# When are statements in propositional logic true or false?

I am studying propositional logic, and have studied how propositions can be combined with logical connectives and such, and truth tables can be used to analyze the resulted truth values, depending on the truth values of involved variables. However, when not talking in the theoretical, how do we know when propositions are actually true or false? For example, "The wall is blue." Is the truth value of this statement solely contingent on our definition of blue? Also, what about mathematical statements? For example, what is the truth value of "1 = 1" dependent on? Do the truth values of statements in mathematics depend on the axioms of the system in question, such as maybe the axioms of arithmetic? How do we "prove" that 1 = 1?

Mark44
Mentor
I am studying propositional logic, and have studied how propositions can be combined with logical connectives and such, and truth tables can be used to analyze the resulted truth values, depending on the truth values of involved variables. However, when not talking in the theoretical, how do we know when propositions are actually true or false? For example, "The wall is blue." Is the truth value of this statement solely contingent on our definition of blue?
I would say so, yes.
Mr Davis 97 said:
Also, what about mathematical statements? For example, what is the truth value of "1 = 1" dependent on? Do the truth values of statements in mathematics depend on the axioms of the system in question, such as maybe the axioms of arithmetic? How do we "prove" that 1 = 1?
There are several axioms involving the = operator. The relevant one here is the reflexive property of equatlity. I.e., a number is equal to itself.

Stephen Tashi
However, when not talking in the theoretical, how do we know when propositions are actually true or false?
That illustrates the difference between mathematics and applying mathematics. As I recall, the mathematics of propositional logic simply assumes or defines it to be the case that a proposition has been assigned a "truth value" of "true" or "false". Technically you could apply this mathematics to any set of things to which such a truth value has been assigned. The assignment of the truth value would not have to correspond to our common language notion of "truth". For example, at a terminal in an electrical circuit the property of "true" might mean to have a voltage of 5V and "false" might mean to have a different voltage. In common language, 5V is no "truer" or "falser" than 3V, but applying the mathematics does not require that we assign the truth values according to common language notions. The mathematics only requires that the values were assigned in some way.

There is nothing in mathematics that says "If you look at this real life situation, you must represent it in the following manner..." So mathematics does not tell you the "actual" truth values of things or even which things you must assign truth values to.

• jim mcnamara
To the best of my limited knowledge equality is an example of a relation. That set of ordered pairs of numbers where (x,x) describes this equality. realtions were taken for granted in school but now this is the msot concrete definition of a relation i have learned. But a relation is also any set of ordered pairs that have no intuitive relation between them.

Ssnow
Gold Member
Do the truth values of statements in mathematics depend on the axioms of the system in question, such as maybe the axioms of arithmetic?
In general in a formal systems (or "theory") you start with a system of axioms that are evidently true, then by logic rules you deduce theorems and propositions from this set of axioms ... (as example you can think to relativity theory and his axiom of constancy of the velocity of the light,the theory predicts new results respect hold theories ...) so yes statements depends on the axioms ...

Stephen Tashi