Conjunctions, Disjunctions, & Disjunctive Syllogisms

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Conjunctions, Disjunctions, & Disjunctive Syllogisms

(1) Conjunctive and Disjunctive Propositions
— “and” propositions and “or” propositions
— exclusive versus inclusive “or”
(2) Disjunctive Syllogisms
— modus ponendo tollens and modus tollendo ponens
(3) Re-expressing Conjunctions into Disjunctions with Negations?
— De Morgan’s law

Conjunctive and Disjunctive Propositions. . .

Conjunction: P and Q
Disjunction: P or Q

Two (or more) categorical propositions can be combined in different ways. A conditional “if …, then” proposition is one example. We can also combine two (or more) categorical propositions, which we will symbolize as P and Q, into conjunctions and disjunctions.

Conjunctions are “and” propositions. It asserts that both propositions, P and Q, are true. That is, P is true and Q is true.

Example. . .
Socrates was born in Athens and Aristotle was born in Stagira.

Common sense is sufficient here. For instance, given the proposition that P and Q, we can infer that P is the case by itself. Alternatively, if premise one states that P is the case and premise two states that Q is the case, we can conclude that P and Q is the case.

Disjunctions are “or” propositions. P is the case or Q is the case.

Example. . .
Either you can have ice cream or you can have cake.


For any disjunction to be true, at least one of its options must be true.
Alas, however, there is ambiguity with the term “or.” Is it “exclusive” or “inclusive”?

An exclusive “or” has it that P can be the case, Q can be the case, but both cannot be the case simultaneously. Hence, there are two possibilities with an exclusive “or”: P or Q.

An inclusive “or” has it that P can be the case, Q can be the case, or both can be the case simultaneously. Hence, there are three possibilities with an inclusive “or”: P, Q, or both P and Q. Mathematical logic always — always! — takes on the inclusive “or” by default.

It’s a matter of context whether “or” should be interpreted as exclusive or inclusive. Consider the example. . . “Either Donald Trump is currently president or Joe Biden is.” It’s clear that both cannot be currently president; thus, we should view the disjunction as an exclusive “or.”

When dealing with an exclusive “or,” the modus ponendo tollens argument is valid.
The modus tollendo ponens argument is valid with both types of “or.”

Disjunctive Syllogisms. . .

Modus ponendo tollens only works with the exclusive “or.” Note that with the exclusive “or,” only one of the options is the case (P or Q). Thus, if we know which one (P), we can conclude the other must not be the case (not Q).

Example. . .
Premise 1: “Either Donald Trump is president or Joe Biden is.”
Premise 2: “Joe Biden is president.“
Conclusion: “Therefore, Donald Trump is not president.“

Mathematical logic generally considers modus ponendo tollens invalid because it doesn’t deal with an exclusive “or.” The question, then, is how can we symbolize “Either Donald Trump is president or Joe Biden is”? Using symbols, we want “P or Q and not both P and Q.” That would be (PvQ)&[-(P&Q)].

To be sure, sometimes you can find an exclusive “or” as symbolized as ⊕.
So, we can have P⊕Q.

Modus tollendo ponens is always valid, regardless of the type of “or.” In any disjunction, at least one of its choices must be true (P or Q). So, if we know one is not the case (not P), it must be the case that the other choice is true (Q).

Example. . .
Premise 1: “Either Socrates is eating fish or broccoli.”
Premise 2: “Socrates is not eating fish.“
Conclusion: “Therefore, Socrates is eating broccoli.“

Notice that premise 1 could be exclusive or inclusive. It doesn’t affect validity in this case.

With mathematical logic (more specifically, propositional logic), we can only work with modus tollendo ponens. So, we could re-work the modus ponendo tollens example into a modus tollendo ponens argument.

1: (PvQ)&[-(P&Q)] premise
2: Q premise
3: -(P&Q) by simplification of the conjunction with line 1
4: (-P)v(-Q) by De Morgan’s law on line 3
5: -(-Q) by double negation of line 2
6: Thus, -P by modus tollendo ponens with lines 4 and 5

Typically, modus tollendo ponens is called a “disjunctive syllogism” in propositional logic.

Re-expressing Conjunctions into Disjunctions with Negations?

All conjunctive propositions can be “reduced” to disjunctive propositions — or, at least, propositions with disjunctions in them. In fact, we can go back-and-forth. The example with mathematical or propositional logic actually illustrated this with the use of De Morgan’s law. Interestingly enough, this law was known well before Augustus De Morgan (19th century). William of Ockham (14th century) makes use of this principle.

Example. . .
“It is not the case that both Donald Trump and Joe Biden are president.”

Symbolized: -(P&Q)

How can this be turned into a disjunction? We just have to consider what it would mean for it to be false that both Donald Trump and Joe Biden are president.

The answer is that at least one of them must not be president! Thus, “it is not the case that both Donald Trump and Joe Biden are president” is the same as “either Donald Trump is not president or Joe Biden is not president.”

Symbolized: -(P&Q) is logically equivalent to (-P)v(-Q)

Then the question is: What about P&Q? How can that be turned into something with a disjunction? It would have to be -[(-P)v(-Q)]. Why you would want to do that is another question, to be sure, given how convoluted the answer is.

In general, De Morgan’s Law tells us. . .
–(PvQ) is logically equivalent to (-P)&(-Q)
-(P&Q) is logically equivalent to (-P)v(-Q)

To be clear, why is P&Q logically equivalent to -[(-P)v(-Q)]?
1: -[(-P)v(-Q)] premise
2: -(-P)&-(-Q) De Morgan’s law on line 1
3: P&Q by double negations on line 2
So, this is not too tricky… It’s just that double negation can throw us off.

A more useful and common use of De Morgan’s Law is to find the negation of a conjunctive or disjunctive proposition. Given P or Q, it’s negation must be not P and not Q. Given P and Q, it’s negation must be not P or not Q.

Example 1 . . .
“Jacob will walk or take the bus to work.” What if that’s false?
We figure out it’s negation: “Jacob will not walk and not take the bus to work.”

Example 2. . .
“Michele is a theologian and is young.” What if that’s false?
Negation: “Michele is not a theologian or is not young.”

Example 3. . .
“It is not cold and not dark outside.” What if that’s false?
Negation: “It is cold or dark outside.”