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## Logic

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**In Part 2 Modules 1 through 5, our topic is symbolic logic.**We will be studying the basic elements and forms that provide the structural foundations for critical reasoning. Symbolic logic is a topic that unites the sciences and the humanities. Researchers in logic may come from philosophy, mathematics, linguistics, or computer science, among other fields. Logic**In logic, a statement or proposition is a declarative**sentence that has truth value. When we say that a sentence has truth value, we mean that it makes sense to ask whether the sentence is true or false. “Today is Monday” is a statement. “1 + 1 = 3” is a statement. Statements in logic**In logic, terms like “all,” “some,” or “none”**are called quantifiers. A statement based on a quantifier is called a quantified statement or categorical statement. “All bad hair days are catastrophes.” “No slugs are speedy.” “Some owls are hooty.” are examples of quantified or categorical statements. Quantifiers and categorical statements**Quantified or categorical statements state a relationship**between two or more classes of objects or categories. In the previous examples, bad hair days catastrophes slugs speedy (things) owls hooty (things) are all categories. Categories**A statement of the form “Some A are B” or “Some A**aren’t B” asserts the existence of at least one element (in logic, “some” means “at least one”). Categorical statements having those forms are called existential statements. “Some owls are hooty” “Some wolverines are not cuddly” are examples of existential statements. Existential statements**“Some owls are hooty” asserts that there exists at least**one thing that is both an owl and hooty. That is, the intersection of the categories “owls” and “hooty things” is not empty. We can convey that information by making a mark on a Venn diagram. We place an “X” in a region of a Venn diagram to indicate that that region must contain at least one element. Existential statements**Diagramming existential statements**The existential statement “Some wolverines are not cuddly” asserts that there must be at least one element who is a wolverine (W) but is not cuddly (C ).**“All bad hair days are catastrophes”**“No slugs are speedy” are examples of universal statements. Universal statements**A statement of the form “No A are B” is called negative**universal. It asserts that there is no element in both category A and category B at the same time. In other words, “No A are B” asserts that categories A and B are disjoint, which means that the intersection of the two categories is empty. “No slugs are speedy” is a negative universal statement. Negative universal statements**In logic, we use shading to indicate that a certain region**of a Venn diagram is empty(contains no elements). The negative universal statement “No slugs are speedy” asserts that the region of the diagram where “Slugs” and “Speedy things” intersect must be empty. Diagramming negative universal statements**A statement of the form “All A are B” is called positive**universal. It asserts that there is no element in category A that isn’t also in category B. “All bad hair days are catastrophes” is an example of a positive universal statement. Positive universal statements**The positive universal statement “All bad hair days are**catastrophes” asserts that it is impossible to be a bad hair day (B) without also being a catastrophe (C). This means that the region of the diagram that is inside B but outside C must be empty. Diagramming positive universal statements**We will use Venn diagrams (typically three-circle diagrams)**to convey the information in propositions about relationships between various categories. Interpreting Venn diagrams in logic**In logic, when a region of a Venn diagram is shaded, this**tells us that that region contains no elements. That is, a shaded region is empty. Suppose that we are presented with the marked Venn diagram shown below and on the following slides. We should be able to interpret the meaning of the marks on the diagram. Shading means “nothing here…”**In logic, when a region of a Venn diagram contains an**“X”, this tells us that that region contains at least one element. An “X” means “something is here…”**In logic, when an “X”, appears on the border between two**regions, this tells us that there is at least one element in the union of the two regions, but we are not certain whether the element(s) are in the first region, the second region, or both regions. An “X” means “something is here…”**In logic, when a region of the Venn diagram contains no**markings, it is uncertain as to whether or not that region contains any elements. No marking means “uncertain…”**Suppose we will use a three-circle Venn diagram to convey**information about the relationships between these three categories: Angry apes (A); Blissful baboons (B); Churlish chimps (C). Select the diagram whose markings correspond to “No blissful baboons are angry apes.” Assume that we do not know of any other relationships between categories. Example**For tutorials on diagramming categorical propositions, see**The DIAGRAMMER on our home page. http://www.math.fsu.edu/~wooland/diagrams/diagramming.html More exercises