# Truth sets in mathematics

The definition of mathematical truth is confusing.

Let P and Q denote the statements 'I'm a man' and '$$1', respectively.

P is truly (naturally) true because it is really true.

In natural language, we cannot determine whether Q is true or false. However in math, as far as I understand, Q has a truth set that is not empty; therefore it is mathematically true.

That there are two ways to determine whether something is true or false confuses me.

• Can you give a reference for this claim "q is mathematically true because it has truth set". [Seriously, I have no idea what a truth set is.] Mar 31 at 14:37
• I've never heard of this term "truth set", but I assume some author uses it to mean the set of values for $x$ which make the statement $1<x<2$ true. A statement with free variables in it is neither true nor false, but there will usually be values of the variables that make it true, and other values that make it false. Mar 31 at 15:03
• You can bind free variables with quantifiers. A statement without only bound variables will have a truth value (in a given context--see my next comment). For example, $\forall x(1<x<2)$ is false, and $\exists x(1<x<2)$ is true (assuming the quantifiers range over all the real numbers). Mar 31 at 15:07
• The context for a mathematical statement includes the domain of discourse, that is, the set of values that the variables range over. Also the intended meaning of relations, functions, operators, and constants. (For example, <, 1, and 2 in $1<x<2$.) Mar 31 at 15:10
• In the special case where a statement with free variables is true for all values of the variable (e.g., $x+y=y+x$ for the real numbers), some authors would say it is true. Others would demand that you put in the implicit universal quantifiers: $\forall x\forall y(x+y=y+x)$. Mar 31 at 15:12

Let P and Q denote the statements 'I'm a man' and '$$1', respectively.

P is truly (naturally) true because it is really true.

In other words, P is synthetically true.

In natural language, we cannot determine whether Q is true or false. However in math, as far as I understand, Q has a truth set that is not empty; therefore it is mathematically true.

On the other hand, $$Q$$ is not a statement, and has no definite truth value.

When you say that $$Q$$ has a truth set, you mean that it is satisfied by every real number strictly between $$1$$ and $$2.$$ However, this does not mean that $$Q$$ is mathematically true:

1. Q is (mathematically) false for $$x=7.$$
2. In Number Theory, the domain of discourse might comprise integers; in this universe, Q is always (mathematically) false.

As a helpful addition, you may imagine your propositional variable to be a function from elements to truth.

In the case of any propositional variable with no free variables, we have a constant function, which (barring concerns of indexicality), always maps to a true or false (or whatever sort of truth values you have in your domain).

in the case of a propositional variable with n free variables, we obtain a n-ary function, such that, when substituted, maps to true or false. your Q for example maps 1.5 to true and 1 to false.

Now you can view a truth set as the preimage of "true" under the induced function. So these two methods are really instances of a more general method.

Note that Q is not mathematically true (or that your author is using a very strange definition). Rather Q(1.5) might be true, or Q(1) might be false.

I would say that both P and Q are "open sentences". Whether P is true or false depending on the variable "I". I presume that "I" refers to the person who wrote the sentence but I do not know whether or not that person is a man or a woman. Whether Q is true or false depends on the variable "x".

• I'd say that P ("I'm a man") is an atomic sentence (so, a closed formula, not an open formula)—for example, in the sentences "Mary likes skating" and "I like skating" both subjects are intended as constants rather than variables (which Mary? which I?)—as such, P's truth value is definite within each interpretation. Apr 1 at 8:59