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I have been going back over some of my PhD research and have discovered an interesting example I didn’t consider. It shows up some odd consequences of the way we understand truth predicates. Consider the following:

B:             Sentence A in this post is true.

There is no A in this post so intuitively we can rule out B being true. That means that whatever else we might say about it B is not true. However this leads to an odd argument:

1. B is not true.

2. B iff B is true.                   (T Schema)

3. A is true iff B is true.         (Spelling out B)

4. A is not true iff B is not true.  (Consequence of 3)

5. A is not true.                     (From 1 and 4)

However, it is somewhat ridiculous to conclude that A is not true since there is no A.

So where does this argument go wrong? It can only be in the initial argument that B is not true, or in the assumption of the T Schema.

The only way I can see that we could argue that it is not the case that B is not true is to argue that as there is no A, we can’t say anything about the truth of B. However, that means that it is not the case that “B is true” – which is equivalent to saying that “B is not true”.

So something odd is going on here with the assumption of the T-Schema.

In my previous post I outlined my conjecture that natural languages partially determine a system of logic, based on the observation that there is debate about some aspects of the correct logic, but not others. For example, no-one disputes the correct logic for the AND connective and there are not serious arguments against Modus Ponens.

There is stronger reason to suppose that natural languages do not fully determine a system of logic than simply that no-one seems to be able to agree what it is. If a natural language completely determines its relevant logic, then that natural language determines a formal system that is expressive enough for the arguments that Goedel used in his Incompleteness Theorems to work in that system. This means firstly that the natural language is incomplete (“There is some statement that is true that cannot be proven to be so in that language”) and it cannot be proven in that language that that language is consistent.

These conclusions are difficult to accept.

While it is plausible that natural languages are incomplete, it runs counter to ordinary use that the exact sentences identified in Goedel’s argument are true but not provably so. We can run Goedel’s argument in a natural language for that language, can understand what the sentence means or represents and that the sentence must be true. It is difficult to see how we can accept that the sentence must be true when Goedel’s argument shows that this cannot be proven in the natural language.

To be more precise, Goedel showed that if we can prove the relevant sentence, then the relevant system (in this case language) is inconsistent. So we seemingly have an argument that if natural languages fully determine a system of logic then, following Goedel, they must be inconsistent.

One thing we can never accept is that natural languages are inconsistent and that therefore everything is provable in them. If this is the case, then the whole of human knowledge is a lie as everything is in fact true.

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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