Logic is part of the deep universal structure of all languages.
Logic is a thinking process where a result is derived from a thought. Thus, logic is a derivative thinking process, and both mathematics and languages are only a tool of expression. We should very much understand the boundaries of the terms we use and how we use them.
Logic began with Aristotelian logic. Aristotle discovered logic, the syllogism by analysis of language. Aristotle was perhaps the first philosopher of language (discovered philosophy of language). If logic was discovered through an analysis of language, logic must be part of the language. Language is larger than logic in that context; it can express illogical and irrational things as well as non-logical or non-propositional things (examples, interrogatives, and commands). When Frege improved Aristotelian logic with an actual mathematical-logical language, the predicate calculus, and quantificational predicate calculus and proved that true/false sentences of language can all be expressed in this propositional calculus, he proved that true/false statements in language are based on logic. Logic is a part of the language, a deep structure of language. Language as narrowly defined by academic linguists requires logic to comply with rules of grammar and syntax, thus, such languages depend upon logic, so logical abilities must exist upstream from every speech act depending on the logical structuring of language parts. Such language depends on logic, not the other way around. Language has a deep structure. All languages have subjects, objects, verbs, tenses, nouns, adjectives, adverbs, connectives, etc., and how you put those features together, the grammar of the language varies widely. Many languages are inflectional, and so endings of words indicate their function or case in the sentence, some are highly non-inflectional, are hence word order dependent. Connectives, which are a part of all languages are logical, if, and, or, not exists in all and helps give the propositional nature of sentences. Even in the area of computing called natural language understanding, sentences in a particular language are given a deeper abstract formulation. You need this propositional or deeper language representation to answer questions about texts. So in short, logic is part of the deep universal structure of all languages.
The answer to the puzzle depends on the meaning you give the word "logic." Formal logic, even reflective informal logic, is very much dependent on language, especially written the language. If you consider the periods following the introduction of written language, they were periods of rapid development of formal logic. Certain aspects of language such as the subject/predicate relationship seem embedded in language and subsequently only reflectively present in formal logic. If on the other hand, "logic" refers instead to unanalyzed elements of semantics or grammar, then the two, language and logic, would be codependent. By logic, it’s not necessarily mean a system of logic such as the formal logic. It logically constructed mind that is imposed on us and defines our way of thinking. The map of mind from which we can't escape. We can't "understand" a contradiction. This kind of logic is tried to be shaped and understood through the different logical systems such as formal logic.
You cannot conceive of a language without context. If we look at all languages, there are certain UNIVERSAL features of all languages, like subjects and objects, verbs, nouns, adjectives, and adverbs. These universal features say something about the world. Verbs describe ACTIONS, something in the world, nouns describe OBJECTS, something in the world, adjectives describe properties of objects and adverbs describe properties or qualities of actions. You cannot conceive of a language without logic, for how would such a language express true/false sentences? You cannot conceive of a natural language without logic, and you certainly cannot conceive of an artificial language like a programming language or a mathematical language like Fregean predicate calculus without logic. It seems that people generally take language are a way that enables it to be talked about like the activities of a machine. This is the descent into technicality and the associated empty talk. Humans are not machines. They are not of the same nature as machines. There, in their essence, an ontological totality and not just a calculating device. For practical purposes, people are modeled as machines, but they are not machines. It might be easier to understand humans as machines, but just because it is easier, it doesn't mean it can be justified.
Logic was discovered, much as we discover mathematical truths. Logic was discovered through an analysis of language. In that sense, language is prior to logic, but more precisely, logic is a PART of language, a deep part of the language that was always there and is discovered by an analysis of language, or doing philosophy of language as Aristotle did to discover the syllogism. You could say that Frege created the quantificational predicate calculus to express true/false statements of language in a mathematical form. But the expression is only possible because logic is embedded in, or a part of the language, a universal part of the language as are subjects and objects, verbs and nouns, true of ALL languages. Other modes of language: using terms, assertions, communication, logic, etc. are derivative from this essential nature. As this is the case, it makes no sense to say that logic is prior to language in any sense because logic is just a mode of language. Worse still, essential language starts disclosing with a contradiction and the contradiction is only later rectified. At its most essential level, language is the disclosing of things from the ontological totality that is our essence. If one wants to find the essential meaning of a term, one must be prepared to go beyond thinking that confines itself to just things; one must be prepared to go to the ontological ground about which that term speaks. In the case of language, we can start from a conventional meaning: language as a method of communicating thoughts. Remember, actions or words are said to be the second order. Thoughts are the first order. However, once thoughts have been accessible to others through language or action, we reach a stage to assess the logic of everything; thoughts, language, actions.
Logic is part of the deep universal structure of all languages. Once thoughts have been accessible to others through language or action, we reach a stage to assess the logic of everything; thoughts, language, actions.
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