Question

When we use 'terms' such as words it is generally clear however, if we have a formula: And I write: 'x+1=2 is true for x=1' is this 'using' or 'mentioning'? If a formula contains variables, it has no truth value or particular meaning, how do I 'use' a free variable? When using variables, how much of it is the syntax of the string or the semantics of the variable? Is it possible to 'use' them like words? Another question is that of intent, If I write: Some would say 'two is a prime number', is this an example of use or 'mention'? I am using the 'assertion' (string of symbols), yet I do not wish to personally make the assertion it just make the reader aware of the assertion, in this case I'd say as long as the 'assertion' is being communicated it is 'use' but I'm not sure. In general, the term "use" refers to the act of employing something for a specific purpose or in a specific context. When we use words or formulas, we are employing them to convey a particular meaning or to perform a specific function. In the case of the formula "x+1=2", it is being used to express the fact that, for a particular value of x (in this case, 1), the formula is true. This is an example of using the formula, as it is being employed to convey a specific meaning. As for free variables, they can be used in a similar way to other variables. For example, if we have a formula with a free variable x, we can use that formula to represent a relationship between x and other variables in the formula. For example, the formula "x+1=2" can be used to represent the relationship between x and the other variables in the formula (in this case, 1 and 2). In terms of the syntax and semantics of variables, the syntax refers to the way in which the variables are written or arranged, while the semantics refers to the meaning of the variables. When using variables, both the syntax and the semantics are important. For example, in the formula "x+1=2", the syntax of the formula (i.e., the way in which the variables and operators are arranged) conveys the mathematical operation being performed, while the semantics of the variables (i.e., the meaning of the individual symbols) determines the specific values being used in the formula. As for the question of intent, whether something is considered "use" or "mention" can depend on the context and the intention of the speaker or writer. In the case of the statement "Some would say 'two is a prime number'", it could be considered either use or mention depending on the context and the speaker's intention. If the speaker is simply mentioning the statement without expressing their own opinion on its truth or falsehood, it could be considered a mention. However, if the speaker is using the statement to make a point or to express their own opinion, it could be considered a use. Ultimately, the determination of whether something is "use" or "mention" depends on the context and the speaker's intent. Long comment If we write "Formula 'x+1=2' has one free variable" we are asserting a property of the formula and not the result of an arithmetical operation. Thus, it is MENTION. If we agree thattruth is a property of sentences, when we write 'x+1=2 is true for x=1' we are asserting a property of the (interpreted) formula. Thus, it is MENTION; we are "describing" in plain languageTarski's rule: "The assignment 1 → x satisfies the formula ‘(x+1=2)’ ". Usual mathematical practice is semi-formal, because humans are intelligent being that can use implicit information from the context in order to correctly understand statements. If I'm uttering "Rose has four letters" there is no need to explicitly insert quotation marks because we can understand that I'm speaking of "rose" and not of Rose (a girl). In mathematics, we use formulas to express mathematical facts: as already discussed,1+1=2express a computation. The issue with a formula with free variables is slightly more convoluted, but the basic principle stands. In order to perform a computation with a formula likex+1=2we have to decide what is the value ofx. So when we give the instruction: assign to variable x the value1into formula "x+1=2" we arementioningthe formula. After the "substitution", what we get is the new formula1+1=2that we willuseto perform the computation. It is interesting to note that also philosopher can sometimes be sloppy. Leibnizexpressed the paramount principle definingidentitywith the dictum:Eadem sunt quorum unum potest substitui alteri salva veritate. As noted by contemporary logician Alonzo Church (Introduction to Mathematical Logic (1956), page 300, footnote 302): "In this form there is a certain confusion of use and mention:thingsare identical if thenameof one can be substituted for that of the other without loss of truth. A. Discussing 'Use' and 'Mention' in Mathematical Formulas B.Differentiating Use and Mention in Asserting Properties C. The Role of Variables and Context in Use and Mention D. Exploring 'Use' and 'Mention' in Philosophy and Logic

Answer 1

In mathematics, formulas are used to convey a specific meaning or perform a specific function. Free variables can be used to represent relationships in formulas. Whether something is considered 'use' or 'mention' depends on the context and the speaker's intent.

Step-by-step explanation:

In mathematics, when we use formulas, we are employing them to convey a particular meaning or perform a specific function. In the case of the formula 'x+1=2', it is being used to express the fact that, for a particular value of x (in this case, 1), the formula is true. This is an example of using the formula, as it is being employed to convey a specific meaning.

Free variables can be used in a similar way to other variables. For example, if we have a formula with a free variable x, we can use that formula to represent a relationship between x and other variables in the formula.

Whether something is considered 'use' or 'mention' depends on the context and the speaker's intent. If a statement like 'two is a prime number' is mentioned without expressing a personal opinion, it could be considered a mention. However, if the statement is used to make a point or express an opinion, it could be considered a use.