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1.3: Atomic Sentences

Our language will be concerned with declarative sentences, sentences that are either true or false, never both, and never neither.  Here are some example sentences.

2+2=4.

Malcolm Little is tall.

If Lincoln wins the election, then Lincoln will be President.

The Earth is not the center of the universe.

These are all declarative sentences.  These all appear to satisfy our principle of bivalence.  But they differ in important ways.  The first two sentences do not have sentences as parts.  For example, try to break up the first sentence.  “2+2” is a function.  “4” is a name.  “=4” is a meaningless fragment, as is “2+”.  Only the whole expression, “2+2=4”, is a sentence with a truth value.  The second sentence is similar in this regard.  “Malcolm Little” is a name.  “is tall” is an adjective phrase (we will discover later that logicians call this a “predicate”).  “Malcolm Little is” or “is tall” are fragments, they have no truth value.[2] Only “Malcolm Little is tall” is a complete sentence.

The first two example sentences above are of a kind we call “atomic sentences”.  The word “atom” comes from the ancient Greek word “atomos”, meaning cannot be cut.  When the ancient Greeks reasoned about matter, for example, some of them believed that if you took some substance, say a rock, and cut it into pieces, then cut the pieces into pieces, and so on, eventually you would get to something that could not be cut.  This would be the smallest possible thing.  (The fact that we now talk of having “split the atom” just goes to show that we changed the meaning of the word “atom”.  We came to use it as a name for a particular kind of thing, which then turned out to have parts, such as electrons, protons, and neutrons.)  In logic, the idea of an atomic sentence is of a sentence that can have no parts that are sentences.

In reasoning about these atomic sentences, we could continue to use English.  But for reasons that become clear as we proceed, there are many advantages to coming up with our own way of writing our sentences.  It is traditional in logic to use upper case letters from P on (PQRS….) to stand for atomic sentences.  Thus, instead of writing

Malcolm Little is tall.

We could write

P

If we want to know how to translate P to English, we can provide a translation key.  Similarly, instead of writing

Malcolm Little is a great orator.

We could write

Q

And so on.  Of course, written in this way, all we can see about such a sentence is that it is a sentence, and that perhaps P and Q are different sentences.  But for now, these will be sufficient.

Note that not all sentences are atomic.  The third sentence in our four examples above contains parts that are sentences.  It contains the atomic sentence, “Lincoln wins the election” and also the atomic sentence, “Lincoln will be President”.  We could represent this whole sentence with a single letter.  That is, we could let

If Lincoln wins the election, Lincoln will be president.

be represented in our logical language by

S

However, this would have the disadvantage that it would hide some of the sentences that are inside this sentence, and also it would hide their relationship.  Our language would tell us more if we could capture the relation between the parts of this sentence, instead of hiding them.  We will do this in chapter 2.

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