Axiomatic Theory

For a theory to be axiomatic, it must follow a deductive logic flow, from major premise, to minor premise, to logically deduced consequences. Adapted from Euclidian mathematics and later enhancements, applications of axiomatic theory to education and the social sciences express a model as a set of formally presented linguistic statements, in which new concepts are introduced through definitions, assertions about the model are stated as theorems, and those assertions then are proved from the axioms, which may be considered to be either truths or just assumptions.

A classic example of axiomatic logic is the set of three statements:

• 1.
  All humans are mortal (major premise).
• 2.
  Socrates is a human (minor premise).
• 3.
  Therefore, Socrates is mortal (logically deduced conclusion).

This syllogistic set of self-contained, logical statements leads ineluctably to the conclusion that any person you can name is going to die at some point. Although that may seem to state only the obvious, the point is that an axiomatic approach to theory is based on logical deduction, rather than on empirical support or proof. In contrast, an empirical theory would be one grounded in observation, generalizing inductively, from the “bottom-up,” rather than deductively, from the “top down.” Even more to the point, axiomatic logic holds that even if the major and minor premises are not necessarily true, what really matters is whether [Page 65]the axiomatic system leads generally to correct predictions or conclusions.

In social science and education research, axiomatic theories may be applied in almost any context. For example, an axiomatic theory of how members of the electorate might vote on a school bond referendum might go something like this:

Major premises:

• A1: Voting-age residents of the school district are more likely to be interested if they have children in school.
• A2: Residents with children in school are more likely to be interested in the school bond referendum.
• A3: Upper-income people are more likely to have the time to engage in political activity.
• A4: Upper-income people are less likely to support a tax increase, because they have the most to lose and the most to want to protect.

Minor Premises:

• a1: Residents with children in school are more likely to show up and vote on election day.
• a2: Voter turnout is likely to be highest among higherincome residents who have schoolchildren.
• a3: Wealth may make a voter less likely to vote for the bond issue, but residents with children in school are likely to support the referendum.

Logically deduced conclusion:

Axiomatic theories may or may not lead to conclusions that have a high degree of predictive validity. In this example, it remains to be seen if higher-income residents really do oppose the tax or whether this thinking leads to a winning strategy.

The structures of axiomatic theories have been applied to a wide array of social science areas of research, including game theory and models of bargaining economics, linguistics, and educating gifted children.

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