The Cognitive Processes of Formal Inferences

The Cognitive Processes of Formal Inferences

Yingxu Wang (University of Calgary, Canada)
Copyright: © 2009 |Pages: 14
DOI: 10.4018/978-1-60566-170-4.ch006
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Abstract

Theoretical research is predominately an inductive process, while applied research is mainly a deductive process. Both inference processes are based on the cognitive process and means of abstraction. This chapter describes the cognitive processes of formal inferences such as deduction, induction, abduction, and analogy. Conventional propositional arguments adopt static causal inference. This chapter introduces more rigorous and dynamic inference methodologies, which are modeled and described as a set of cognitive processes encompassing a series of basic inference steps. A set of mathematical models of formal inference methodologies is developed. Formal descriptions of the 4 forms of cognitive processes of inferences are presented using Real-Time Process Algebra (RTPA). The cognitive processes and mental mechanisms of inferences are systematically explored and rigorously modeled. Applications of abstraction and formal inferences in both the revilement of the fundamental mechanisms of the brain and the investigation of next generation cognitive computers are explored.
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Introduction

Inferences are a formalized cognitive process that reasons a possible causal conclusion from given premises based on known causal relations between a pair of cause and effect proven true by empirical observations, theoretical inferences, and/or statistical regulations (Bender, 1996; Wilson and Keil, 2001; Wang, 2007a). Formal logic inferences may be classified as causal argument, deductive inference, inductive inference, abductive inference, and analogical inference (Schoning, 1989; Sperschneider and Antoniou, 1991; Hurley, 1997; Tomassi, 1999; Smith, 2001; Wilson and Keil, 2001; Wang et al., 2006).

Theoretical research is predominately an inductive process; while applied research is mainly a deductive process. Abstraction is a powerful means of philosophy and mathematics. It is also a preeminent trait of the human brain identified in cognitive informatics studies (Wang, 2005, 2007c; Wang et al., 2006). All formal logical inferences and reasonings can only be carried out on the basis of abstract properties shared by a given set of objects under study.

  • Definition 1.Abstraction is a process to elicit a subset of objects that shares a common property from a given set of objects and to use the property to identify and distinguish the subset from the whole in order to facilitate reasoning.

Abstraction is a gifted capability of human beings. Abstraction is a basic cognitive process of the brain at the meta cognitive layer according to the Layered Reference Model of the Brain (LRMB) (Wang, 2003a, 2007c; Wang et al., 2003, 2006). Only by abstraction can important theorems and laws about the objects under study be elicited and discovered from a great variety of phenomena and empirical observations in an area of inquiry.

  • Definition 2.Inferences are a formal cognitive process that reasons a possible causality from given premises based on known causal relations between a pair of cause and effect proven true by empirical arguments, theoretical inferences, or statistical regulations.

Mathematical logic, such as propositional and predicate logic, provide a powerful means for logical reasoning and inference on truth and falsity (Schoning, 1989; Sperschneider and Antoniou, 1991; Hurley, 1997; van Heijenoort, 1997).

  • Definition 3. An argument Α is an assertion that yields (□) a proposition Q called the conclusion from a given finite set of propositions known as the premises P1, P2, …, Pn, i.e.:

ΑBL Α (P1BL ∧ P2BL ∧ … ∧ PnBL □ QBL)BL (1)where the argument and all propositions are in type Boolean (BL). Hence, ΑBL = T called a valid argument, otherwise it is a fallacy, i.e. ΑBL = F.

Equation 1 can also be denoted in the following inference structure:

(2)
  • Example 1. The following expressions are concrete arguments:

    • a.

      A concrete deductive argument

      (3)

    • b.

      A concrete inductive argument

      (4)

  • Example 2. The following expressions are abstract arguments:

    • a.

      Abstract deductive arguments

      (5)

      (6)

where N represents the type of natural numbers.
  • b.

    Abstract inductive arguments

    (7)

    (8)

where N represents the type of natural numbers.

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