Both traditional (e.g., syllogistic or dialectical reasoning) and modern symbolic logic are based on equivalent rules of formal inference. What is essential to either case is not the content expressed in the premises or conclusions of the arguments, but the mediation of the premises according to certain abstract laws of thought, which can be expressed either formally or informally.
In a categorical syllogism, for example, the major premise “All men are mortal,” and minor premise “Socrates is a man” is what supplies the content rather than the form of the argument. In this case, the inference rule is categorical since it coordinates the middle term (man) with the major term (mortal) and minor term (Socrates) to yield the conclusion that "Socrates is mortal." In other words, a categorical syllogism is a formal pattern of reasoning that certifies the inference from universal to particular.
The very same rule can also be expressed in predicate logic using, for instance, the law of universal instantiation, as in the following: ∀xP(x); ∴ P(c). Stated in natural language, this argument would be rendered as: all (∀ = the universal quantifier) mortals (x = the subject or variable) are men (P = the predicate); therefore (∴ = the logical consequence or entailment) Socrates is mortal, as Socrates is an instance (c = an element of the domain) of men (P): or more economically ∀xP(x); ∴ P(Socrates).
The point here is that there are always formal laws of logic operative in philosophy. Whether these are expressed implicitly in a natural language, or explicitly using artificial formalisms is more a matter of taste and the particular kinds of problems one has to deal with. A complex empirical theory, for example, might require the precision and mathematical grip made possible by the latter, whereas a more creative or synthetic account may depend on the fluid or rhetorical properties of speech made possible by the former. With either approach, though, you are still accountable to the same logical laws; that is, the distinction is closer to vocabulary or mode of presentation, than of logical rigour or expressive power.