When are statements in propositional logic true or false?

[Mr Davis 97]

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]

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.

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.

 

[Logical Dog]

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]

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]

There is relation between the issues of your other post https://www.physicsforums.com/threads/logic-puzzle-and.894325/#post-5626096 and question of how and whether propositional logic is applied to mathematics. I don't whether you are studying material that takes up this issue - or whether that thread and this thread just came up coincidentally.

In the thread on the logic puzzle, as @haruspex pointed out:

Self-referential statements are not really allowed in a formal study of logic.

There are self-referential statements that do not meet the criteria of being "propositions" because they cannot be assigned a single truth value. ( For example, in your other thread, suppose you had only 1 statement (instead of 100) and that statement said "1. Exactly 1 of these 1 statements is false". )

Do we apply propositional logic when we do mathematics ? Yes, in a informal sense, but not in the formal sense. Propositional logic itself is a field of mathematics. If we applied propositional logic to make statements about propositional logic, that would require making self-referential statements.