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I'm reflecting upon the distinction between a sentence and a proposition in logic and philosophy. I knew that a proposition is the meaning of a declarative sentence (i.e. the abstract content of a declarative sentence, the bearer of truth-value), whereas a sentence is simply a group of words (symbols, signs) ordered according to some grammatical rule in any natural or artificial language. A sentence isn't necessarily meaningful.

According to these definitions, you could say hat a predicate like "is interesting" is a sentence. Of course a sentence like "is interesting" wouldn't express a proposition, but still, it would be a sentence. This seemed ok to me until I remembered that propositional logic is sometimes called sentential logic. But then you cannot say that "is interesting" is a sentence, because it certainly isn't a sentence in the sense of sentential logic! As a result, I'm getting confused about what a sentence is according to sentential logic.

Can someone help?

user405159
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  • Maybe the ambiguity of natural language does not help... When we assert "is interesting" we are asserting more correctly "it is interesting". The meaning is disambiguated contextually: according to the context of the assertion, "it" may stand for the book on the table, the car, the pen. It is like writing in the context of formal logic "$x$ is red". Until we do not assign a denotation to the variable $x$ (through e.f. a variable assigment fucntion, the formula is an "open" one, and thus it is not a sentence. – Mauro ALLEGRANZA Apr 13 '17 at 12:07
  • When I say "is interesting" in my question I'm really thinking about a predicate, as when in predicate logic you specify the meaning of a predicate letter: I = is interesting. I'm not thinking about a closed formula. – user405159 Apr 13 '17 at 12:12
  • Nor an open one. – user405159 Apr 13 '17 at 12:13
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    A "predicate" is not a sentence: "... is interesting" looks exactly like an open formula. – Mauro ALLEGRANZA Apr 13 '17 at 12:19
  • "A sentence isn't necessarily meaningful." In the sense of logic, both propositional and predicate one, a sentence is always meaningful, and it has always a definite truth value. – Mauro ALLEGRANZA Apr 13 '17 at 12:26
  • @Mauro Allegranza: but even if you assign a denotation to the variable x you still get Ix (where I = is interesting), which is still an open formula, isn't it? Formulas like Px or Pa (where "a" is a constant) are open formulas, aren't they? – user405159 Apr 13 '17 at 14:19
  • In fact, I think I'm a bit confused about what exactly is a closed formula. – user405159 Apr 13 '17 at 14:44

2 Answers2

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We need some conventions on the terminology.

A sentence

is a meaningful group of words that express a statement, question, exclamation, request, command or suggestion.

A declarative sentence (stament, assertion) is a sentence stating a fact, like: "The rose is red".

In logic, there are declarative sentences and not e.g. questions or commands.

Unfortunately, we call the propositional calculus also sentential logic.

Thus, in propositional calculus we can replace a sentential variable $p_i$ with the declarative sentence: "The rose is red" and not with the question: "Which is the color of the rose?"

In propositional calculus, sentence and proposition are interchangeable, while in philosophical discourse, a proposition is usually an extra-linguistic entity: the content expressed by, the meaning of, the reference of a linguistc entity (a declarative sentence).

In predicate logic we have formulas with free variables (called open formulas), like: "$x$ is red".

The free variable acts as a pronoun; when we assert "it is red", the meaning of the assertion is disambiguated contextually: according to the context, "it" may stand for the book on the table, the car, the pen, and the truth value of the assertion may change accordingly.

In logic, we have to assign a denotation to the free variable $x$ (through e.g. a variable assigment fucntion) in order that the formula has a meaning (and a truth value).

In predicate logic, a formula without free variables (or closed formula), like "the rose in my hand is red" or "all the roses are red", is called a sentence.

Thus, in conclusion, either in propositional logic or in predicate one, a sentence is always meaningful, and it has always a definite truth value.

Mauro ALLEGRANZA
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  • Can we say that declaractive sentences can also be seen as closed questions (answered by yes or no) ? For instance we can read "The rose is red" as "Is the rose red ?" – Boris Apr 13 '17 at 17:50
  • @ Mauro Allegranza: so, if I understand well, as soon as we assign a denotation to a free variable x (through a variable assignment function) the formula becomes a closed one. I suppose this also works with constants: as soon as we assign a denotation to a constant through an interpretation function the formula becomes a closed one (??). – user405159 Apr 13 '17 at 18:15
  • @Fishermansfriend - constants have their denotation assigned by the interpretation. For free var, yes: as soon as the var assign functions assign a denoatation to all free vars in the formula, then the formula becomes a sentence. "It is red"[s(it)=my book] becomes "My book is red". – Mauro ALLEGRANZA Apr 13 '17 at 20:22
  • @Mauro Allegranza: I found this definition of a sentence in the Blackwell Dictionary of Western Philosophy: http://www.blackwellreference.com/public/tocnode?id=g9781405106795_chunk_g978140510679517_ss1-300 Is it compatible with yours, apart from the fact that according to the Blackwell Dictionary a sentence is not necessarily meaningful? – user405159 Apr 14 '17 at 12:02
  • @Fishermansfriend - maybe they are alluding at the well-known Chonsky's example “Colorless green ideas sleep furiously” of a grammatically correct expression that is meaningless. In logic, we do not have cases like this: a "grammatically correct" expression, i.e. a well-formed formula, is interpretable according to the rules of the semantics, and thus has a meaning and a truth-value. – Mauro ALLEGRANZA Apr 19 '17 at 06:19
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Propositional logic or sentential logic is the branch of logic that studies ways of joining or modifying entire propositions, statements or sentences to form more complicated ones. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement.

Thus, a statement can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value, such as being true or false. So, for example, the following are statements:

George W. Bush is the 43rd President of the United States. Paris is the capital of France.

The term proposition is sometimes used synonymously with statement. However, it is sometimes used to name something abstract that two different statements with the same meaning are both said to "express". In this usage, the two English sentences, "Callisto orbits Jupiter" and "Jupiter is orbitted by Callisto" would also be considered to express the same proposition. However, the nature or existence of propositions as abstract meanings is still a matter of philosophical controversy, thus the phrases "statement" and "proposition" are used interchangeably.

What is distinctive about propositional logic as opposed to other (typically more complicated) branches of logic is that propositional logic does not deal with logical relationships and properties that involve the parts of a statement smaller than the simple statements making it up. Therefore, propositional logic does not study those logical characteristics of the propositions below in virtue of which they constitute a valid argument:

George W. Bush is a president of the United States. George W. Bush is a son of a president of the United States. Therefore, there is someone who is both a president of the United States and a son of a president of the United States.

The recognition that the above argument is valid requires one to recognize that the subject in the first premise is the same as the subject in the second premise. However, in propositional logic, simple statements are considered as indivisible wholes, and those logical relationships and properties that involve parts of statements such as their subjects and predicates are not taken into consideration.

Adapated from http://www.iep.utm.edu/prop-log/

Homaniac
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  • So according to what you say in your answer a predicate like "is interesting" is a sentence (but not a statement, or a proposition), isn't it? – user405159 Apr 13 '17 at 13:03