The discussion revolves around the definition of a meaningless proposition in the context of logic and mathematics. The original poster questions whether expressions like "4+1" can be considered meaningless if they do not evaluate to true or false.
There is an ongoing exploration of different interpretations regarding the terms 'proposition' and 'statement'. Participants are sharing insights and clarifying definitions, but no consensus has been reached on the nuances of these terms.
Some participants note that terminology can vary across different textbooks, which may contribute to confusion in the discussion. The original poster expresses uncertainty due to differing opinions presented in the thread.
andrewkirk said: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.
I see.andrewkirk said:'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.
andrewkirk said:What that thread doesn't make clear is that these terms are used differently in different textbooks. 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.
andrewkirk said: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 textbooks and in some cases that meaning will be the same as that of 'proposition'.