What is This?

This is a supplement to section 2.3 of the Logic Text. You should read that before you read this. Section 2.3 discusses some principles useful for symbolizing complex sentences involving ‘and’, ‘or’, and ‘if and only if’.

Principles and Examples

Each of the following principles is taken verbatim from Chapter 2, Section 3.

First Principle

“When any of these expressions occurs between sentences, it gives rise to a conjunction, disjunction, or biconditional. The constituents of the conjunction, disjunction, or biconditional are symbolizations of sentences immediately to the left and to the right of ‘and’, ‘or’, or ‘if and only if’.”

Examples:

It is raining but it is not snowing.

Here ‘but’ is another way of saying ‘and’. It has a sentence, ‘it is raining’, immediately to its left, and another sentence, ‘it is not snowing’, immediately to its right. So, applying our principle, we treat these two sentences at the two conjuncts:

$$\fbox{It is raining}{\mathord{\wedge}}\fbox{it is not snowing}$$

And so, symbolizing the two conjucts, we get:

R∧∼S

Now consider,

It is not raining unless it is snowing

Remember that ‘unless’ is another way of saying ‘or’. Here ‘unless’ is immediately surrounded on both sides by complete sentences, so

$$\fbox{it is not raining}{\mathord{\vee}}\fbox{it is snowing}$$

And so,

∼R∨S

What about the sentence

It is raining or it is snowing and it is not raining

Here we have two connectives, ‘or’ and ‘and’. Is the ‘or’ immediately surrounded by complete sentences? If so, what are they?

Second Principle: Commas

As in Chapter 1, commas indicate the bigger break. You can use commas to disambiguate

It is raining or it is snowing and it is not raining

How?

Here is another ambiguous sentence:

Roses are red and violets are blue unless colors are an illusion

What are the two readings, and how would you use commas to force one rather than the other?

Here are four more examples. Each is ambiguous. Each can be disambiguated using commas. How?

Randy runs but Wendy walks if Teresa tumbles.

Randy runs if Wendy walks if Teresa tumbles.

Randy runs only if Wendy walks or Teresa tumbles.

Wendy walks unless Randy runs but Teresa doesn’t tumble.

Third Principle: ‘Either’ and ‘Both’

“When ‘either’ occurs with ‘or’, the symbolization of the expression enclosed between ‘either’ and ‘or’ is a disjunct. Likewise, When ‘both’ occurs with ‘and’, the symbolization of the expression enclosed between ‘both’ and ‘and’ is a conjunct.”

These complement the corresponding principle for ‘if…then’: when ‘if’ occurs with ‘then’, the symbolization of the expression between the ‘if’ and the ‘then’ is an antecedent.

Consider the ambiguous sentence:

It is raining and it is not snowing or it is snowing.

You can use commas to disambiguate it:

It is raining, and it is not snowing or it is snowing.

It is raining and it is not snowing, or it is snowing.

But you can also use ‘either’ to force the second reading:

$$

Our principle tells us that the sentence that occurs between ‘either’ and ‘or’ must be a disjunct:

$$\mbox{Either}\ \fbox{it is raining and it is not snowing}\ \mbox{or it is snowing}$$

So only the second reading,

It is raining and it is not snowing, or it is snowing.

is possible. How would you symbolize that?

The following sentence is not ambiguous:

Both penguins are birds or dinosaurs and TRexes are birds

How should it be symbolized?

Fourth Principle: Neither Nor

“‘neither nor ’ is equivalent to ‘not (either or )’.”

It is also equivalent to ∼ ∧∼ . Perhaps the easiest way to see this is to consider some truth tables:

The ‘Neither…nor’ column is filled out by considering our intuitions about what it takes for a ‘Neither…nor’ claim to be true. Intuitively, its true only when both the component sentences are false, and so only on the fourth row of the table.

The next column over shows the truth table for the disjunction. Note that it is the exact opposite of the truth table we want for ‘Neither…nor’.

The next column, then, shows the truth table for the negation of the disjunction, which matches our intuitive table for ‘Neither…nor’.

The last column shows that this is equivalent to the truth table for “both not ◻ and not ○.”

So, provide two equivalent symbolizations for each of the following sentences.

Neither dogs nor cats are dinosaurs

It is neither raining nor snowing.

Fifth Principle: Complex Subjects and Predicates

“Sometimes ‘and’ or ‘or’ occurs inside a simple sentence, where only the subject is conjoined or disjoined, and there is a single predicate, or only the predicate is conjoined or disjoined, and there is a single subject. If you fill in a copy of the shared part, you will get a synonymous sentence that we already know how to symbolize.”

For example, in

Penguins and Quail are birds

the ‘and’ does not join two complete sentences. Note that to its left, we have just a noun, not a sentence,

$$\fbox{Penguins}\ \mbox{and Quail are birds}$$

What is ‘and’ doing here? It is joining the two nouns, ‘Penguins’ and ‘Quail’, into a single subject. But this sentence is synonymous with

$$\fbox{Penguins are birds}\ \mbox{and}\ \fbox{Quail are birds}$$

And we know how to symbolize this, by our first principle, as

P∧Q

The same trick works for

Penguins are birds or dinosaurs

Again, the ‘or’ does not join two complete sentences, because ‘dinosaurs’ is not a sentence:

$$\mbox{Penguins are birds or}\ \fbox{dinosaurs}$$

But it is synonymous with

$$\fbox{Penguins are birds}\ \mbox{or}\ \fbox{Penguins are dinosaurs}$$

which we can symbolize by the first principle.

For each of the following, find the synonymous sentence and symbolize:

Penguins or TRexes are birds

Penguins but not TRexes are birds

Penguins are birds but not dinosaurs

Either penguins and TRexes are birds or penguins are dinosaurs

Sixth Principle

“When connectives occur inside otherwise simple sentences, the symbolizations of the sentences form a unit.”

This is similar to the principle governing ‘not’: when it occurs inside a sentence, next to the verb, it negates the smallest sentence it is a part of.

Consider the ambiguous sentence:

Bill is an excellent adventurer and Ted is an excellent adventurer unless Ted is lazy.

This sentence is ambiguous. Is it,

$$\fbox{Bill is an excellent adventurer}\mbox{, and}\ \fbox{Ted is an excellent adventurer unless Ted is lazy}$$

R∧(T∨Z)

or is it,

$$\fbox{Bill is an excellent adventurer and Ted is an excellent adventurer}\mbox{, unless}\ \fbox{Ted is lazy}$$

(R∧T)∨Z

Commas give us one way to disambiguate. But we can also disambiguate by taking advantage of this sixth principle. Consider:

Bill and Ted are excellent adventurers unless Ted is lazy

Here ‘and’ occurs inside a simple sentence, ‘Bill and Ted are excellent adventurers’. So, by our principle, this forms a unit:

$$\fbox{Bill and Ted are excellent adventurers}\ \mbox{unless Ted is lazy}$$

That unit is synonymous, by our fifth principle, with

$$\fbox{Bill is an excellent adventurer}\ \mbox{and}\ \fbox{Ted is an excellent adventurer}$$

So the sentence is unambiguously,

(R∧T)∨Z

How should we symbolize the following:

Roses and Violets are edible unless all flowers are poisonous

25. Krampus but not Santa Claus is evil if the stories are true.