[ ... still under construction ...]
Files related to the use of an Algorithmic Truth Table Method (ATTM) for proving the validity/invalidity of certain argument forms (singly-quantified and monadic). In particular, the ATTM can be used for evaluating the validity of each of the 256 possible forms of a categorical syllogism (CS).
The method was previously presented in the following scientific meetings:
- Second Lecture Series for SY 2003-2004 of the Mathematical Society of the Philippines – National Capital Region Chapter (November 22, 2003). Seminar Room, Mapua Institute of Technology, Intramuros, Manila.
- 2003 Science and Technology Research Colloquium (November 19, 2003). Thomas Aquinas Research Complex, University of Santo Tomas, Manila, Philippines.
In the PDF file 256-tables.pdf, the table labels indicating the CS form, e.g., Form: I-AAI, for the first table, have been highlighted to indicate whether the form is valid/invalid (Aristotelian logic), or unconditionally/conditionally valid or invalid (Boolean logic). The color coding is as follows:
- dark goldenrod - indicates a valid CS in Aristotelian (with either a loose or an express definition of a CS) and an unconditionally valid CS in Boolean logic (there are 15 such truth tables)
- goldenrod - indicates a valid CS in Aristotelian logic and a conditionally valid CS in Boolean logic (there are 4 such truth tables)
- silver - indicates a valid CS in Aristotelian logic with a loose definition of a CS, an invalid CS in Aristotelian logic with an express definition of a CS, and a conditionally valid CS in Boolean logic (there are 5 such truth tables). The conclusion for each of these five CS is a weakened version of the conclusion of the corresponding CS in the first list of 15 CS.
- white - invalid CS
Thus, in
- Aristotelian logic with an express definition of a CS, exactly 19 (15 dark goldenrod + 4 goldenrod) CS are deemed to be valid;
- Aristotelian logic with a loose definition of a CS, exactly 24 (15 dark goldenrod + 4 goldenrod + 5 silver) CS are deemed to be valid;
- Boolean logic, exactly 15 (only the 15 dark goldenrod) CS are deemed to be unconditionally valid and 9 (4 goldenrod + 5 silver) CS are deemed conditionally valid subject to certain requirements on existential import (please see List of valid argument forms for more information).
The PDF file 256-tables-sum.pdf shows the tables from 256-tables.pdf after some of the steps from the algorithm have been implemented on them. For example, for the first table (Form: I-AAA (bArbArA)), which has a valid form, implementing this ATTM requires object types 2, 3, 4, and 6 to be deleted (yellow highlighting for the rows). For the subuniverse containing the remaining objects types (1, 5, 7, and 8), the two premises and the conclusion are all true, thus the form of this CS is valid in Aristotelian logic and unconditionally valid in Boolean logic (note that each term (minor (S), middle (M), and major (P)) have existential import, i.e., each term is present in the subuniverse of discourse).
This work (syllogisms) is covered by an Apache 2.0 License.