Logic

Foundations of Mathematics

Estimated reading time: 3 min

Statements

To avoid ambiguity when expressing mathematical statements in various languages, which arises before even including slang, modern mathematicatians have moved to using an artifical, formalised language to express theories. This starts by specifying what a statement even is. Considering that a goal is to classify them into statements that are either true or false, we can say that a statement is a combination of natural and formal language that can be assigned a truthiness value of $\top$ (true) and $\bot$ (false).

If $A$ is a statement we write $w(A)$ for the truthiness value of $A$. For example for the statement $A_1 =$ “An ape is a fish” we would have $w(A_1) = \bot$, while the statement $A_2 =$ “A rose is a plant” would yield $w(A_2) = \top$.

The truthiness value of simple-looking statements often cannot be simply determined: One need only consider open, unsolved problems in mathematics:

Note that not any “sentence” results in a statement, since we may not be able to assign a (sensible) truthiness value, for example with “The number 3 is larger” and “If not yellow then red”.

Statements $A, B$ can also be composed together, via a few “elementary” operations:

• $\lnot A$ is the statement “not $A$”, which is true if and only if $A$ is false.
• $A \lor B$ is the statement ”$A$ or $B$”, which is true if and only if either $A$ or $B$ is true (or both).
• $A \land B$ is the statement ”$A$ and $B$”, which is true if and only if both $A$ and $B$ are true.
• $A \implies B$ is the statement ”$A$ implies $B$”, which is true if and only if $B$ is always true if $A$ is also true.

Tautologies and Conclusions

Quantifiers

Proofs