Classical cognitive science and Artificial Intelligence relied on the idea of “knowledge representation”
The Representational-Computational theory of mind • Knowledge consists of mental representations (mental symbols). • Thinking consists of the manipulation of these symbols. • These computations have effects on behavior or on other representations.
THE COMPUTER ANALOGY • The mind is like a computer. • A computer consists of: 1) Symbols or data structures: • A string of letters, like “abc”. • Numbers like 3. • Lists • Trees • Etc. 2) Algorithms (step-by-step procedures to operate on those structures). For instance, a procedure may “reverse” the order of elements in a list.
Computers and mind COMPUTERS MIND Data structures Mental representations Algorithms Computations Running programs Thinking
“The fundamental working hypothesis of AI is that intelligent behavior can be precisely described as symbol manipulation and can be modeled with the symbol processing capabilities of the computer.” Robert S. Engelmore and Edward Feigenbaum
A cognitive theory describes the mental representations (symbols) and the procedures or computations on these representations. • What kinds of representations are there?
Different theories have different views about how the mind represents knowledge. The “symbols” could include: IMAGES LOGICAL SYMBOLS RULES CONCEPTUAL SYMBOLS (frames, scripts).
IMAGES • Empiricists and other philosophers believed that mental representations are mainly visual images.
It is clear that we sometimes think in terms of images. • In the exercise below, for ex., we must manipulate (rotate) images.
Most philosophers, however, believe that images cannot express many important features of abstract human thinking. • For instance, logical relations like “if… then”.
LOGIC • Another option involves using formal logic to model human thinking. • There are several systems of logic. • One system is the propositional calculus, also known as sentential logic. • Formulas like “P” or “Q” represent propositions like “Peter is in school” and “Mary is in school”. • A proposition is a statement that refers to a fact.
An expression is any sequence of sentence letters, connectives or parentheses. • For example: P —> Q A) PQE —> (QQF <—> P)))) (( • 22 —> 5 is not an expression
Not all expressions are well-formed. Many expressions in the previous page were not well-formed. A well-formed formula (wff) is an expression that follows certain rules:
A sentence letter by itself is a wff Example: P
2. If we add the negation symbol ~ to any well-formed expression, the result is a also well-formed. Example: ~P Note: Since ~P is well-formed, it follows from rule 2 that ~~P is also well-formed. Note: ~ goes together with only one expression. ~PQ is not well-formed.
3. Given two well-formed expressions, the result of connecting them by means of &, —>, <—>, or V is also a well-formed formula. Example: P, Q, and ~Q are all well formed, so the following are also well-formed: P & ~Q P <—> Q P —> Q Note: &, —>, <—>, and V must always go together with two wwfs. Otherwise, the expression is not well-formed. Sample expressions that are not well-formed: P <—> &Q
Exercise: Which of these are well-formed? Why, or why not? A & B ~P —> Q ~ (A & B) A V —> ~A ~
A —> B can be translated as: • If A, then B • B, if A • A is a condition of B • A is necessary for B • A is sufficient for B • B, provided that A • Whenever A, B • B, on the condition that A
A & B • A and B • Both A and B • A, but B • A, although B • A, also B
A V B • A or B • Either A or B
A <—> B • A if and only if B • A is equivalent to B • A is a necessary and sufficient condition for B • A just in case B
How would you write “neither A nor B” in the propositional calculus? • The sentence can be written in two ways: ~(A V B) (~A & ~B)
Exercise--Translate the following into the propositional calculus: Maggie is smiling but Zoe is not smiling If Zoe does not smile, then Janice will not be happy Maggie’s smiling is necessary to make Janice happy. If Maggie smiles although Janice is not happy, then Zoe will smile. Use the following translation scheme: A: Maggie is smiling B: Zoe is smiling C: Janice is happy
If Maggie smiles although Janice is not happy, then Zoe will smile. (A & ~C) —> B
The truth (T) or falsehood (F) of propositions are called TRUTH VALUES. • The logical systems that we are studying today only have two possible truth values: T or F. • Note: there are several systems of logic that involve three or more values.
A Truth Table (TT) gives every possible combination of truth values between propositions.
A Truth Table gives the meaning (the grammar) of logical sentences. • If we want to known the meaning of ~A, we just make a Truth Table. A ~A T F F T If A is true, then ~A is false. If A is false, then ~A is true.
A B A&B T T T T F F F T F F F F A & B is only true if both A and B are true. Otherwise, it is false.
A B A V B T T T T F T F T T F F F A V B is false if both A and B are false. Otherwise, it is true.
A B A —> B T T T T F F F T T F F T A —> B is true, except when A is true and B is false.
A B A <—> B T T T T F F F T F F F T A <—> B is only true if A and B are both true or both false.
Logic is concerned with truth. • Its concern is how the truth or falsehood of one proposition depends on the truth or falsehood of one or more propositions. • For instance, if A and B are both true, then A → B must also be true.
A complex proposition, such as A <—> B is a truth function of simple propositions A and B. • Its truth values depend on the truth values of its components. • These components are simple propositions.
We can consider a well-formed formula like A —> B as an expression, and then use the previous rules to construct a new truth table.
Example: Construct a TT for the expression (P —> Q) V (~Q & R) First write down all the possible combinations for P, Q, and R. Construct first the table for P —> Q, then for ~Q, then for ~Q & R. Now you can do a TT for the whole formula!
Construct a TT for the expressions • P V (~P V Q) • ~ (P & Q) V P • R <—> ~P V (R & Q)
A TAUTOLOGY is a formula that is always true. • For instance, P —> (~P —> Q) To prove this, please construct a truth table, and you will see that for every value of P and Q the whole formula comes out true!
Is P V ~ P a tautology? • What about ((P—> Q) —> P) —> P)?
An INCONSISTENT FORMULA is always false. Example: P & ~P
A wff that is neither tautologous nor inconsistent is contingent. • Is this formula tautologous, inconsistent, or contingent? (P <—> Q) —> (P V ~R)
A proposition that is always true (a tautology) is so general that it says nothing in particular. • Tautologies contain no information about the world. • Only contingent propositions give information about the world.
To repeat: • All propositions that assert some particular information about the world are contingent.
The modern theory of truth tables for propositional logic was developed by… …the philosopher Ludwig Wittgenstein in his book Tractatus Logico-Philosophicus.
Logicians are mainly interested in reasoning. • Logical reasoning begins with some assumptions or premises. • The philosopher then applies certain rules of reasoning to reach conclusions. • We are now going to study several rules of valid reasoning.
Two important rules The Modus Ponens P → Q P Therefore Q The Modus Tollens P → Q ~ P Therefore ~ Q