formal semantics

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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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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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