5.2 Venn Diagrams

Technique for Testing the Validity of Categorical Syllogisms

  • The section outlines a systematic method to assess the validity of categorical syllogisms using various techniques and conventions.

Conventions to Follow

  • A series of conventions must be established to properly utilize a Venn Diagram for testing syllogisms:

    • Labeling the Diagram:

    • Label seven areas starting from the top circle of the diagram.

    • Assign labels to each circle as follows:

      • Lower Left Circle: This represents the subject (S) of the conclusion.

      • Lower Right Circle: This represents the predicate (P) of the conclusion.

      • Top Circle: This represents the middle term (M).

    • Numeric Labels for Areas:

    • 1 = M only

    • 2 = S & M (not P)

    • 3 = S, M, & P

    • 4 = P & M (not S)

    • 5 = S only

The Task

  • The main task comprises two essential steps in utilizing the Venn diagrams:

    1. Depict Information:

    • Illustrate the information contained within each premise proposition onto the Venn Diagram.

    1. Determine Conclusion Validity:

    • Assess whether the conclusion can be implied or inferred from the represented premises. If the conclusion is indeed present, the syllogism is valid; otherwise, it is flagged as invalid.

The Markings

  • Marking the Venn diagram accurately is crucial for validity assessment. The following guidelines are provided:

    1. Shading and Marking:

    • Use shade or a cross (X) for the premises, with no markings made for the conclusion.

    1. Universal Premises First:

    • Enter universal premises before particular ones during marking.

    1. Focus on Relevant Circles:

    • Concentrate on the two circles that contain the primary terms, giving the third term minimal attention.

    1. Inspect the Venn Diagram:

    • Examine the diagram to confirm whether it supports the conclusion. If it does, the syllogism is valid; if not, the syllogism is invalid.

Additional Marking Guidelines

  • Further instructions on how to perform specific markings:

    1. Shading Area of Interest:

    • Shade all areas that are relevant to the premises in question.

    1. Placing the Cross (X):

    • The area designated for the cross (X) is always split into two parts.

      • If one part has been shaded, place the X in the unshaded section.

      • If both parts are not shaded, position the X on the line separating them.

    1. Avoiding Incorrect Placement:

    • Ensure that an X does not hang outside of the diagram and is not placed in the intersection of the circles incorrectly.

Testing for Validity from the Boolean Standpoint

  • This section outlines the process for testing syllogisms using Boolean logic:

    • Begin with shading the appropriate areas in the diagram before other steps.

    • When placing an X, check the general area where it is supposed to go:

    • If neither of the two sub-areas is shaded, place the X on the boundary line (arc of the circle) separating the two sub-areas.

    • Conclusion Representation:

    • The conclusion is never to be entered in the diagram at this stage.

    • If the conclusion is already represented in the diagram, the syllogism is deemed valid without further checks.

Testing for Validity from the Aristotelian Standpoint

  • This section explains the considerations when existential import is a factor:

    • Begin by reducing the syllogism to its standard form and test it using the Boolean viewpoint.

    • If the syllogism is invalid according to Boolean logic, check for shading in the circles:

      • If a circle is entirely shaded except for one region, indicate this by entering a circled X in the unshaded region and retest the validity of the syllogism's form.

    • If the form is confirmed to be syllogistically valid, and the circled X aligns with something that exists, it implies a stronger affirmation of the conclusion's validity.