4.3 Venn Diagrams And The Modern Square Of Opposition

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Course lecture I developed over section 4.3 of Patrick Hurley\'s "A Concise Introduction to Logic".

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4.3 Venn Diagrams And The Modern Square Of Opposition

  1. 1. 4.3 Venn Diagrams and the Modern Square of Opposition
  2. 2. Overview <ul><li>Existential import </li></ul><ul><li>Venn diagrams </li></ul><ul><li>Modern square of opposition </li></ul><ul><li>Immediate inferences </li></ul><ul><ul><li>Modern square </li></ul></ul><ul><ul><li>Venn diagrams </li></ul></ul><ul><li>Existential fallacy </li></ul>
  3. 3. Existential Import <ul><li>A statement is said to have existential import if it implies that the thing being talked about actually exists. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>Some bears are brown. </li></ul></ul></ul><ul><ul><ul><li>Some people are not irresponsible. </li></ul></ul></ul><ul><ul><ul><li>All faeries are able to fly. </li></ul></ul></ul><ul><ul><li>These two statements have existential import since they talk about bears and people, which are both things that exist. </li></ul></ul><ul><ul><li>The third statement does not have existential import since faeries do not actually exist. </li></ul></ul><ul><li>Particular statements can always provide existential import, if their subjects are things that actually exist. </li></ul><ul><ul><li>I and O type statement </li></ul></ul><ul><li>The question we have to deal with is this: should universal claims imply that the things they talk about actually exist? </li></ul><ul><ul><li>Two schools of thought on this: </li></ul></ul><ul><ul><ul><li>Aristotelian standpoint: Yes, universal propositions imply that their subjects actually exist. </li></ul></ul></ul><ul><ul><ul><li>Boolean standpoint: No, universal propositions don’t have existential import. </li></ul></ul></ul>
  4. 4. Aristotelian versus Boolean <ul><li>Aristotelian logic indicates that universal claims have existential import. </li></ul><ul><ul><li>All bears are brown. </li></ul></ul><ul><ul><ul><li>Implies that bears actually exist. </li></ul></ul></ul><ul><ul><li>No red cars are blue. </li></ul></ul><ul><ul><ul><li>It is implied that red cars exist. </li></ul></ul></ul><ul><ul><li>A and E type statements have existential import. </li></ul></ul><ul><li>Boolean logic does not provide support for universal claims having existential import. </li></ul><ul><ul><li>All people are happy. </li></ul></ul><ul><ul><ul><li>It is not implied that people actually exist. </li></ul></ul></ul><ul><ul><li>No dogs are cats. </li></ul></ul><ul><ul><ul><li>Does not imply the existence of dogs. </li></ul></ul></ul><ul><ul><li>A and E type statements do not have existential import. </li></ul></ul>
  5. 5. Venn Diagrams <ul><li>Example: </li></ul><ul><ul><li>All people are happy. </li></ul></ul><ul><ul><li>1: People who aren’t included in being happy. </li></ul></ul><ul><ul><li>2: People who are happy. </li></ul></ul><ul><ul><li>3: Anything that is happy, but not a person. </li></ul></ul><ul><ul><li>4: Anything that is neither a person or happy. </li></ul></ul>
  6. 6. Boolean Venn diagrams of the four types of categorical propositions <ul><li> All S are P. No S are P. </li></ul><ul><li> Some S are P. Some S are not P. </li></ul>
  7. 7. Modern square of opposition <ul><li>The relationship between these propositions contradict each other in several ways, as can be illustrated here. </li></ul><ul><li>Contradictory means they have opposite truth values. </li></ul><ul><li>The other unlabelled relationships (like A to I, E to O, etc.) are logically undetermined, meaning their truth values can’t be determined by the relationship between them. </li></ul>
  8. 8. Immediate inferences (using the square) <ul><li>A claim that can be made based on a truth value that you have about a proposition. </li></ul><ul><li>They are “immediate” inferences because they have only one premise and lead directly to the conclusion. </li></ul><ul><ul><li>Examples: </li></ul></ul><ul><ul><ul><li>All people are happy. (A - true) </li></ul></ul></ul><ul><ul><ul><li>Therefore, it is false that some people are not happy. (I - false) </li></ul></ul></ul><ul><ul><ul><li>(All S are P), therefore (It’s false that some S are not P). </li></ul></ul></ul><ul><ul><ul><li>No pigs can fly. (E – true) </li></ul></ul></ul><ul><ul><ul><li>Therefore, it is false that some pigs can fly. (I – false) </li></ul></ul></ul><ul><ul><ul><li>(No S are P), therefore (It’s false that some S are P). </li></ul></ul></ul><ul><li>Claims made from the Boolean standpoint are unconditionally valid , since they are true regardless of whether they refer to existing things. </li></ul><ul><li>Note that the conclusion in these examples is not in standard form, but we fix that by simply assuming that the statement itself is false and inserting that truth value into the modern square. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>“ It’s false that some S are P” can be rephrased as “Some S are P” is false. </li></ul></ul></ul>
  9. 9. Immediate inferences, continued <ul><li>Some inferences aren’t as easy to figure out. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>It is false that all people are happy. </li></ul></ul></ul><ul><ul><ul><li>Therefore, no people are happy. </li></ul></ul></ul><ul><ul><li>The premise is an A type and the conclusion is an E type. This inference doesn’t work since their relationship is logically undetermined . Thus, this inference is invalid. </li></ul></ul><ul><li>However, we can use Venn diagrams to illustrate truth values too. </li></ul>
  10. 10. Immediate inferences (using Venn diagrams) <ul><li>To test inferences with Venn diagrams, we need to draw two circles, one for the premise and one for the conclusion, then we illustrate the claims made in the argument. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>Some people are happy. </li></ul></ul></ul><ul><ul><ul><li>Therefore, it is false that no people are happy. </li></ul></ul></ul><ul><ul><li>We can see here that the argument is valid. </li></ul></ul><ul><li> Premise Conclusion </li></ul>
  11. 11. Showing falsity <ul><li>To show falsity in a statement we do the exact opposite of what is stated. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>All S are P. </li></ul></ul></ul><ul><ul><ul><li>Therefore, it is false that no S are P. </li></ul></ul></ul>All S are P. No S are P. It is false that no S are P.
  12. 12. Existential fallacy <ul><li>This is a type of fallacy that is committed when an argument is made invalid because the claim in the premise does not have existential import. </li></ul><ul><ul><li>Example: </li></ul></ul><ul><ul><ul><li>All people are happy. </li></ul></ul></ul><ul><ul><ul><li>Therefore, some people are happy. </li></ul></ul></ul><ul><li>All people are happy. Therefore, some people are happy. </li></ul>
  13. 13. Final point <ul><li>A statement’s premises don’t have to be identical with the conclusion, they just have to claim at least as much as the conclusion. </li></ul><ul><ul><li>This becomes more clear in section 4.5. </li></ul></ul>

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