syntax

S ::=Object | Agent | Self @ Agent | ActionType | Action v Event |


f ::=










⇤ : Agent ! Self


t ::= x : S | c : S | f (t1 , . . . , tn)

f ::=

t : Boolean | ¬f | f^y | f_y |


B(a, t,f) | D(a, t,holds( f , t0)) | I(a, t,happens(action(a⇤ ,a), t0))


Rules of Inference

C(t,P(a, t,f)! K(a, t,f))[R1 ] C(t,K(a, t,f)! B(a, t,f))

[R2 ]

C(t,f) t t1 . . . t t

n

K(a1 , t1 , . . .K(an

, tn

,f) . . .)[R3 ]

K(a, t,f)

f[R4 ]

C(t,K(a, t1 ,f1 ! f2))! K(a, t2 ,f1)! K(a, t3 ,f2)[R5 ]

C(t,B(a, t1 ,f1 ! f2))! B(a, t2 ,f1)! B(a, t3 ,f2)[R6 ]

C(t,C(t1 ,f1 ! f2))! C(t2 ,f1)! C(t3 ,f2)[R7 ]

C(t,8x. f ! f[x 7! t])[R8 ] C(t,f1 $ f2 ! ¬f2 ! ¬f1)

[R9 ]

C(t, [f1 ^ . . .^fn

! f]! [f1 ! . . .! fn

! y])[R10 ]

B(a, t,f) f ! y

B(a, t,y)[R11a

]B(a, t,f) B(a, t,y)

B(a, t,y^f)[R11b

]

S(s,h, t,f)

B(h, t,B(s, t,f))[R12 ]

I(a, t,happens(action(a⇤ ,a), t0))


B(a, t,f) B(a, t,O(a⇤ , t,f,happens(action(a⇤ ,a), t0)))


K(a, t,I(a⇤ , t,happens(action(a⇤ ,a), t0)))[R14 ]

f $ y

O(a, t,f,g)$ O(a, t,y,g)[R15 ]

1

3

always position some particular work he and likeminded collaboratorsare undertaking within a view of logic that allows a particularlogical system to be positioned relative to three dimensions, whichcorrespond to the three arrows shown in Figure 2. We have positionedDCEC ⇤within Figure 2; it’s location is indicated by the black dottherein, which the reader will note is quite far down the dimensionof increasing expressivity that ranges from expressive extensionallogics (e.g., FOL and SOL), to logics with intensional operators forknowledge, belief, and obligation (so-called philosophical logics; foran overview, see Goble 2001). Intensional operators like these arefirst-class elements of the language for DCEC ⇤. This language isshown in Figure 1.

Syntax

S ::=Object | Agent | Self � Agent | ActionType | Action � Event |


f ::=










⇤ : Agent ! Self


t ::= x : S | c : S | f (t1 , . . . , tn)

f ::=

t : Boolean | ¬f | f^� | f�� | 8x : S. f | �x : S. f


B(a, t,f) | D(a, t,holds( f , t�)) | I(a, t,happens(action(a⇤ ,a), t�))


Rules of Inference

C(t,P(a, t,f) ! K(a, t,f))[R1 ]

C(t,K(a, t,f) ! B(a, t,f))[R2 ]

C(t,f) t t1 . . . t tn

K(a1 , t1 , . . .K(an , tn ,f) . . .)[R3 ]

K(a, t,f)

f[R4 ]

C(t,K(a, t1 ,f1 ! f2) ! K(a, t2 ,f1) ! K(a, t3 ,f3))[R5 ]

C(t,B(a, t1 ,f1 ! f2) ! B(a, t2 ,f1) ! B(a, t3 ,f3))[R6 ]

C(t,C(t1 ,f1 ! f2) ! C(t2 ,f1) ! C(t3 ,f3))[R7 ]

C(t,8x. f ! f[x �! t])[R8 ]

C(t,f1 � f2 ! ¬f2 ! ¬f1)[R9 ]

C(t, [f1 ^ . . .^fn ! f] ! [f1 ! . . . ! fn ! �])[R10 ]

B(a, t,f) B(a, t,f ! �)

B(a, t,�)[R11a ]

B(a, t,f) B(a, t,�)

B(a, t,�^f)[R11b ]

S(s,h, t,f)

B(h, t,B(s, t,f))[R12 ]

I(a, t,happens(action(a⇤ ,a), t�))


B(a, t,f) B(a, t,O(a⇤ , t,f,happens(action(a⇤ ,a), t�)))


K(a, t,I(a⇤ , t,happens(action(a⇤ ,a), t�)))[R14 ]

f � �

O(a, t,f,�) � O(a, t,�,�)[R15 ]

1

Fig. 1. DCEC ⇤Syntax and Rules of Inference

Fig. 2. Locating DCEC ⇤in “Three-Ray” Leibnizian Universe

The final layer in our hierarchy is built upon an even more expres-sive logic: DCEC ⇤

CL. The subscript here indicates that distinctiveelements of the branch of logic known as conditional logic are

U

ADR M

DCEC ⇤

DCEC ⇤CL

Moral/Ethical Stack

Robotic Stack

Fig. 3. Pictorial Overview of the Situation Now The first layer, U, is, assaid in the main text, inspired by UIMA; the second layer is based on whatwe call analogico-deductive reasoning for ethics; the third on the “deonticcognitive event calculus” with a indirect indexical; and the fourth like thethird except that the logic in question includes aspects of conditional logic.(Robot schematic from Aldebaran Robotics’ user manual for Nao. The RAIRLab has a number of Aldebaran’s impressive Nao robots.)

included.8 Without these elements, the only form of a conditionalused in our hierarchy is the material conditional; but the materialconditional is notoriously inexpressive, as it cannot represent coun-terfactuals like:

If the robot had been more empathetic, Officer Smith would have thrived.

While elaborating on this architecture or any of the four layersis beyond the scope of the paper, we do note that DCEC ⇤(and afortiori DCEC ⇤

CL) has facilities for representing and reasoning overmodalities and self-referential statements that no other computationallogic enjoys; see (Bringsjord & Govindarajulu 2013) for a more in-depth treatment.

B. Augustinian Definition, Formal VersionWe view a robot abstractly as a robotic substrate rs on which wecan install modules {m1,m2, . . . ,mn}. The robotic substrate rs wouldform an immutable part of the robot and could neither be removednor modified. We can think of rs as akin to an “operating system”for the robot. Modules correspond to functionality that can be addedto robots or removed from them. Associated with each module miis a knowledge-base KBmi that represents the module. The substratealso has an associated knowledge-base KBrs. Perhaps surprisingly,we don’t stipulate that the modules are logic-based; the modulescould internally be implemented using computational formalisms (e.g.neural networks, statistical AI) that at the surface level seem far awayfrom formal logic. No matter what the underlying implementation ofa module is, if we so wished we could always talk about modulesin formal-logic terms.9 This abstract view lets us model robots that

8Though written rather long ago, (Nute 1984) is still a wonderful intro-duction to the sub-field in formal logic of conditional logic. In the finalanalysis, sophisticated moral reasoning can only be accurately modeled forformal logics that include conditionals much more expressive and nuancedthan the material conditional. (Reliance on conditional branching in standarprogramming languages is nothing more than reliance upon the materialconditional.) For example, even the well-known trolley-problem cases (inwhich, to save multiple lives, one can either redirect a train, killing oneperson in the process, or directly stop the train by throwing someone in frontof it), which are not exactly complicated formally speaking, require, whenanalyzed informally but systematically, as indicated e.g. by Mikhail (2011),counterfactuals.

9This stems from the fact that theorem proving in just first-order logic isenough to simulate any Turing-level computation; see e.g. (Boolos, Burgess& Jeffrey 2007, Chapter 11).

1666

Leibniz

“Universal Computational Logic”

The Future?

20201956

Logic Theorist(birth of modern

logicist AI)(birth of agent-based/

behavioral econ)

Simon

1.5 centuries < Boole

CC

1854

Propositional Calculus!

Variables & Connectives

Variables & ConnectivesVariables to represent declarative statements.


E.g., K to represent ‘There is a king in the hand’.



And five simple Boolean connectives (with the truth tables that define them):




not ¬ and ^ or (inclusive) _ if ... then ... ! ... if and only if ... $




not ¬ and ^ or (inclusive) _ if ... then ... ! ... if and only if ... $

� ¬�T F

F T

� � ^ T T T

T F F

F T F

F F F

� � _ T T T

T F T

F T T

F F F

� �!

T T T

T F F

F T T

F F T

� �$

T T T

T F F

F T F

F F T

MOLLY: It’s all well and clever having a Mind Palace, but you’ve only three seconds of consciousness left to use it. So, come on – what’s going to kill you? SHERLOCK: Blood loss. MOLLY (quietly, intensely): Exactly. (Sherlock looks at her, frowning a little.) MOLLY: So, it’s all about one thing now. MOLLY: Forwards, or backwards? MOLLY (offscreen): We need to decide which way you’re going to fall. ANDERSON: One hole, or two? SHERLOCK (frowning and turning to look over his shoulder at him): Sorry? MOLLY: Is the bullet still inside you ... or is there an exit wound? (The perspective changes and she is no longer in front of him, though Anderson is still behind him.) MOLLY (voiceover): It’ll depend on the gun. (Sherlock turns his head to the left and now he can see diagrams of many different pistols in front of his eyes. He zooms in on one – which changes from a blue outline to a yellow one – and a tag appears above it reading, “Cat-0208”.) SHERLOCK: That one, I think. (He looks across the diagrams and another pistol identified as “Cat-077839” turns yellow. He moves on to another gun which changes to yellow. We can’t see the first part of the identification tag but its number ends “173634”.) SHERLOCK: Or that one. MYCROFT (offscreen): Oh, for God’s sake, Sherlock. (Sherlock turns his head to the right and sees his brother sitting at his desk in his office at The Diogenes Club.) MYCROFT: It doesn’t matter about the gun. Don’t be stupid. (Sherlock turns and walks towards him. Mycroft leans forward and folds his hands on the table in front of him.) MYCROFT: You always were so stupid. (Sherlock continues towards the desk, but now he’s a young boy – about eleven years old – and wearing dark trousers and a shirt with a buttoned dark green cardigan over it. He walks slowly towards his big brother.) MYCROFT: Such a disappointment. YOUNG SHERLOCK (angrily): I’m not stupid. MYCROFT (sternly): You’re a very stupid little boy. (He stands up and walks around the table.) MYCROFT: Mummy and Daddy are very cross … because it doesn’t matter about the gun. YOUNG SHERLOCK (frowning up at him): Why not? MYCROFT: You saw the whole room when you entered it. What was directly behind you when you were murdered? YOUNG SHERLOCK (sounding petulant): I’ve not been murdered yet. MYCROFT (leaning down to him): Balance of probability, little brother. (In Magnussen’s room, adult Sherlock also turns around to where a row of panelled mirrors is behind him on the wall. Mycroft can be seen fuzzily reflected in the mirrors as if he is standing some distance away. Sherlock walks closer to the mirrors and looks in them.) MYCROFT (walking closer): If the bullet had passed through you, what would you have heard? SHERLOCK: The mirror shattering. MYCROFT: You didn’t. Therefore ...? (Sherlock turns and slowly walks past him.) SHERLOCK: The bullet’s still inside me. (He walks back to his original position.) ANDERSON (offscreen): So, we need to take him down backwards. MOLLY (standing in front of Sherlock again): I agree. Sherlock ...

(1) ¬Inside ! Cracked

(2) ¬Cracked) (3) ¬¬Inside

) (4) Inside

(5) Inside ! ShldFallBackwards

rule?

rule?

) (6) ShldFallBackwards rule?

How Sherlock Uses the Propositional Calculus to Save His Life

modus tollens

�! ,¬ ¬�

modus ponens

�! ,�

disjunctive syllogism

� _ ,¬�

� _ ,¬ �

DeMorgan’s Laws

¬(� _ )¬� ^ ¬

¬(� ^ )¬� _ ¬

double negation

¬¬��

�

¬¬�

Some Helpful Inference Schemas for the Propositional Calculus

modus tollens

�! ,¬ ¬�

modus ponens

�! ,�


� _ ,¬�

� _ ,¬ �

DeMorgan’s Laws

¬(� _ )¬� ^ ¬

¬(� ^ )¬� _ ¬

double negation

¬¬��

�

¬¬�


explosion

� ^ ¬�

contrapositive

�!

¬ ! ¬�and elimination

� ^ �

� ^

Let’s see how this works on “NYS 1” …

“NYS 1”Given the statements

¬a ∨ ¬bbc → a

which one of the following statements must also be true?

c¬b¬chanone of the above


¬a ∨ ¬bbc → a



First, from b to ¬¬b by double negation.


¬a ∨ ¬bbc → a




Second, from ¬¬b and ¬a ∨ ¬b to ¬a by disjunctive syllogism.


¬a ∨ ¬bbc → a





Third, from ¬a and c → a to ¬c by modus tollens.


¬a ∨ ¬bbc → a





Third, from ¬a and c → a to ¬c by modus tollens.

QED

And how about how this works on “NYS 2” …

“NYS 2”Which one of the following statements is logically equivalent to thefollowing statement: “If you are not part of the solution, then youare part of the problem.”

If you are part of the solution, then you are not part of the problem.

If you are not part of the problem, then you are part of the solution.

If you are part of the problem, then you are not part of the solution.

If you are not part of the problem, then you are not part of thesolution.






?






?QED

And Finally, A Crucial Concept & Its Notation:Provability


� `pc �


� `pc �set of formulae in the propositional calculus


� `pc �set of formulae in the propositional calculus indicates that the

formula on the right can be proved from the set of formulae on the left




individual formula in the propositional calculus





So, the equation reads like this:





“Formula � is provable in the propositional calculus from the set � of formula.”






“Formula � is provable in the propositional calculus from the set � of formula.”


` � simply means, then, that � can be proved with only temporary suppositions.

modus tollens

�! ,¬ ¬�

modus ponens

�! ,�


� _ ,¬�

� _ ,¬ �

DeMorgan’s Laws

¬(� _ )¬� ^ ¬

¬(� ^ )¬� _ ¬

double negation

¬¬��

�

¬¬�


explosion

� ^ ¬�

contrapositive

�!

¬ ! ¬�

modus tollens

�! ,¬ ¬�

modus ponens

�! ,�


� _ ,¬�

� _ ,¬ �

DeMorgan’s Laws

¬(� _ )¬� ^ ¬

¬(� ^ )¬� _ ¬

double negation

¬¬��

�

¬¬�


explosion

� ^ ¬�

contrapositive

�!

¬ ! ¬�

reductio ad absurdum

{�} ` ( ^ ¬ )¬�

key elements of the propositional calculuskryten.mm.rpi.edu/courses/ahr/prop_calc.pdf · key...

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