# Definition of a meaningless proposition?

1. Jul 12, 2016

### errwrsysalan

1. The problem statement, all variables and given/known data
I know that if a proposition can not be evaluated then it is meaningless, but how about statement like this? 4+1.

2. Relevant equations

3. The attempt at a solution
I think "4+1" itself is meaningless because it can't be evaluated.

Thanks!

2. Jul 12, 2016

### andrewkirk

In logic, a proposition is defined to be a sequence of symbols that evaluates to True or False.

4+1 does not evaluate to True or False, so it is not a proposition. In logic, it would be called a 'term', which is something that evaluates to an object in what is called the 'universe of discourse'. In this case the universe of discourse is the set of numbers, and the term '4+1' evaluates to the number '5'.

Both terms and propositions (also called 'well-formed formulas' or WFFs) are important concepts in logic. WFFs are made up of finite sequences of applied 'predicates', where each predicate takes one or more terms as input and gives the value TRUE or FALSE as output. The predicates in a WFF must be joined by logical connectors such as AND, OR or IMPLIES.

The term '4+1' is not meaningless because it can be used as part of a proposition. But it is not itself a proposition.

3. Jul 12, 2016

### errwrsysalan

Thank you for the reply! :)

A few more questions:
1. People say that in maths logic there is not difference between statement and proposition. Is this correct?
2. They say a meaningless statement is a statement that can't be evaluated as true of false. Does such statement exist? If it does, why is 4+1 not categorized as a meaningless statement?

Thank you!

4. Jul 12, 2016

### andrewkirk

'Proposition' is a more precise term than 'statement'. Propositions are always statements, but it's a matter of interpretation whether the converse is true. If I hurt my foot and say 'Ow!' that is not a proposition and some would call that a statement but others wouldn't. If you want to avoid ambiguity, use the word 'proposition'.

When people say a sequence of written symbols or a vocal utterance is meaningless, they usually mean that it appears to be intended to be a proposition, but it does not qualify as a proposition, typically because it contains a syntactic error.
With that interpretation, '4+1' is not a meaningless statement because it is clearly not intended to be a proposition. An example of a meaningless statement would be '$4+1\times=3$'. This is meaningless because it contains a syntactic error - the multiplication symbol is missing one of its arguments.

Some people also say that attempted propositions can be meaningless because of semantic errors. But others say that those statements are meaningful but evaluate to false. The classic example is Bertrand Russell's 'The present king of France is bald'. Some say this is meaningless because France currently has no king. But Russell gives an interesting way of analysing the sentence, bringing out the hidden assumptions, so that it evaluates to false.

5. Jul 12, 2016

### errwrsysalan

I see.
Thank you for that!

Apparently people are saying different things, that's why I'm confused. Like this onehttp://math.stackexchange.com/questions/440952/claim-vs-statement-vs-proposition :P

6. Jul 12, 2016

### andrewkirk

What that thread doesn't make clear is that these terms are used differently in different text books. The only ones that nearly always mean the same thing are 'proposition' and 'wff'. My advice is to, when working in logic, avoid using similar-seeming words such as 'claim' or 'statement' except when in a context where the meaning of that term is clearly understood by all involved. When others use them, ask them what they mean by them if it is not perfectly clear from the context.

7. Jul 12, 2016

### errwrsysalan

Hum. I see. Thanks for the advice.
So you are saying what people say is true, i.e. there is no difference between statement and proposition in contemporary mathematical logic?

8. Jul 12, 2016

### andrewkirk

No I did not say that. I said there is a difference. 'Proposition' is a term that has a universally accepted meaning in logic. 'Statement' does not have a universally accepted meaning, but will have local meanings in certain text books and in some cases that meaning will be the same as that of 'proposition'.

9. Jul 12, 2016

### errwrsysalan

Okay. I think that is all I want to ask. Thank you for helping me! :)