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Can you use multiple connectives in atomic sentences? For example, consider the following transcription guide:

A. The New England Patriots are the best team in the NFL. B. The New England Patriots will win the Super Bowl next year. C. Jacob will be furious.

Now consider the following sentence:

The New England Patriots are the best team in the NFL and they will win the Super Bowl next year and Jacob will not be furious.

In terms of atomic sentences, would the following be a correct transcription of this sentence?:

A ∧ B ∧ ¬C

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Can you use multiple connectives in atomic sentences?

No; in propositional logic, an atomic sentence is a proposition symbol.

We can use connectives with atomic sentences, to "connect" them in order to form complex sentences.

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Connectives such as ∧ can connect only 2 sentences. If you wish to connect more than 2 sentences you need to use parentheses. So you should write either

A ∧ (B ∧ ¬C)

which connects the sentence A to the sentence B ∧ ¬C, or

(A ∧ B) ∧ ¬C

which connects the sentence A ∧ B to the sentence ¬C.

Since both these formulations have exactly the same truth conditions, however, no harm is done if you just write A ∧ B ∧ ¬C, even though that is not strictly correct. In other cases you must use parentheses to avoid ambiguity, for example if you had A ∧ B ∨ ¬C.

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As Mauro ALLEGRANZA notes an "atomic sentence is a proposition symbol". You cannot have more than one atomic sentence contained within an atomic sentence. That is why they are called "atomic".

However, more generally a sentence (not an atomic sentence) can contain multiple sentences (some of which may be atomic sentences) which are connected by logical operators. In the example provided, A ∧ B ∧ ¬C would be such a sentence containing the sentences A, B and ¬C separated by the conjunction logical operator. Even these are not all atomic sentences. ¬C contains the atomic sentence C with the negation operator.

One side issue is operator precedence. To not allow ambiguity in assigning a truth value to these sentences, parentheses are often used. According to the "Operator Precedence" chapter of Stanford's Introduction to Logic, one needs to identify rules for whether to write A ∧ B ∧ ¬C as (A ∧ B) ∧ ¬C or A ∧ (B ∧ ¬C).

Here is how that source describes this situation:

In unparenthesized sentences, it is often the case that an expression is flanked by operators, one on either side. In interpreting such sentences, the question is whether the expression associates with the operator on its left or the one on its right. We can use precedence to make this determination. In particular, we agree that an operand in such a situation always associates with the operator of higher precedence. When an operand is surrounded by operators of equal precedence, the operand associates to the right.

So in this case the association would be to the right. That is, if one wanted to avoid ambiguity one would write A ∧ B ∧ ¬C as A ∧ (B ∧ (¬C)).


"Operator Precedence" Introduction to Logic http://intrologic.stanford.edu/glossary/operator_precedence.html

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