Computational Logic both as a descriptive theory of human thinking and as a normative theory of how humans can think and communicate more effectively

• The selection task

• Switching and truncation of conditionals

• The relationship between natural language and the language of thought

• The University of Michigan Lease termination clause

• The British Nationality Act

Dual process theory: human thinking involves two cognitive systems (from Daniel Kahneman, Nobel Prize in Economics 2002)

Intuitive thinking Reflective thinking

Automatic Controlled

Effortless Effortful

Associative Deductive

Rapid, parallel Slow, serial

Process opaque Self-aware

Skilled action Rule application

Intuitive thinking “quickly proposes intuitive answers to judgement problems as they arise”,

Reflective thinking “monitors the quality of these proposals, which it may endorse, correct, or override”

Computational logic can help to explain intuitive thinking in the selection task

Determine whether given data/observations satisfy the following rule:If D is on one side, then 3 is on the other side.

Interpretation “descriptively” as a clause/belief inhibits the contrapositive:If 3 is not on one side, then D is not on the other side.

Interpretation of if as if-and-only-if justifies the converse:If 3 is on one side, then D is on the other side.

Abduction also justifies the converse.

Computational logic can help to explain intuitive thinking in the selection task

Determine whether given data/observations satisfy the following rule:

If a person is drinking beer in a bar, then the person should be over eighteen.

Here it is natural to interpret the conditional deonticallyas a goal or integrity constraint:

It is not the case thata person is drinking beer in a bar andthe person is not over eighteen (i.e. the person is eighteen or under)

The classical logic interpretation of the rules is compatible with the consistency semantics of integrity satisfaction.

An abstract formulation of the selection task

Given some incomplete information/observations, what conclusions can be derived using the conditional if P then Q?

Suppose the conditional is interpreted descriptively as a belief (LP clause).

Given an observation Pforward reasoning can be used to derive Q

Given an observation QBackward reasoning can be used to derive P as an explanation of Q.

These are the classic responses to Wason’s original selection task.

The modus tollens derivation of not P from not Q is also possible, but more difficult.

The modus tollens derivation of not P from not Q, assuming the conditional is interpreted descriptively.

Because observations are atomic sentences, not Q needs to be derived by means of an integrity constraint:

if Q’ and Q then false i.e.not(Q’ and Q)

where Q’ is the positive observation or some generalization of the observation (such as the card has a consonant on the face).

In the ALP agent model, given an observation O that leads by forward reasoning to the conclusion Q’, a further step of forward reasoning is needed to derive not Q. Backward reasoning with the conditional Q if P (inside not Q) derives not P.

As Sperber, Cara, and Girotto (1995) argue, the longer the derivation of not Q, and the greater the number of irrelevant, alternative derivations, the less likely the subject will be able to perform the derivation of not P.

Suppose the conditional is interpreted deontically as a goal (integrity constraint)The integrity constraint if P then Q is used to reason forwards, to derive Q from P. It is not used backwards to derive P from Q.

Negative premises such as not Q and not P need to be derived by forward reasoning from positive atomic observations, using integrity constraint such as:

if Q’ and Q then falseif P’ and P then false

Given positive observations that imply Q’ and P’ , it is possible to derive:

if Q then false, i.e. not Qif P then false, i.e. not P.

The only further inference that is possible is:From if P then Qand if Q then falsederive if P then false, i.e. not P.

This inference step is needed for the consistency view of integrity constraint satisfaction.

Determining the logical meaning of natural language conditionalsGiven the two sentences:

If an object looks red, then it is red.This object looks red.

it is natural to draw the conclusion:This object is red.

However, given the additional sentence:

If an object is illuminated by a red light, then it looks red.it is natural to withdraw the conclusion.

This can be explained by interpreting the two natural language conditionals as having the underlying logical meaning:

An object looks red if it is red.An object looks red if it is illuminated by a red light.

Here the first natural language conditional is the switched form of the logical meaning.

Switching and truncation of conditionalsConsider the two clauses:

A if B.A if C.

whose completion isA if and only if B or C.

It is possible to write the two clauses in the switched form: B or C if A.

The disjunction in the conclusion of the switched form can be eliminated by using negative conditions, yielding the switched clauses:

B if A and not C.C if A and not B.

It is also common to truncate certain conditions, obtaining:B if A C if A

In AI applications such as fault diagnosis, the standard representation models causality in the form effect if cause. But this requires the use of abduction to explain observations. The switched representation, cause if effect and not other-causes, is also common because it requires only the use of deduction.

Computational Logic can help to improve reflective thinking

Thinking and human communication can be made

clearer simpler more coherent more effective

The relationship between language, thought and the world. Two kinds of meaning:

Natural languagesentences (surface structure)

Meaning Meaning as as logical form reference in a language of thought to the real world(deep structure)

Computational logic for human language

Clarity: Minimise the distance between thesurface structure of natural language and deep structure of the meaning in logical form.

Simplicity: Use logically simple forms of sentences.

Coherence: Use logical relationships to link sentences (e.g. “in particular”, “in general”. “therefore”. “in order to”, “old-new”….)

Effectiveness: Make relations between goals and subgoals explicit.

Coherence

Place old, familiar ideas at the beginning of a sentence.

Place new ideas at the end of the sentence. e.g.

 

 A. If A then B. If B then C. Therefore C.

C? C if B. B if A. A.Therefore C.

 

Coherence

Rules and exceptions

 

Birds fly. But penguins don't fly.

Not Penguins don't fly. But (most other) birds do.

 

Coherence

Object-orientation 

  The prime minister stepped off the plane. She was immediately surrounded by journalists.

 

Not The prime minister stepped of the plane. Journalists immediately surrounded her.

University of Michigan Lease Termination Clause

The University may terminate this lease when the Lessee, having made application and executed this lease in advance of enrolment, is not eligible to enrol or fails to enrol in the University or leaves the University at any time prior to the expiration of this lease, or for violation of any provisions of this lease, or for violation of any University regulations relative to residence or for health reasons, by providing the student with written notice of termination 30 days prior to the effective time of termination,

unless life, limb or property would be jeopardised, the Lessee engages in the sale or purchase of controlled substances in violation of Federal, state, or local law, or the Lessee is no longer enrolled as a student, or the Lessee engages in the use of firearms, explosives, inflammable liquids, fireworks or other dangerous weapons within the building or turns in a false alarm in which case a maximum of 24 hours notice would be sufficient.

The Lease Termination Clause has the ambiguous form

A if B and B’, C or D or E or F or G or Hunless I or J or K or L or Min which case A’.

Intended meaning

A if {[B and B’ and [C or D]] or E or F or G or H]} and not A’

A’ if H or I or J or K or L or M.

The University may terminate this lease by providing the student with written notice of termination 30 days prior to the effective time of termination

if the Lessee has made application and executed this lease in advance of enrolment and [the Lessee is not eligible to enrol or the Lessee fails to enrol in the University] or

the Lessee leaves the University at any time prior to the expiration of this lease or

the Lessee violates any provisions of this lease or

the Lessee violates University regulations regarding residence or

there are health reasons

and it is not the case that the University may terminate this lease with a maximum of 24 hours notice of termination.

The University may terminate this lease by … ? with maximum 24 hours notice

if life, limb or property would be jeopardised or

the Lessee engages in the sale or purchase of controlled substances in violation of Federal, state, or local law or

the Lessee is no longer enrolled as a student or

the Lessee engages in the use of firearms, explosives, inflammable liquids, fireworks or other dangerous weapons within the building or

the Lessee turns in a false alarm.

In general

An agent does X if the agent may do X by doing Y and the agent does Y

i.e.

The University terminates a lease if The University may terminate the lease by providing the student with written notice of termination 30 days prior to the effective time of termination and the University provides the student with written notice of termination 30 days prior to the effective time of termination

Subsection 1.-(1)

1.-(1) A person born in the United Kingdom after commencement shall be a British citizen if at the time of the birth his father or mother is –

(a) a British citizen; or

(b) settled in the United Kingdom.

The logic of subsection 1.-(1)

A person shall be a British citizen by 1.-(1)if the person was born in the United Kingdomand the person was born after commencementand a parent of the person was a British citizen

at the time of the person’s birth ora parent of the person was settled in the United Kingdom at the time of the person’s birth.

Subsection 1.-(2)

(2) A new-born infant who, after commencement, is found abandoned in the United Kingdom shall, unless the contrary is shown, be deemed for the purposes of subsection (1) –

(a) to have been born in the United Kingdom after commencement; and

(b) to have been born to a parent who at the time of the birth was a British citizen or settled in the United Kingdom.

The logic of subsection 1.-(2)

The conditions of 1.-(1) hold for a personif the person was found new-born

abandoned in the United Kingdomafter commencement

and it can not be shown that it is not the case thatthe conditions of 1.-(1) hold for the person.

Or in more conventional English:

(2) A person who is found abandoned in the United Kingdom after commencement shall be deemed to satisfy the conditions of subsection (1), unless the contrary is shown.

Rules and exceptions

40.-(1) Subject to the provisions of this section, the Secretary of State may by order deprive any British citizen to whom this subsection applies of his British citizenship if the Secretary of State is satisfied that the registration or certificate of naturalisation by virtue of which he is such a citizen was obtained by means of fraud, false representation or the concealment of any material fact.

40.-(5) The Secretary of State -(a) shall not deprive a person of British citizenship under this section unless he is satisfied that it is not conducive to the public good that that person should continue to be a British citizen; ...

Rules and exceptions simplified

40.-(1) The Secretary of State may deprive any British citizen to whom this subsection applies of his British citizenship if the Secretary of State is satisfied that the registration or certificate of naturalisation by virtue of which he is such a citizen was obtained by means of fraud, false representation or the concealment of any material factand it is not the case that section 40.-(5) applies. Section 40.-(5) applies If the Secretary of State is satisfied that it is not conducive to the public good that that person should continue to be a British citizen; ...

Conclusions

The Language of Thought versus Natural Language.

Computational logic is a candidate forthe language of thought.

Natural language is an imperfect expression of the language of thought.

Even seemingly logical use of natural language needs to be interpreted into its intended logical form.