Is an argument in natural language as logically valid as in formal logic?

Is a natural language philosophical argument which is argued strictly from first principles widely considered equally as valid as a proof written in formal logic?

• Nobody would write a formal logic proof to make a philosophical point. – Jishin Noben Feb 18 at 21:40
• Ignore the first comment posted here, they are incorrect. Usually arguments are given in natural language but it is clear that they could be fully transformed into a formal logic argument if someone sat down and worked it out. They are considered to have the same level of validity when it is clear that that translation process would be unproblematic. – Not_Here Feb 18 at 21:45
• @Not_Here I think not. The aspect convertible into formal logic is a triviality. The hard part is arguing for premises, and that is formally invalid if it is of any substantive interest. Balancing concerns and plausibility plays do not fit into chaining logical forms, informal validity is something else. Btw, Robinson, a designer of automated proof checkers in math, writes:"The translation from informal to formal is by no means merely a matter of routine. In most cases it requires considerable ingenuity, and has the feel of a fresh and separate mathematical problem in itself." – Conifold Feb 18 at 22:25
• @Conifold The question asks about validity, not soundness, so your comment agrees with me. – Not_Here Feb 18 at 22:42
• @Not_Here In informal reasoning the distinction between validity and soundness is essentially meaningless, so no. Material validity is both and neither. – Conifold Feb 18 at 22:57

In the context of logic and reasoning, 'valid' is a property of an argument whereby the conclusion follows from the premises by necessity, or where it is impossible for the premises to be true and the conclusion false. It is possible to state valid arguments in English, or in other natural languages, provided one is careful with the terms and the syntax. An argument, such as "All elephants are mammals; all mammals are warm-blooded; therefore, all elephants are warm-blooded" is recognisable as valid, without being rendered into symbols.

But there is a serious issue with more complex arguments. Natural languages evolved for performing various tasks, but they are not ideally suited to expressing complex relationships of the kind that turn up in reasoning. Mathematicians and scientists prefer to express themselves using a symbolic language for just this reason. One could state equations using English, but it would be clumsy, long-winded, and it would risk a lack of rigour. For the same reason, logicians often prefer to use a formal language. It doesn't make arguments more valid, indeed, it is not clear what 'more valid' or 'equally as valid' could mean, but formal languages in logic can be very helpful in avoiding errors.

For example, formal languages such as predicate logic are particularly helpful in avoiding errors of scope. In English we may use a negation, or a quantifier, or a modal operator, without it being clear exactly what part of a sentence it applies to. One example of this is the quantifier shift fallacy. "Every boy loves some girl" is potentially ambiguous between there is one particular girl whom every boy loves, and there being a different girl for each boy. In English we usually understand from the context which is meant, but the ambiguity exists and is commonly exploited in a formula joke, such as "This newspaper says that a person is run over by a car every ten minutes; he should just stay in bed". Or, "According to this report, every hour a married man commits adultery. Really? Does it give his name?" Thankfully for logicians, but unfortunately for comedians, this kind of syntactic ambiguity is readily eliminated in formal languages. Another example is the modal fallacy. In English we often say something of the form, "If A then necessarily B", when the 'necessarily' part really applies to the conditional as a whole, not the consequent part of it. "If X knows that P, then necessarily P is true", expresses the fact that if something is known then it is true, because one cannot know a falsehood. But if for some X and P, X does know that P, it does not follow that "necessarily P is true", but rather that "P is true" follows necessarily. Again, this ambiguity is easily spotted and eliminated in a formal logic.

All of which is to say that formal logics are useful, including within philosophy, but they exist to make arguments clearer, not to make them more valid.

A good illustration of your question and a possible answer is the ontological argument for the existence of god.

The original argument was given by Anselm of Canterbury in the 11th century using ordinary langugage (Proslogion, Chap. 2):

Therefore, Lord, you who give knowledge of the faith, give me as much knowledge as you know to be fitting for me, because you are as we believe and that which we believe. And indeed we believe you are something greater than which cannot be thought.

Or is there no such kind of thing, for "the fool said in his heart, 'there is no God'" (Ps. 13:1, 52:1)? But certainly that same fool, having heard what I just said, "something greater than which cannot be thought," understands what he heard, and what he understands is in his thought, even if he does not think it exists.

For it is one thing for something to exist in a person's thought and quite another for the person to think that thing exists. For when a painter thinks ahead to what he will paint, he has that picture in his thought, but he does not yet think it exists, because he has not done it yet. Once he has painted it he has it in his thought and thinks it exists because he has done it.

Thus even the fool is compelled to grant that something greater than which cannot be thought exists in thought, because he understands what he hears, and whatever is understood exists in thought.

And certainly that greater than which cannot be understood cannot exist only in thought, for if it exists only in thought it could also be thought of as existing in reality as well, which is greater. If, therefore, that than which greater cannot be thought exists in thought alone, then that than which greater cannot be thought turns out to be that than which something greater actually can be thought, but that is obviously impossible. Therefore something than which greater cannot be thought undoubtedly exists both in thought and in reality.

In the 20th century Kurt Goedel presented an ontological argument in formal logic. More specifically, he used a calulculus of modal logic:

Ax. 1. ( P ( φ ) ∧ ◻ ∀ x ( φ ( x ) ⇒ ψ ( x ) ) ) ⇒ P ( ψ )

Ax. 2. P ( ¬ φ ) ⇔ ¬ P ( φ )

Th. 1. P ( φ ) ⇒ ◊ ∃ x φ ( x )

Df. 1. G ( x ) ⇔ ∀ φ ( P ( φ ) ⇒ φ ( x ) )

Ax. 3. P ( G )

Th. 2. ◊ ∃ x G ( x )

Df. 2. φ ess x ⇔ φ ( x ) ∧ ∀ ψ ( ψ ( x ) ⇒ ◻ ∀ y ( φ ( y ) ⇒ ψ ( y ) ) )

Ax. 4. P ( φ ) ⇒ ◻ P ( φ )

Th. 3. G ( x ) ⇒ G ess x

Df. 3. E ( x ) ⇔ ∀ φ ( φ ess x ⇒ ◻ ∃ y φ ( y ) )

Ax. 5. P ( E )

Th. 4. ◻ ∃ x G ( x )

Apparently, only experts in modal logic are able to follow the formal proof and to discuss its different steps. But the formalization requires that one gives a precise meaning to each term. As a consequence, one can decide whether to accept the proposed meaning or not.

References

1. Anselm von Canterbury: Proslogion. Hrsg. P. Franciscus Salesius Schmitt O.S.B. fromann-holzboog, Latein/Deutsch. Stuttgart-Bad Canstatt (1984)

2. The quote in my answer is from wikipedia. It follows Gödel's original version from his heritage. The following paper quotes and comments the steps from the heritage:

3. For an explication of the steps from Gödel's proof see also @jqxxxx https://math.stackexchange.com/questions/248548/g%C3%B6dels-ontological-proof

• This is an interesting example, but one should note that Canterbury's argument is of quite different nature than Gödel's. The argument can be understood by anyone putting some time into it, especially since there are nice expositions of it, like "Types, Tableaux's and Gödel's God", by Fitting. If you look into that book, you will clearly see in what sense natural language has precedence. The logic has to be constructed, only so that the argument can be formalised. – Jishin Noben Feb 18 at 22:38
• @Jishin Noben Do you mean that Goedel's formalization does not meet Anselm's argument? – Jo Wehler Feb 18 at 23:07
• There are several different versions of the ontological argument, e.g. those of Anselm, Descartes, Gödel and Plantinga. They all work from somewhat different premises, so I wouldn't say that Gödel's version is a formalisation of Anselm as such. Also, it is worth noting that Gödel's argument has been shown to have inconsistent premises, so it is valid for all the wrong reasons! – Bumble Feb 19 at 14:24
• Nowhere to the best of my knowledge has such a thing been shown @Bumble. Jo Wehler, Gödel's argument is not a formalisation of Canterbury's. Both are classified as ontological, but the details are vastly different. For example you won't find any reference to mind or thought in Gödel's argument and in Canterbury's goodness is not mentioned. – Jishin Noben Feb 19 at 17:18
• This paper: ijcai.org/Proceedings/16/Papers/137.pdf demonstrates a modal collapse in Gödel's premises, within S5, or any modal system that includes T. – Bumble Feb 19 at 17:28

Humans don't think in symbolic logic, so if a human can understand an argument presented in symbolic logic, then they could express it with some other notation.

The main problem with expressing logical arguments in natural human languages is polysemy. For example, the word or can mean both inclusive and exclusive OR. The word if sometimes means "if and only if", and sometimes it doesn't. But it is possible to develop subsets of natural language where these words are defined unambiguously. Or new words are coined, such as iff.

Although logical arguments was not the reason why the Natural Semantic Metalanguage (NSM) was developed, it is interesting how many logical constructs have been identified as semantic primes: THERE IS, ALL, NOT, BECAUSE, IF, TRUE. For these to have been accepted as primes means that there is good empirical research that these are universal concepts shared by all languages. This means that not only can most (perhaps all) logical arguments be expressed in natural language, they can be expressed in all natural human languages.

It can be. Take the definition of deduction by Aristotle

A kind of discourse, that something having being stated, some other statement follows that is different from the preceding by neccessity.

Aristotle says ‘discourse’ and there are many kinds of discourse and not just the purely formal, that many think of when they hear the terms logic or deduction. It’s interesting, to me at least, that Aristotle writes that a deduced statement should be different from the preceding.

This careful, and at first sight - pedantic - language marks him, at least in my eyes, as being of a more scientific bent than his predecessor, Plato. I would say, even as a modern. Except of course that is wrong. We write like this because he wrote like that and his influence went deep. It merely highlights the scientific inheritance from Aristotle. It’s also interesting, because in the plentiful superficial literature about him, I mean in the anti-Aristotelian literature he is belittled as being anti-science when in fact he was laying the foundations stones of science.

It depends on how you define "valid". If you are using the formal, technical definition of valid as certifying an argument in a system of formal logic, then no, a natural language argument cannot be valid in that sense. It must first be translated into a a formal argument, and then evaluated for validity.

There are cases where a natural language argument seems identical to a formal language argument --it may even be expressed in the same words --but there are inherent ambiguities in natural language that are explicitly removed in formal languages.

It is quite possible for a natural language argument to be so readily translatable into a formal argument that they are essentially treated as interchangeable, but calling a argument of this type "valid" is a distinguishably different use of the term "valid" --it is, to be exact, a natural language usage of "valid," not a formal language one.

• +1 I agree that the description valid should only apply to arguments in a formal logic. There is too much ambiguity in natural language for the relationship between premises and conclusion to be necessarily so that validity requires. – Frank Hubeny Feb 21 at 23:16