PHL 3305 Lecture 18

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Chapter 10: Divisions of the Proposition

Three basic divisions:

simple: "one-to-one" - says one thing about one thing (e.g., a hexagon is a figure); predicate can potentially be more than one word (e.g., man is a rational, sensitive, living, bodily substance)
compound: "one-to-many" - says one thing about many things (e.g., Christopher, Michael, and Thomas are wise); "many-to-one" - says many things about one thing (e.g., Andrew is young, humorous, and energetic); "many-to-many" - says many things about many things (e.g., Mary, Martha, and Elizabeth are kind, courageous, and prudent); may be broken into multiple simple propositions (e.g., Christopher is wise. Michael is wise. Thomas is wise)

Propositions may be joined by connective words and still retain unity:

conditional (if … then …): if man is rational, then man is able to make works of art
disjunction (either … or …): either the suspect is innocent or he is guilty

All simple propositions are either affirmative or negative:

composition: man is rational unites the terms man and rational in an affirmative way.
division: no spider is an insect (or equivalently a spider is not an insect) divides the terms spider and insect in a negative way.
- depends on the verb copula: if the sentence can be equivalently rewritten using "is not", then it is negative

Aristotle's "2nd law of propositions":

for every affirmation, there is an ~~equal and~~ opposite negation, and vice versa.

not necessarily logically equivalent.

these pairs are called contradictories.

The quantity of a proposition is determined by the quantity of its subject.

Recall universal terms versus singular terms:

e.g., man and horse are universals, but this man and that horse are singulars.
May be negated: no horse is rational
The propositions no horse is rational and every horse is rational are contraries
How do contradictories and contraries differ?

Something may be said particularly of a universal subject:

some men are astronomers and some men are not astronomers
can best be interpreted as "at least some" or "there exists a" ( $\exists a\in A:P(a)$ )

A proposition is called indefinite if it does not have a quantifier (e.g., every, all', no, etc.)

In logic, these are treated as particular (when in doubt, interpret the statement to say the least)

Sometimes difficult to determine quantity:

not every man is just → $\neg \left(\forall a:a{\text{ is just}}\right)$ $\neg \left(\forall a:a{\text{ is just}}\right)$
- logically equivalent to some man is not just → $\exists a:\neg \left(a{\text{ is just}}\right)$ (De Morgan's Law)
every man is not just → $\forall a:\neg \left(a{\text{ is just}}\right)$ $\forall a:\neg \left(a{\text{ is just}}\right)$
- ambiguity forces us to treat as some man is not just → $\exists a:\neg \left(a{\text{ is just}}\right)$ ???