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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In choosing the design for this site, I was unable to go past the Tarski Theme. It both captures the type of look I was after and is named after the logician Alfred Tarski, whose work was foundational to much of my research. Tarski’s work, particularly “The Concept of Truth in Formalized Languages” (Der Wahrheitsbegriff in den formalisierten Sprachen – in the original German), is central to all following work on formal semantics and formal work on Truth, and worth the little effort it takes to understand. The influence of this work across logic, maths and philosophy is huge.

However, having come to know this work very well, in two languages, I have to say that Tarski’s logic is brilliant and insightful, but his philosophy and philosophical interpretation is limited by his formal mindset. To put it a bit more bluntly, on philosophical matters he is often clearly wrong. More of that in another post.

 

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This is a quick post to say welcome to my completely redesigned site. I intend to start posting some observations and comments shortly. If you happen to visit in the meantime, please come back soon.

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