The document discusses an approach to probabilistic quantifier logic called indefinite probabilities. It uses third-order probabilities to assign truth values to expressions with unbound variables, extending standard quantifier logic. For universal quantifiers, it calculates the probability that the true envelope of distributions is contained within an interval representing 'essentially 1'. For existential quantifiers, it calculates the probability that the true envelope is not contained within an interval representing 'essentially 0'. It also explains how this approach can handle fuzzy quantifiers using proxy confidence levels.

1. Dr. Matthew Ikl é Department of Mathematics and Computer Science Adams State College Probabilistic Quantifier Logic for General Intelligence: An Indefinite Probabilities Approach

2. Probabilistic Logic Networks A Probabilistic Logic Inference System: unifies probability and logic key component of the Novamente Cognition Engine, an integrative AGI system BUT independent of Novamente -- designed with ability to be incorporated into other systems

3. Probabilistic Logic Networks supports the full scope of inferences required within an intelligent system, including e.g. first and higher order logic, intensional and extensional reasoning, and so forth lends itself naturally to methods of inference control that are computationally tractable, and able to make use of the inputs provided by non-logical cognitive mechanisms

4. Weight of Evidence What is it? Why is it important? E.g. belief revision One approach: weight of ev. = interval width [.2,.8] means less evidence than [.4,.6] Pei Wang’s NARS system Walley’s Imprecise Probabilities Heuristic approaches

5. primary measure of uncertainty utilized within PLN hybrid of Walley’s Imprecise Probabilities and Bayesian credible intervals provides a natural mechanism for determining weight of evidence PLN’s logical inference rules are associated with indefinite truth value formulas or procedures Prior papers have given indefinite truth value formulas for a number of PLN inference rules, but have not dealt with quantifiers Indefinite Probabilities

6. Indefinite Probabilities Review truth-value takes the form of a quadruple ([L, U], b, k) There is a probability b that, after k more observations, the truth value assigned to the statement S will lie in the interval [L, U] Given intervals, [Li,Ui] , of mean premise probabilities, we first find a distribution from the “second-order distribution family” supported on [L1i,U1i ]so that these means have [L i,Ui] as (100*bi)% credible intervals For each premise, we use Monte-Carlo methods to generate samples for each of the “first-order” distributions with means given by samples of the “second- order” distributions. We then apply the inference rules to the set of premises for each sample point, and calculate the mean of each of these distributions.

7. Goals: logical and conceptual consistency agreement with standard quantifier logic for the crisp case (for all expressions to which standard quantifier logic assigns truth values) gives intuitively reasonable answers in practical cases compatibility with probability theory in general and PLN in particular handles fuzzy quantifiers as well as standard universal and existential quantifiers comprehensive, conceptually coherent, probabilistically grounded and computationally tractable approach Quantifiers Via Indefinite Probabilities

8. Quantifiers in Indefinite Probabilities utilize third-order probabilities non-standard semantic approach we assign truth values to expressions with unbound variables, yet without in doing so binding the variables unusual but not contradictory expression with unbound variables, as a mathematical entity, may be mapped into a truth value without introducing any mathematical or conceptual inconsistency approach reduces to the standard crisp approach in terms of truth value assignation for all expressions for which the standard crisp approach assigns a truth value. our approach also assigns truth values to some expressions (formulas standard crisp approach assigns no truth value

9. Suppose we have an indefinite probability for an expression F(t) with unbound variable t, summarizing an envelope E of probability distributions corresponding to F(t) How do derive from this an indefinite probability for the expression “ForAll x, F(x)”? we consider the envelope E to be part of a higher-level envelope E1, which is an envelope of envelopes given that we have observed E, what is the chance (according to E1) that the true envelope describing the world actually is almost entirely supported within [1-e, 1], where the latter interval is interpreted to constitute “essentially 1” Quantifiers in Indefinite Probabilities: ForAll

10. For “ThereExists x, F(x),” what is the chance (according to E1) that the true envelope describing the world actually is not entirely supported within [0, e], where the latter interval is interpreted to constitute “essentially zero” Quantifiers in Indefinite Probabilities: ThereExists

11. By almost entirely (in ForAll case) we mean that the fraction contained is at least proxy_confidence_level (PCL) the interval [PCL, 1] represents the fraction of bottom-level distributions completely contained in the interval [1-e, 1] Quantifiers in Indefinite Probabilities: The proxy_confidence_level parameter

12. indefinite probabilities provide a natural method for “fuzzy” quantifiers such as AlmostAll and Afew In analogy with the interval [PCL, 1] we introduce the parameters lower_proxy_confidence (LPC) and upper_proxy_confidence (UPC) Letting [LPC, UPC] = [0.9, 0.99], the interval could now naturally represent AlmostAll the same interval could represent AFew by setting LPC to a value such as 0.05 and UPC to, say, 0.1. Quantifiers in Indefinite Probabilities: Fuzzy Quantifiers

13. incorporating a third level of distributions, as perturbations, into the indefinite probabilities framework allows for extension of indefinite probabilities to handle a sliding scale of fuzzy and crisp quantifiers computationally tractable Summary