Chapter 7: categorical logic Flashcards
What does chapter 7 and 8 cover
¬ Last two chapters examine precise rules of logic that can help to eliminate some of the interpretive aspects of deductive reasoning
¬ These rules of logic do not apply to inductive reasoning or reasoning from analogy, nor do they work with all forms of deductive arguments—in those cases you must rely on the lessons discussed in earlier chapters.
What is the rule of logic used for
only used strictly for determining whether a deductive argument’s premises are sufficiently supportive
Categorical logic
o developed by Aristotle. Deals with the relationship between real or abstract groups.
♣ A subfield of formal logic that looks at the relationships between categories or groups
¬ Categorical logic generates new knowledge. By knowing one thing about the world, you can know more.
Categorical logic deals with categorical statements
Categorical statements
o Categorical statement: a claim about whether the members of one category are, are not, or may be members of another category.
¬ Some categories may overlap and some may only have one member
Subject category
the group that a categorical statement says something about
Predicate category
¬ the group that is related to the subject category in a categorical statement
A statement
¬ a categorical statement of the form “All S are P”. This states that all members of S are members of P. Also called a “universal affirmative”.
How many possible relationships between groups in categorical logic?
Four possible relationships between groups
Statement forms
A, E, I, and O
A statement form:
all S are P
E statement form
No S are P
I statement form
Some S are P
O statement form
Some S are not P
A statement
A categorical statement of the form “All S are P”. This states that all members of S are members of P. Also called a “universal affirmative”.
E statement:
a categorical stamen of the form “no S are P”. This states that no members of S are members of P. Also called a “universal negative”
I Statement
A categorical statement of the form “Some S are P”. this states that there exists at least one member of S that is also a member of P. Also called a “particular affirmative”
O statement
¬ A categorical statement of the form “S are not P”. This states that there exists at least one member of S that is not also a member of P. Also called a “particular negative”
Some peculiarities of categorical statements
o Categorical statements can’t capture all of the nuances of our language. “A friend” is translated in exactly the same way as “a few friends” or “most friends”—any part of a group greater than none and less than all is translated as “some”.
o Since I and O statements refer to existing particular—“some” means at least one—you can’t move from A to I or from E to O
A and E statements
¬ are both universal—they say something about all members of the subject category. A and E statements are contraries. For any given subject and predicate, A and E cannot both be true at the same time, but both can be false at the same time
I and O statements
are particular, in that they say something about some but not all members of the subject category. I and O are sub-contraries—for any subject and predicate, I and O cannot both be false at the same time, but both can be true at the same time.
The pair A and O and the pair E and I
are contradictories. If one is true, the other is false and vice versa. Ex, If “all dogs are mammals” is true, then “some dogs are not mammals” must be false (and vice versa).
Distribution
¬ In any categorical statement, a category is distributed if the statement tells you something about each and every member of that category.
A property of categories within categorical statements. A category is distributed within a categorical statement if the statement indicated something about each and every member of that category. The subject category is distributed in A and E statements; the predicate category is distributed in E and O statements
Logically equivalent statements
¬ Statements that are true and false under the same conditions—meaning that they could both be true or both be false, but if one were true the other could not be false.
Rules of inference for when a categorical statement is logically equivalent
Two statements are logically equivalent when they are true or false under the same conditions. Two statements are equivalent if it’s impossible for one to be true and the other false.
Venn diagram
A diagram of overlapping circles, representing a relationship between two or more categories. The intersection represents shared or common features between the two groups.
Categorical syllogism
¬ the argument you come up with when you combine three categorical statements to create an argument
o A form of argument in which one categorical statement is deduced fom two other categorical statements.
o These are distinguished from other types of arguments in that they always have exactly two premises, one conclusion, and three categories, and they use each category only twice. These terms are called
1. The minor term
2. The major term
3. The middle term
Minor term
o in a categorical syllogism, the subject of the conclusion
Major term
in a categorical syllogism, the predicate in the conclusion
Middle term
in a categorical syllogism, the term that appears in both premises, but not the conclusion.
Using venn diagrams to evaluate categorical syllogism
- Assign the term for each of these categories (S,P, M) to a circle
- Proceed to fill in the circles with shading or X’s using the info in the premises
- Look to the conclusion and see which X’s or shading it would require
♣ If that information has already been filled in by the premises, then the argument is valid
♣ If the conclusion would require additional X’s (X=some) or shading, or if it contradicts the X’s and shading that you’ve already filled in then the argument is invalid.
♣ X means an area is not empty
Using rules to evaluate categorical syllogisms
o Checks your arguments for deductive validity, aside from Venn diagrams
- The middle term, which is in both premises but not the conclusion, must be distributed at least once.
- If a term is distributed in the conclusion, it must be distributed in a premise
- There must be at least one affirmative premise (i.e, a statement of the A or I form)
- There must be a negative (O or E form) premise if the conclusion is negative, and there must be a negative conclusion if one of the premises is negative.
- If there are two universal (A or E form) premises, the conclusion cannot be a particular (O or I form)
Missing statements
o If a categorical syllogism has an implicit premise or conclusion, you can combine your knowledge of what a categorical syllogism is with the five rules above to determine what the missing statement is. This won’t work in all cases, but more often than not it does. The only potential concern here is that you have to assume that the argument is valid. If the argument is invalid, and the missing premise wasn’t intended, then you will not be able to use this technique without misinterpreting the argument.
