Can all sentences be represented logically?

Question

I am wondering if every sentence (in English or any other language) can be "translated" to a logic notation. For example, consider the following.

If you go out you'll get sick.

This is an hypothetical sentence which I can represent logically:

P -> Q

Where P is "Go out" and Q is "get sick".

But what about other sentences like "I am hungry"?

Update: thanks a lot for your answers, they were all very helpful! They all answered my question and I've learned from each one. So, I don't think I should mark one as the definitely answer since they're all good and have my upvote.

Answer 1

If your question means, is it possible to express any sentence of a natural language such as English in a logical formalism, such as first order predicate logic (FOPL), then the answer is no. Natural languages are richer and more expressive than formal ones.

Consider FOPL, such as you might learn it from an introductory text book. It does not have the resources to express indexicals such as "I", "you", "here", "now", etc. So you cannot express "I am hungry". The nearest you can get is Hungry(john) where john is your name. It only allows quantification over things, not properties, propositions or classes, so you cannot express, "everything John says is true". It does not as standard quantify over entities such as events, so you cannot say "John ran quickly so John ran" - although Davidson made a good showing of how we might remedy that limitation. It does not allow you to quantify over fictional or hypothetical entities. It does not have the resources to express temporal relations such as "it was raining yesterday but not today" - although again there are temporal extensions to logic that can help. It does not express modal relations such as "it is possible that...", "it is obligatory that...", "you must...". Again, modal logics attempt to fill in this gap with partial success. It does not express speech acts such as questions, promises, threats, expostulations, etc., e.g. it cannot express the simple sentence "Damn!", let alone something subtle like "I now pronounce you husband and wife". It does not express the attributive nature of some adjectives, e.g. "this is a good knife" does not simply mean this is good and this is a knife. It does not attempt to express pragmatic features of language use, such as conversational implicatures, e.g. it does not express the fact that "A and B" might carry the implicature of ordered events. It cannot cope with intensional contexts such as the propositional attitudes, e.g. "Mary thinks that...", "John hopes that...", "Carol fears that...". It is bivalent, so it does not cope well with statements where one might say there is a degree or truth, or no truth of the matter, such as vague statements.

Some of the above limitations can be overcome by using extensions to logic, e.g. temporal logic, modal logics, higher order logics, etc., but (a) none of these work completely, (b) the more complex you make the logical apparatus, the less likely you are to have a proof system or model system for it, so there is a diminishing return in utility, (c) there is still a substantial residue of English usage that does not fit.

FOPL and its extensions are still very useful. I hope I haven't put you off studying it. But it is well to understand their limitations.

Answer 2

According to Bertrand Russell, logical propositions are of the form "p implies q," where p and q are propositions containing one or more variables. (See Chapter 1 Principles of Mathematics) Note that Russell used mathematics in the place of logic because Russell demonstrated that mathematics and logic were one and the same.

Wittgenstein later defined logical propositions as tautologies. By tautologies Wittgenstein meant sentences that are always true regardless of the truth-value of its constituent propositions. Bertrand Russell later agreed with Wittgenstein's definition. This definition was not drastically different from Russell's.

It follows that logic is not a source of human knowledge of the external world because it only tells you what follows from your premises but does not tell you whether you premises are true.

Accordingly, not all statements are logical statements. "I am hungry" is a matter of brute fact, not a logical statement. One of Ramsey's attacks on W&R's Principia Mathematica 1st ed was that the axiom of reducibility was a matter of brute fact, not a tautology. Russell acknowledged the severity of this attack: if the statement is about a fact, then it should be luminously self-evident which the axiom of reducibility lacks.

This is an answer from the the logicist school. I'm curious how a formalist would answer this question.

Answer 3

Since "I am hungry, so I am going to have dinner" can be formalised as P -> Q, I suppose that "I am hungry" can be formalised as P.

For instance:

1. When I am hungry I think of chocolate.
2. I am hungry.
3. From 1. and 2.: I am thinking of chocolate.

can be formalised as

1. P -> Q
2. P
3. From 1. and 2.: Q

There are two big problems with formalisation, though. First is that, as implied in jobermark's comment to the question, atomic facts do not exist, and consequently atomic propositions don't exist either. So the formalisations are going to be either incomplete or circular.

Second problem is context:

(Setargew Kenaw Fantaw): The third feature we need to consider is context-dependent ambiguity reduction. This means the human brain, unlike computers, interprets things on the basis of information that the context of perception provides. When using language or performing any task, we do it within a context. When we hear a certain sentence, we capture its meaning because it is uttered in certain context. Otherwise, were we to hear them free of any context there would always be ambiguities. Sentences are heard

(Hubert Dreyfus): in the appropriate way because the context organizes the perception; and since sentences are not perceived except in context they are always perceived with the narrow range of meanings the context confers.

But formalisations rarely even attempt to capture context. Plus, they ignore a very basic fact: "I am hungry" in the context of everyday conversation, is a sentence within a real life context - usually signaling an attempt to postpone a given conversation ("I am hungry" = "can we discuss the issue of formalisation of simple sentences after lunch?"), or functioning as an answer to a given question. It isn't intended as a statement on the "state of affairs" of the world. "I am hungry" in the context of a logico-philosophicus discussion is a sentence in an abnormal context, very different from any of those in which it is normally used. Elsewhere, I have put this as

To take Wittgenstein's insight of "language games" seriously, the problem with logic classes is that they completely remove "propositions" from any actual context.

I don't think "X is tall" where X is the name of a human being is a common kind of sentence in common discourse. It seems ackward in any living context that I can imagine. Of course, sentences such as, "Phil is the tallest of them", "Phil is taller than Mark", "The suspect is a tall man", "Phil is tall for his age", etc., are common sentences, and they may imply the idea that Phil is tall. But a sentence like "Phil is tall" seems to only belong in two not too "ordinary" "language games": the one that is played in English language classes for foreigners, and the one that is played in Logic classes.

Again, if we stick with Wittgenstein, there is a problem with the game that is played in Logic classes: it mistakenly assumes that removing sentences from their usual context, or "language games", is a neutral operation, that has no effect on their meaning.

If Wittgenstein is right, such assumption is not only false, but it accounts for many of the problems with philosophy. If he is right, Logic classes are like a kind of morgue for sentences, where they are subjected to procedures that are analogue to forensic anatomy procedures. But while coroners understand that they are analysing corpses, not living people, logicians fail to make the distinction.

Or you could think of Wittgenstein's insight as the linguistic equivalent to quantum physics: when you observe a sentence, just like when you observe a particle, observation itself affects the observed item (but this in turn requires a much more careful use of words than his "look how words are used").

Answer 4

Logic concerns itself with the validity of arguments, so the propositions which make up those arguments must, at least, be capable of bearing a truth value. That is, it has to make sense to say that a given statement is true or false. Questions or interjections don't qualify in this respect.

Concerning the example you gave, "I am hungry," it could be expressed as a simple proposition such as "H", or it could be expressed with predicate logic as a function such as "Ha". Deciding between the two depends on the nature of the argument:

"The programme of formal logic consists in the attempt to exhibit logical forms of argument and, further, to capture every valid form in a single formal language." (Paul Tomassi, Logic)

When arguments involve universal or existential quantifiers; or when the argument depends on the internal grammatical structure of the sentence, predicate logic is probably the better choice. However, simplicity is preferred when possible:

"If an argument's validity depends only on propositional logic relations —structure between, not within, propositions— then don't bother translating it into predicate notation." (Peter Suber, "Translation Tips")

There are other types of logic that try to capture more of the sentence structure in different ways. Perhaps the most important among these alternative forms is modal logic which attempts to maintain valid arguments involving possibility and necessity.