natural languages

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In my previous post, I outlined the argument that the meanings of certain words, especially connectives, provides natural languages like English with an inbuilt logic, or system of reasoning. The point is that the meaning of many words fix reasoning involving sentences that include that word.

To take a different example from the last post, if I know that the sentence “The cake is a chocolate mud cake and it has white icing on it” is true, then I am completely justified in concluding that “The cake has white icing on it” by virtue of the meaning of the word ‘and’.  This seems unremarkable but that is an example of one of the AND rules in classical logic.

However, if natural languages do have an inbuilt logic, why is there debate and a lack of consensus about what the correct logic is?

I am not aware of any literature that has explored this question in this form, and so what follows are purely my ideas.

My basic thesis is that the meanings of words in natural languages, like English, partially determine a system of logic, not completely determine one – at least in the sense that logic is commonly used by philosophers and logicians. The main evidence for this, which there is not space to justify here, is that all viable logics agree on many of the rules and principles, and all the disagreement is on a few points. I take it then that logic is determined by languages for the rules and principles that are agreed on, and only partially determined for those where there is disagreement.

To date, I have identified four areas of disagreement, which I plan to explore in later posts:

1) rules of inference  – e.g. over introducing the conditional/implication

2) meanings of connectives – e.g. one inference rule can have various semantic interpretations

3) basic principles of reasoning – e.g. rejecting or accepting the principle of non-contradiction

4) the grammatical (non-logical) rules of a logic  – e.g. what counts as a sentence

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In my posts to date on Tarski, I have noted that he argued that natural languages were inconsistent. This claim needs some explanation as it does not simply mean that inconsistent statements can be stated in natural languages, or that people can hold inconsistent beliefs.

Consistency in this context is a defined property of logical or mathematical systems. Such a system is consistent if there is no statement in the system that is both demonstrably true and demonstrably not true. In claiming that natural languages are not consistent, Tarski assumes that natural languages have (at least partially) a logical structure that allows statements to be demonstrated to be true or not true.

This assumption is highly plausible. For example, if you understand the meaning of the connectives “If … then …” in English, then you  almost certainly have to accept the Modus Ponens rule of inference. You can go through similar exercises with connectives like “and” and “or” and pretty soon you get to a set of inference rules that would determine a full system of logic.

If English has an inbuilt logic, then it makes sense that natural languages like English must be (logically) consistent or inconsistent.

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In my previous post, I noted the most obvious problem with Tarski’s (or almost anybody’s) argument that natural languages are inconsistent – if natural languages are in fact inconsistent, then everything is provable within natural languages and so the argument they are inconsistent does not establish anything.

Tarski’s response to his belief that natural languages are inconsistent was to define a formal concept of truth that could be used in place of the natural language concept of truth. There is a further problem with this  approach that follows from the more obvious point above. If natural languages are in fact inconsistent, then any proof within a natural language will not establish any truth. This means that the proofs necessary to define, and determine the properties of, any defined formal language will not establish any truths about those languages. In other words, if natural languages are inconsistent, then it is not possible to define any meaningful formal languages that embody anything true.

The premise of Tarski’s approach, that it is possible to replace natural language use with a formal language, is not consistent with Tarski’s basic assumption – that natural languages are inconsistent.

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More on Tarski

I probably need to justify my claim in the last post that Tarski was brilliant on logical analysis but pretty average when it comes to philosophy. I will argue this based on Tarski’s claims about the consistency of natural languages, in two papers of his: “The Concept of Truth in Formalized Languages” and “The Semantic Conception of Truth and the Foundations of Semantics”.

In “The Concept of Truth…..”, Tarski offers a philosophical justification for pursuing the logical task of a formal definition of truth. To paraphrase, the philosophical justification is that natural languages are necessarily inconsistent when they come to use the concept of truth. The argument is the familiar argument from the Liar Paradox. In “The Semantic Conception of Truth…”, Tarski offers a clear logical argument for this claim: any language that is ‘semantically closed’, i.e. the language contains a properly defined concepttrue, and which contains the normal laws of logic, is inconsistent. Natural languages satisfy both of these and so are inconsistent.

Tarski’s argument about the inconsistency of semantically closed languages is clear and insightful. The conclusion that natural languages are inconsistent is philosophically disastrous. The most obvious problem is that, if English (or German) is inconsistent when it uses the concept of truth, then nothing can be proven using those languages. This is because everything is provable in inconsistent languages (which obey the normal laws of logic). The result of this is that, if natural languages are inconsistent, Tarski’s argument occurs in an inconsistent language so does not actually prove anything.

Any argument in a natural languages that natural languages are inconsistent is self-defeating. It cannot establish that natural languages are in fact inconsistent. This does not mean that natural languages are consistent though.

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