Compound Propositions - Amateur Logician

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Compound Propositions

Compound propositions are composed of multiple sub-propositions.
This can be so either explicitly or implicitly.

Sometimes these compound propositions can be re-expressed into a single proposition in standard categorical form; other times, strictly speaking, that is impossible.

There’s no one classification scheme here. An Aristotelian-Scholastic approach doesn’t merely approach this from a “formal” aspect, but also from a “grammatical” aspect.

Logic is an art, not just a science.

The so-called “matter” or so-called “informal” context of propositions is important. This is not just a game of moving around arbitrary symbolism. What matters is the meaning of what any given proposition conveys.

Openly Compound

First, there is what I’m calling “openly compound” propositions. They are “formally” structured, and have been extensively studied especially in propositional logic.

There are conjunctions, disjunctions, conditionals, and biconditionals.

Since they are so important, these will be treated in more detail elsewhere.

Review these pages. . .
– “Conditional Propositions & Conditional Syllogisms”
– “Conditional Contrapositions, Reductions, & Biconditionals“
– “Conjunctions, Disjunctions, & Disjunctive Syllogisms”

S and P both respectfully symbolize two different propositions.
They are sub-propositions of the following openly compound propositions.

Conjunction: S and P
Disjunction: S or P
Conditional: If S, then P
Biconditional: S if and only if P

Note that conjunctions combine two propositions with the “and” operator. “S and P” is true when S is true and P is true. Disjunctions combine two propositions with the “or” operator. An inclusive disjunction like “S or P” is true if at least one of its sub-propositions is true. A conditional is of the form “if P, then Q.” P being the case is a sufficient condition for Q being the case. And a biconditional is of the form “P if and only if Q.” P is both sufficient and necessary for Q (and vice versa).

Plainly Composed

Second, there are three main “plainly composed” propositions. They are copulative, remotive, and discretive or adversative. Their grammatical structures indicate that multiple propositions compose them.

Copulative propositions are affirmative propositions with more than one subject and/or predicate. We can “resolve” them into independent affirmative propositions. Alternatively, we could treat the copulative proposition as a conjunction.

“Leibniz and Newton discovered the calculus.”
In this example, we have two subjects.

We can “resolve” it into two propositions:
“Leibniz discovered the calculus.”
“Newton discovered the calculus.”

Alternatively, we can have the conjunction:
“Leibniz discovered the calculus and Newton discovered the calculus.”
Note that a conjunction can be “resolved” into two propositions.

Remotive propositions are the negative version of copulative propositions.

“No Aristotelian will agree with idealism or relativism.”
In this example, there are two predicates.

It can be “resolved” into two propositions:
“No Aristotelian will agree with idealism.”
“No Aristotelian will agree with relativism.”

Alternatively, we can have the conjunction of the two negative propositions: “No Aristotelian will agree with idealism and no Aristotelian will agree with relativism.”

To be sure, there was an “or” in the original propositions; but, context matters when interpreting. An Aristotelian will agree with neither; thus, it is more accurate to think of it as a conjunction, not a disjunction.

Discretive or adversative propositions can be “reduced” into either two affirmative propositions or an affirmative and a negative proposition. Such propositions contain an adversative conjunction, e.g., a “but,” “yet,” or “although.”

“Today is a sunny but drizzling day.”

Clearly this can be resolved into two propositions:
“Today is sunny.”
“Today is a drizzling day.”

Alternatively, we can have a conjunction:
“Today is sunny and today is a raining day.”

Occultly Composed

Third, there are three main “occultly composed” propositions: exclusive, exceptive, and inceptive or destitute. There’s nothing in their grammatical structure, unlike “plainly composed” propositions, that indicates their composition of multiple propositions.

Exclusive propositions contain a word like “only” or “alone.” A predicate is attributed to the subject in such a way that it is excluded from being predicated of anything else.

Although exclusive propositions can be “reduced” into two propositions, there is strictly no need to. That’s because, in the general case, “only S is P” can be legitimately converted into “all P is S.”

“Only God is omnipotent.”

On the one hand, this can be re-expressed:
“All that is omnipotent is God.”

Predicate logic can also treat it as a single proposition with the usage of identity. It can be “translated” as follows: Og & (∀x)(Ox → x = g). In other words, God is omnipotent and for all that is omnipotent it is identical to God.

The proposition can be resolved into two propositions, on the other hand:
“God is omnipotent.”
“No non-God [non-divine being] is omnipotent.”
One is affirmative, the other negative.

Exceptive propositions contain a phrase like “all except” or “save for.” Thus the subject, in some way, is restricted such that its entire denotation is not being referred to. In general, “all except S is P” resolves into “all non-S is P” and “no S is P.”

“All except God is finite.”

On the one hand, let’s resolve this into two propositions:
“All non-divine beings are finite.”
“No God is finite.”
One is affirmative, the other negative.

And let’s re-express this as one proposition, on the other hand:
“The being who is non-finite [infinite] is God.”

In predicate logic: (∀x)[(x ≠ g) → Fx].
In other words, for anything that is not God, it is finite.

“Every planet save Earth has no intelligent life.”
(The truth of this proposition is another question! I wonder. . .?)

On the one hand, we can resolve this to two propositions:
“Earth has intelligent life.”
“No non-Earth planets have intelligent life.”

This can be re-expressed as one proposition, on the other hand:
“The planet in which there is intelligent life is Earth.”

Predicate logic allows this “translation”: Pe & Ie & (∀x)[(Px & x ≠ e) → -Ix].
In other words, Earth is a planet, Earth has intelligent life, and if there is a planet that is not identical to Earth, then it does not have intelligent life.

Inceptive or desitive propositions make judgements about something that has started or ended. These, too, will resolve into two propositions.

“E-mail became customary after 1990.”

This can be resolved into:
“E-mail was not customary before 1990.”
“E-mail was customary after 1990.”