# 2a Logic

## Introduction to Logic

### Logic: principles of reasoning.

Deductive: e.g. modus ponens.

If p then q.

p

So q.

Inductive: uncertainty, e.g.

All UW students are under 7 feet tall.

### History of logic:

1. syllogism

All horses are mammals.

All mammals have lungs.

So, all horses have lungs.

2. 19th century, end: rise of formal logic

Frege, Russell

3. 20th century -> theory of computation (Turing).

So both through the study of reasoning and the theory of computation, logic is one path to cognitive science. (But not the only path!)

### Propositional representations

1. syntax: p&qvr -> ~s.

2. semantics

p q p&q

T T T

T F F

F T F

F F F

note exponential growth: n propositions require 2 to the n rows in truth tables.

### Predicate calculus.

1. Add quantifiers, all and some.

All horses are mammals.

(x) (Hx -> Mx)

Some horses are brown

(Ex)(Hx & Bx)

All horses have lungs

(x)(Hx -> (Ey)(Lx & Have(x, y)))

Note difference between bite (boy, dog)

and bite (dog, boy).

### Probability theory.

P(q) is a number between 0 and 1.

P(q/r) is the probability of q given r.

Bayes theorem tells you how to update the probability of evidence based on evidence.

## The Power of Logic

### Representational power

Much can be represented using the above machinery and its extensions.

### Computational Power

#### Decision making

Make decisions by maximizing expected utility: use probabilities and expected outcomes to calculate best course of action. Raiffa story.

#### Explanation

Explanation is deduction from general principles such as scientific laws.

E.g. Why do baseballs fall?

All objects fall toward the earth because of gravitational force.

Therefore, baseballs fall toward the earth.

#### Learning

Inductive generalization is a matter of using probabilities to infer from properties of samples to properties of whole populations.

#### Language

Understanding language is a matter of making deductions or probabilistic inferences from spoken or written input.

# 2b Logic - Evaluation

## The advantages of logic as a representational scheme: review

Neats (logic) versus scruffies (other kinds of representation).

1. Long history, well understood formalism

Use in philosophy to evaluate arguments.

Heroes: Leibniz, Frege, Russell, Carnap + logical positivists, Church, Turing, Quine, Kripke.

2. Clear principles, rigourous

3. Much representational power

4. Computational power, as shown by Prolog, theorem provers.

1. Natural language is much more flexible than formal logic: not

easy to formalize.

2. Exotic logics needed to handle possibility, propositional attitudes (e.g. knows).

3. Restricted to verbal information.

1. Much reasoning is nonmonotonic: you can't just add more beliefs deductively, but must subtract as well.

Minsky's critique of logic:

Tweety is a bird. Does Tweety fly?

Reasoning is nonmonotonic: sometimes you have to take things back. Minsky concluded that logic was irrelevant to reasoning, but others have proposed "nonmonotonic logic." Key question: is logic fundamental to an intelligent processing system? Psychologists tend to deny it.

2. Logical deduction is potentially computationally explosive: you can't just generate inferences by, e.g.: p, q, therefore p&q. Need constraints on what gets inferred. Similar problem for truth tables.

3. The most interesting kinds of reasoning are non-deductive, e.g. inductive learning.

4. Visual reasoning easier for some problems: see week 6.

Some psychologists (Braine, Rips) think that people use something like logical reasoning. But experiments suggest otherwise.

### Wason experiment

Cards with letters on one side and numbers on other.

[A] [B]  

Question: What cards do you need to turn over to determine whether it is true that: If there is a vowel on one side of the card, then there is an even number on the other side.

Right answer: A for confirmation, 3 for refutation (modus tollens).

But compare a known case:

If someone to drink in a bar, he or she is over 19. Cards with location on one side and age on other.

[in bar] [not in bar]  

How is this done? Pragmatic reasoning schemas, e.g. permission schema.

### Explanations of Wason results:

• People misunderstand question
• Permission schema (Cheng)
• Cheater module (Cosmides)
• Mental models (Johnson-Laird)

### Mental Models

A mental model is a representation that has the same structure as what it represents.

Examples:

• Visual: maps, diagrams
• Abstract: concrete examples, e.g. cards in Wason task
• General: rules to describe dynamics

How Brains Make Mental Models (Thagard, 2010)

### Johnson-Laird's experiments on syllogistic reasoning

All biologists are scientists.

All scientists are intelligent.

What follows?

So, all biologists are intelligent.

biologist=scientist=intelligent

biologist=scientist=intelligent

(scientist) (intelligent)

Check model: conclusion ok.

Johnson-Laird's basic point: deductive reasoning is not done with formal rules, but with mental models.

Visual example:

Fork is to the left of the spoon, the knife is to the right of the spoon.

What is the relation of the fork and knife?

Could do this with logic using ssymmetry of left and right and transitivity of left.

But it's much easier to do it with a spatial mental model:

fork --- spoon ---- knife

The human mind is set up to do reasoning in lots of different ways, specific to reasoning tasks, without the generality of logical deduction but with potentially much more efficiency on tasks that matter.

## Key points in Johnson-Laird & Byrne:

• People are rational in principle but fallible in practice.
• There are three main classes of theory about the process of deduction: formal rules, content-specific rules, and mental models.
• The formal rule account is psychologically implausible because people are affected by the content of deductions.
• But the content-specific rules view ignores the fact that people are able to make valid deductions based solely on logical connectives and quantifiers.
• Mental models form the basis for various kinds of reasoning.

Phil/Psych 256

Computational Epistemology Laboratory.

Paul Thagard