Dictionary of Arguments


Philosophical and Scientific Issues in Dispute
 
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Entry
Reference
Quantifiers Cresswell I 137f
Quantifiers/everyday language/Quine/Kaplan/Geach/Cresswell: not 1st order: E.g. some critics only admire each other
2nd order:

(Eφ)(Exφx u (x)(φx > x is a critics) u (x)(y)((φx u x admires y) > (x ≠ y u φy))).

That is not equivalent to any 1st order sentence - involves plural noun phrases (plural quantification).
The following is not correct: "two Fs are G".
One would have to assume that "admire" should be valid in both directions - (then

x is a K u y is a K u x ≠ y ... ").

Better: "admire each other" is a predicate that is applied to pairs.

139
Correct: "Smart and Armstrong are present" for "S. is a and A is a". Problem: "King and Queen are a lovable couple", then "The King is an adorable ..." analog: E.g. "similar", e.g. "lessen".
Solution/Cresswell: applying predicate to quantities.

I 140
.. "admires another linguist" must be a predicate which is applied to all logicians. - This shows that quantification of higher level is required. >Second order logic.
Problem: this leads to the fact that the possibilities to have different ranges are restricted.
I 142
Higher order quantifiers/plural quantifiers/Boolos: Thesis: these do not have to go via set theoretical entities, but can simply be interpreted as semantically primitive. ((s) basic concept). Cresswell: perhaps he is right. Hintikka: game theory.
>Game-theoretical semantics.
CresswellVsHintikka: only higher order entities. 2nd order quantification due to reference to quantities.

I 156
Branching quantifiers/Booles/Cresswell: "for every A there is a B".
(x)(Ey)
(x = z ⇔ y = w) u (Ax > By)
(z)(Ew)

2nd order translation: EφEψ(x)(z)((x = z ⇔ φ(x) = ψ(z)) u (Ax > Bφ(x)).
Function/unique image/assignment/logical form/Cresswell: "(x = z ⇔ φ(x) = ψ (z)" says that the function is 1: 1.
Generalization/Cresswell: If we replace W, C, A, B, and R by predicates that are true of all, and Lxyzw by Boolos ((x = z ⇔ y = w) u Ax> By) we have a proof of non-orderability of 1st order.
>Orderability.

Cr I
M. J. Cresswell
Semantical Essays (Possible worlds and their rivals) Dordrecht Boston 1988

Cr II
M. J. Cresswell
Structured Meanings Cambridge Mass. 1984

Utility Theory Norvig Norvig I 611
Utility theory/AI research/Norvig/Russell: Intuitively, the principle of Maximum Expected Utility (MEU) seems like a reasonable way to make decisions, but it is by no means obvious that it is the only rational way. After all, why should maximizing the average utility be so special? What’s wrong with an agent that
Norvig I 612
maximizes the weighted sum of the cubes of the possible utilities, or tries to minimize the worst possible loss? Could an agent act rationally just by expressing preferences between states, without giving them numeric values? Finally, why should a utility function with the required properties exist at all? Solution: constraints on rational preferences.
Possible preferences: a) the agent prefers A over B, b) he is indifferent between A and B, c) he prefers A over B or is indifferent between them.
The primary issue for utility theory is to understand how preferences between complex lotteries are related to preferences between the underlying states in those lotteries. To address this issue we list six constraints that we require any reasonable preference relation to obey:
1. Orderability: Given any two lotteries, a rational agent must either prefer one to the other or else rate the two as equally preferable. That is, the agent cannot avoid deciding.
2. Transitivity: Given any three lotteries, if an agent prefers A to B and prefers B to C, then the agent must prefer A to C.
3. Continuity: If some lottery B is between A and C in preference, then there is some probability p for which the rational agent will be indifferent between getting B for sure and the lottery that yields A with probability p and C with probability 1 − p.
4. Substitutability: If an agent is indifferent between two lotteries A and B, then the agent is indifferent between two more complex lotteries that are the same except that B
Norvig I 613
is substituted for A in one of them. 5. Monotonicity: Suppose two lotteries have the same two possible outcomes, A and B. If an agent prefers A to B, then the agent must prefer the lottery that has a higher probability for A (and vice versa).
6. Decomposability: Compound lotteries can be reduced to simpler ones using the laws of probability. This has been called the “no fun in gambling” rule because it says that two consecutive lotteries can be compressed into a single equivalent lottery (…).
These constraints are known as the axioms of utility theory. >Preferences/Norvig, >Rationality/AI research, >Certainty effect/Kahneman/Tversky, >Ambiguity/Kahneman/Tversky.

Norvig I
Peter Norvig
Stuart J. Russell
Artificial Intelligence: A Modern Approach Upper Saddle River, NJ 2010

Utility Theory Russell Norvig I 611
Utility theory/AI research/Norvig/Russell: Intuitively, the principle of Maximum Expected Utility (MEU) seems like a reasonable way to make decisions, but it is by no means obvious that it is the only rational way. After all, why should maximizing the average utility be so special? What’s wrong with an agent that
Norvig I 612
maximizes the weighted sum of the cubes of the possible utilities, or tries to minimize the worst possible loss? Could an agent act rationally just by expressing preferences between states, without giving them numeric values? Finally, why should a utility function with the required properties exist at all? Solution: constraints on rational preferences.
Possible preferences: a) the agent prefers A over B, b) he is indifferent between A and B, c) he prefers A over B or is indifferent between them.
The primary issue for utility theory is to understand how preferences between complex lotteries are related to preferences between the underlying states in those lotteries. To address this issue we list six constraints that we require any reasonable preference relation to obey:
1. Orderability: Given any two lotteries, a rational agent must either prefer one to the other or else rate the two as equally preferable. That is, the agent cannot avoid deciding.
2. Transitivity: Given any three lotteries, if an agent prefers A to B and prefers B to C, then the agent must prefer A to C.
3. Continuity: If some lottery B is between A and C in preference, then there is some probability p for which the rational agent will be indifferent between getting B for sure and the lottery that yields A with probability p and C with probability 1 − p.
4. Substitutability: If an agent is indifferent between two lotteries A and B, then the agent is indifferent between two more complex lotteries that are the same except that B
Norvig I 613
is substituted for A in one of them. 5. Monotonicity: Suppose two lotteries have the same two possible outcomes, A and B. If an agent prefers A to B, then the agent must prefer the lottery that has a higher probability for A (and vice versa).
6. Decomposability: Compound lotteries can be reduced to simpler ones using the laws of probability. This has been called the “no fun in gambling” rule because it says that two consecutive lotteries can be compressed into a single equivalent lottery (…).
These constraints are known as the axioms of utility theory.
>Preferences/Norvig, >Rationality/AI research, >Certainty effect/Kahneman/Tversky, >Ambiguity/Kahneman/Tversky.

Russell I
B. Russell/A.N. Whitehead
Principia Mathematica Frankfurt 1986

Russell II
B. Russell
The ABC of Relativity, London 1958, 1969
German Edition:
Das ABC der Relativitätstheorie Frankfurt 1989

Russell IV
B. Russell
The Problems of Philosophy, Oxford 1912
German Edition:
Probleme der Philosophie Frankfurt 1967

Russell VI
B. Russell
"The Philosophy of Logical Atomism", in: B. Russell, Logic and KNowledge, ed. R. Ch. Marsh, London 1956, pp. 200-202
German Edition:
Die Philosophie des logischen Atomismus
In
Eigennamen, U. Wolf (Hg) Frankfurt 1993

Russell VII
B. Russell
On the Nature of Truth and Falsehood, in: B. Russell, The Problems of Philosophy, Oxford 1912 - Dt. "Wahrheit und Falschheit"
In
Wahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996


Norvig I
Peter Norvig
Stuart J. Russell
Artificial Intelligence: A Modern Approach Upper Saddle River, NJ 2010


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