Handbook of Logical Validity
For more information on the logical concepts presented on this web page, consult the Elementary Logic page.
This book is 273 pages long.
Further details about the statements made on this web page, as well as a full listing of Valid Argument Forms, are presented in Handbook of Logical Validity. No longer can an Intermediate Logic Course be taught without this book as a required text. There are several examples of Venn Diagrams on the Venn Diagram Examples page of this website.
Introducing 'The Syllogistic Calculus'
A computer program was run in the BASIC Programming Language that tested for over 50,000 Permutations of Logical Argument Forms. Of these forty thousand Logical Possibilities, 1,936 were found to be Formally Valid Argument Forms. Since computer programs and checking for Valid Argument Forms are both based on the same Binary Logic, this project was eminently successful.
All of these Argument Forms can be proved for oneself using nothing but a pencil and piece of paper. What would take thousands of hours writing out these thousands of Venn Diagrams by hand, the computer can output in a matter of seconds, once the correct algorithm had been solved. This is an important innovation both in Logic as well as Computer Science.
The Present State of Formal Logic
The present state of Formal Logic recognizes only fewer than a hundred or so established Valid Argument Forms. There are 15 Categorical Syllogisms, which are taken by the Syllogistic Calculus as a limiting case. Given the several established Symbolic Logic Argument Forms such as Disjunctive Syllogism, Modus Ponens, Modus Tollens, and Hypothetical Syllogism added to DeMorgan's Theorems, the number of hitherto known deductive argument forms is shewn to be quite paltry.
A Plethora of New Argument Forms
Thousands of new argument forms have been discovered by the use of computerized technology. This is an exponential expansion of known Valid Argument Forms. These are Argument Forms that can be utilized in any form of Deductive Reasoning, irrespective of the subject matter of the writing.
The entire Logical System of Handbook of Logical Validity contains over 2,700 established Deductively Valid Argument Forms. These Argument Forms can be utilized to prove Conclusions in any Field of Knowledge whatsoever.
The biggest discovery in logic since Gödel's Incompleteness Theorem
In 1930, the Austrian-born mathematician Kurt Gödel proved (allegedly) that all Logico-Mathematical systems of thought are either Incomplete or Inconsistent, or both. Perhaps his methodology may have been wrong.
For instance, it is proved in Handbook of Logical Validity that the following Inference is a Fallacy:
- ASSERTION OF .w.; THEREFORE, EITHER .w. AND/OR .v. (Invalid Inference).
- .w.; ∴ .w.∨14 .v. (Fallacy).
However, the following demonstration is a Valid Inference:
- .w. CONJUNCTION .v.; THEREFORE, ASSERTION OF .w. (Valid Inference).
- .w. ∧08 .v.; ∴ .w. (Correct Inference).
What this means is that inferring INCLUSIVE DISJUNCTION (∨14) simply by virtue of the Assertion of one of its Disjuncts is, in fact, not a Valid Inference. However, the Assertion of CONJUNCTION (∧08) does in fact imply the Validity of each of its Conjuncts. For more information on how this is proven in Handbook of Logical Validity, go to the Buy the Books page and order the books!
By the way, Gödel's Conclusion, according to the Symbology of Handbook of Logical Validity, is written:
[∀([.Logico-Mathematical Systems.]→13[(.Consistent.)↓07(.Complete.)])].
Well Formed Formulae (WFF's) of the Syllogistic Calculus
Out of the 64 Proposition Types (see: Elementary Logic), 48 of them are significant for our present purposes; namely the 16 Universally Quantified (∀), the 16 Existentially Quantified (∃), and the 16 Compound Proposition Types.
Classical Syllogistic Theory tests for 256 permutations of Argument Forms, of which 15 are found to be Formally Valid by Venn Diagram. The Syllogistic Calculus possesses 48 Well-Formed Formulae (WFF's) and tests for over 50,000 permutations of Argument Forms. Of these 50,000 permutations of Argument Forms, 1,936 were found to be Formally Valid. (A Well-Formed Formula is a Proposition Type allowable within a given system).
For instance, according to The Syllogistic Calculus, there are 64 Argument Forms that can prove "Some .S. is .P." (∃[.S.∧08.P.] and 24 Argument Forms for proving "All .S. is .P." (∀[.S.→13.P.]). Classical Syllogistic Theory can only prove "Some .S. is .P." four ways and prove "All .S. is .P." one way. The Syllogistic Calculus takes Classical Syllogistic Theory as a limiting case.
How the Syllogistic Calculus algorithm was solved
The Syllogistic Calculus was constructed utilizing 12 computer programs written in the BASIC Programming Language. These computer programs consisted of several 4 by 16 arrays and three For-Next loops from 1 to 16. Nested within these For-Next loops was the code that found these hundreds of new Formally Valid Argument Forms.
The Three Types of Syllogistic Calculii
The Propositional Syllogistic Calculus = 688 Formally Valid Argument Forms.
The Universal Quantifier Syllogistic Calculus = 688 Formally Valid Argument Forms.
The Mixed Quantifier Syllogistic Calculus = 560 Formally Valid Argument Forms.
The Universal Quantifier Syllogistic Calculus is essentially equivalent to the Propositional Syllogistic Calculus. The Propositional Syllogistic Calculus handles Compound Proposition Types (.w.#16.v.) and the Universal Quantifier Syllogistic Calculus is comprised of Universally Quantified Proposition Types ([∀(.W.#16.V.)]).
About The Truth-Functional Calulus
The Logical Schemata of Handbook of Logical Validity are arranged in a hierarchical manner. As such, The Syllogistic Calculus (TSC) is a Set of Propositions nested within the Set of Propositions which consist of The Truth-Functional Calculus. The Truth-Functional Calulus (the TFC) is comprised of types of Argument Forms other than those of the Syllogistic Case (TSC).
In all, the TFC possesses over 2,700 Argument Forms which were hitherto undiscovered by logicians, including hundreds of Logical Equivalencies which takes all prior logical thought, including, for example Modus Ponens, Modus Tollens, and DeMorgan's Theorems as a limiting case. All that was required to achieve this summit of Deductibility was the recognition that all 16 Binary Truth-Functional Operators (TFO's) are "logically interesting."
Prior Logical Thought held that all attention ought to be constricted to several of the TFO's [IF-THEN (→13), AND/OR (∨14), CONJUNCTION, (∧08)] because all of the 16 TFO's could be obtained by various combinations of them. However, it greatly simplifies the case when all 16 TFO's are considered. When combined with the three Quantifiers [(∀), (∃), and Compound Propositional], there exist 48 WFF's within the Logical System instead of perhaps nine or ten.
These are over 2,700 firmly established Forms of Deductive Reasoning, the vast majority of which were previously unrecognized. This Theory incorporates almost all prior Formal Logic as a limiting case.
New concept in decidability
The generally recognized Law of Formal Validity states "An Argument is Formally Valid if and only if it is logically impossible for both its Premises to be True while its Conclusion is False." It was overtly assumed by prior logicians that the Law of Formal Validity was equivalent to the concept of "Provability by Venn Diagram."
This equivalence is not the case. For some reason the discrepancy was never realized by the prior logical theory. Further details are provided in Handbook of Logical Validity.
Potential Practical Applications of the Truth-Functional Calculus
There are enormous possibilities as to the potential applications of the Truth-Functional Calculus, from disciplines as varied as computerized technology, electrical engineering (simplifying circuits), Jurisprudence, Game Theory, Physics; et cetera.