The official answer is that any deduction from a false premise is true. It is false that you have the triangle, so anything about it is true.

At the same time, the official answer is that any truth is implied by any other truth.

One of the reasons for moving away from classical logic in mathematics, in general, is that both of those positions are preposterous.

And the problem does not hinge on the problem of modal existence.

Just as it is not true that there is a sum of the angles of your nonexistent triangle, it is just not the fact that "If puppies are cute, then the U.S. is a democratic republic." The two have nothing to do with one another, and that 'if' and that 'then' have no business being there.

Positions like https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=intuitionism, https://en.wikipedia.org/wiki/Constructivism_(mathematics), and even https://en.wikipedia.org/wiki/Fictionalism formalized through https://en.wikipedia.org/wiki/Proof_theory take restoring the natural connection between statements seriously, whether they are motivated by distrust of classical foundations, an imperative toward usefulness, or a feeling of basic integrity.

The problem is that it is so convenient to render formal logic into an algebra, that thinking through the real rules is, by comparison, a muddle. Intuitionism, for instance, proposes that the real numbers, in the form of an endless sequence of digits actually exist. But constructivists respond, "Not in the real world, they don't. If you are actually interested in such things, you might well have already gone around the bend that leads down the path to untrustworthy conventions." Even if you back off to the modified formalism of proof theory, can proofs be arbitrarily long? By arbitrarily, can we mean implicitly infinitely long with big infinite gaps? Unfortunately, it matters.

So, yes, there are many arguments against this kind of logic. But there are so many it is hard to choose one. And very few people are worried enough about it to fight for a given solution, even if they find winding out special solutions can be fun, (and sometimes helps improve engineering tasks like computer science.)

The official answer is that any deduction from a false premise is true. It is false that you have the triangle, so anything about it is true.

At the same time, the official answer is that any truth is implied by any other truth.

One of the reasons for moving away from classical logic in mathematics, in general, is that both of those positions are preposterous.

And the problem does not hinge on the problem of modal existence.

Just as it is not true that there is a sum of the angles of your nonexistent triangle, it is just not the fact that "If puppies are cute, then the U.S. is a democratic republic." The two have nothing to do with one another, and that 'if' and that 'then' have no business being there.

Positions like Intuitionism, Constructivism, and even Fictionalism formalized through proof theory take restoring the natural connection between statements seriously, whether they are motivated by distrust of classical foundations, an imperative toward usefulness, or a feeling of basic integrity.

The problem is that it is so convenient to render formal logic into an algebra, that thinking through the real rules is, by comparison, a muddle. Intuitionism, for instance, proposes that the real numbers, in the form of an endless sequence of digits actually exist. But constructivists respond, "Not in the real world, they don't. If you are actually interested in such things, you might well have already gone around the bend that leads down the path to untrustworthy conventions." Even if you back off to the modified formalism of proof theory, can proofs be arbitrarily long? By arbitrarily, can we mean implicitly infinitely long with big infinite gaps? Unfortunately, it matters.

So, yes, there are many arguments against this kind of logic. But there are so many it is hard to choose one. And very few people are worried enough about it to fight for a given solution, even if they find winding out special solutions can be fun, (and sometimes helps improve engineering tasks like computer science.)

The official answer is that any deduction from a false premise is true. At the same time the official answer is that any truth is implied by any other truth. One of the reasons for moving away from classical logic in mathematics in general is that both of those positions are preposterous.

And the problem does not hinge on the problem of modal existence. Just as it is not true that there is a sum of the angles of your nonexistent triangle, it is just not the fact that "If puppies are cute, then the U.S. is a democratic republic." The two have nothing to do with one another, and that 'if' and that 'then' have no business being there.

Positions like intuitionistic logic, constructivism, and even mathematical fictionalism via proof theory take restoring the natural connection between statements seriously, whether they are motivated by distrust of classical foundations, an imperative toward usefulness, or a feeling of basic integrity.

The problem is that it is so convenient to render formal logic into an algebra, that thinking through the real rules is, by comparison, a muddle. Intuitionism, for instance, proposes that the real numbers, in the form of an endless sequence of digits actually exist. But constructivists respond, "Not in the real world, they don't. If you are actually interested in such things, you might well have already gone around the bend that leads down the path to untrustworthy conventions." Even if you back off to the modified formalism of proof theory, can proofs be arbitrarily long? By arbitrarily, can we mean implicitly infinitely long with big infinite gaps? Unfortunately, it matters.

So, yes, there are many arguments against this kind of logic. But there are so many it is hard to choose one. And very few people are worried enough about it to fight for a given solution, even if they find winding out special solutions can be fun, (and sometimes helps improve engineering tasks like computer science.)

The official answer is that any deduction from a false premise is true. It is false that you have the triangle, so anything about it is true.

At the same time, the official answer is that any truth is implied by any other truth.

One of the reasons for moving away from classical logic in mathematics, in general, is that both of those positions are preposterous.

And the problem does not hinge on the problem of modal existence.

Just as it is not true that there is a sum of the angles of your nonexistent triangle, it is just not the fact that "If puppies are cute, then the U.S. is a democratic republic." The two have nothing to do with one another, and that 'if' and that 'then' have no business being there.

Positions like https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=intuitionism, https://en.wikipedia.org/wiki/Constructivism_(mathematics), and even https://en.wikipedia.org/wiki/Fictionalism formalized through https://en.wikipedia.org/wiki/Proof_theory take restoring the natural connection between statements seriously, whether they are motivated by distrust of classical foundations, an imperative toward usefulness, or a feeling of basic integrity.

The problem is that it is so convenient to render formal logic into an algebra, that thinking through the real rules is, by comparison, a muddle. Intuitionism, for instance, proposes that the real numbers, in the form of an endless sequence of digits actually exist. But constructivists respond, "Not in the real world, they don't. If you are actually interested in such things, you might well have already gone around the bend that leads down the path to untrustworthy conventions." Even if you back off to the modified formalism of proof theory, can proofs be arbitrarily long? By arbitrarily, can we mean implicitly infinitely long with big infinite gaps? Unfortunately, it matters.

So, yes, there are many arguments against this kind of logic. But there are so many it is hard to choose one. And very few people are worried enough about it to fight for a given solution, even if they find winding out special solutions can be fun, (and sometimes helps improve engineering tasks like computer science.)