November 2012

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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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