Thread: Do math textbooks use "If-then" in their theorems in their logical sense?

I'm not a native English speaker so the distinction (if any) between the colloquial usage and mathematical usage (which is more 'rigid' so to speak') is not obvious to me. It makes more sense that a math text would use if-then statements (and similarly other words in the study of logic) in their logical sense.

If this is the case, can't people just use logic to justify math concepts e.g., "Math statement A can't exist because according to Theorem 1.2: If A then B. and we observe not B therefore by modus tollens, not A"? I've seen a classmate do it but don't know what to make of it since it doesn't address any math concept. It's as if he's using it as someone would use a "black box". But is he justified in using logic since the foundation of math is supposedly built on logic?

Originally Posted by Elusive1324

I'm not a native English speaker so the distinction (if any) between the colloquial usage and mathematical usage (which is more 'rigid' so to speak') is not obvious to me. It makes more sense that a math text would use if-then statements (and similarly other words in the study of logic) in their logical sense.

If this is the case, can't people just use logic to justify math concepts e.g., "Math statement A can't exist because according to Theorem 1.2: If A then B. and we observe not B therefore by modus tollens, not A"? I've seen a classmate do it but don't know what to make of it since it doesn't address any math concept. It's as if he's using it as someone would use a "black box". But is he justified in using logic since the foundation of math is supposedly built on logic?

I am not at all sure about what point you are making.
But the 'mathematical' use of the if-then construction is exactly the same as that of formal logic.

To say If an integer is even then it is a multiple of two can be done as saying If an integer is not a multiple of two then it is not even.

There is no black-box there.

Originally Posted by Elusive1324

I'm not a native English speaker so the distinction (if any) between the colloquial usage and mathematical usage (which is more 'rigid' so to speak') is not obvious to me. It makes more sense that a math text would use if-then statements (and similarly other words in the study of logic) in their logical sense.

If this is the case, can't people just use logic to justify math concepts e.g., "Math statement A can't exist because according to Theorem 1.2: If A then B. and we observe not B therefore by modus tollens, not A"? I've seen a classmate do it but don't know what to make of it since it doesn't address any math concept. It's as if he's using it as someone would use a "black box". But is he justified in using logic since the foundation of math is supposedly built on logic?

it IS logic- a statement is true if and only if its contrapositive is true.

"if A then B" is true if and only if "if (not B) then (not A)".

Originally Posted by Plato

I am not at all sure about what point you are making.
But the 'mathematical' use of the if-then construction is exactly the same as that of formal logic.

To say If an integer is even then it is a multiple of two can be done as saying If an integer is not a multiple of two then it is not even.

There is no black-box there.

Sorry for the confusion. I think I'm making a point regarding linguistic use of the if-then statement. Perhaps a brief narrative will clarify what I was getting at:

While reading theorems from a math text, I was wondering if the writers used the if-then statement in a colloquial or logical context. Depending on their choice of usage, I figured I would know how to "treat the theorems" (understand & use the theorems).

By colloquial usage I'm referring to the example: Bill, an average guy, might say "If I'm hungry then I eat." and assert that to be true when in reality, there are instances when Bill is hungry but he doesn't eat.

By logical context I'm referring to the usage of if-then statements in the context of mathematics or formal study of logic.

Now if the math text used 'if-then' in a colloquial context to describe a theorem (say the math theorem was 'If A then B'), I would treat the math statement as if it were Bill telling me "If I'm hungry then I eat". That is, I would take that if-then statement with a grain of salt -- Bill may or may not eat if he's hungry.
In contrast, if the math text used 'if-then' in the logical context, I would treat their description of theorems as though the rules of inferences applies. For example, if the text asserts 'If A then B', then I would not doubt that 'If not B, then not A' is a true statement.

So to summarize, if the math text used if-then in the logical sense and it included "If Bill is hungry, then he eats", I would have no doubt in reasoning that if Bill doesn't eat, he's not hungry.

Originally Posted by Elusive1324

While reading theorems from a math text, I was wondering if the writers used the if-then statement in a colloquial or logical context. Depending on their choice of usage, I figured I would know how to "treat the theorems" (understand & use the theorems).

That simply is not an issue.
There is no use of colloquial language in mathematics.
Any 'if-then' in mathematics is logical language.

Originally Posted by Plato

I am not at all sure about what point you are making.
But the 'mathematical' use of the if-then construction is exactly the same as that of formal logic.

To say If an integer is even then it is a multiple of two can be done as saying If an integer is not a multiple of two then it is not even.

There is no black-box there.

This question came up while I was reading a math book and it mentioned "For a differential equation of the form M(x,y)dx+N(x,y)dy=0, If there exists a function f(x,y) such that the partial derivatives f_x(x,y) = M(x,y) and f_y(x,y) = N(x,y), then that differential equation is called exact."

This appears to be a math definition of "exact ODE". How do we treat definitions logically? Do we treat it as a statement of equivalence?

The book uses an if-then statement to define a math term. Wouldn't it make more sense for the writer to use "if and only if"? Because they didn't, I'm taking this to be a math statement and not a definition.

Personally, I would consider that improper usage. Any definition should be "A if and only if B". In my (not very humble) opinion, this should have been "an equation of the form M(x,y)dx+N(x,y)dy=0, is said to "exact" if and only if there exists a function f(x,y) such that the partial derivatives f_x(x,y) = M(x,y) and f_y(x,y) = N(x,y)".

Originally Posted by Elusive1324

Sorry for the confusion. I think I'm making a point regarding linguistic use of the if-then statement. Perhaps a brief narrative will clarify what I was getting at:

While reading theorems from a math text, I was wondering if the writers used the if-then statement in a colloquial or logical context. Depending on their choice of usage, I figured I would know how to "treat the theorems" (understand & use the theorems).

By colloquial usage I'm referring to the example: Bill, an average guy, might say "If I'm hungry then I eat." and assert that to be true when in reality, there are instances when Bill is hungry but he doesn't eat.

By logical context I'm referring to the usage of if-then statements in the context of mathematics or formal study of logic.

Now if the math text used 'if-then' in a colloquial context to describe a theorem (say the math theorem was 'If A then B'), I would treat the math statement as if it were Bill telling me "If I'm hungry then I eat". That is, I would take that if-then statement with a grain of salt -- Bill may or may not eat if he's hungry.
In contrast, if the math text used 'if-then' in the logical context, I would treat their description of theorems as though the rules of inferences applies. For example, if the text asserts 'If A then B', then I would not doubt that 'If not B, then not A' is a true statement.

So to summarize, if the math text used if-then in the logical sense and it included "If Bill is hungry, then he eats", I would have no doubt in reasoning that if Bill doesn't eat, he's not hungry.

If Bill says "If I'm hungry, then I eat" and there are instances when Bill is hungry, yet doesn't eat, then Bill is lying, what is he says is not true (it is falsified by the "exceptional" instances). People often inaccurately state what they mean. Perhaps Bill INTENDS to say: "If I'm hungry, I usually eat" and by force of habit leaves out what he considers "unnecessary qualifying phrases". This is common with spoken (informal) language, we do not say what we mean (but hope our intention is "generally understood").

If a mathematics text is not clear and concise about stating the author's intentions in writing the text, it is a bad text. The mathematical community has come to expect that mathematical texts are written logically, even if the exposition is informal. Speaking slight and generally ignorable falsehoods in everyday conversation is common-place, and is not worth commenting on. We generally do not hold spoken conversation to a high standard of accuracy or consistency. Whenever we talk about mathematics, it is expected we are making, if not TRUE statements, at least logically consistent ones.

If a math text states something along the lines of: "Now if condition A holds, we must have these consequences B", even though this is in "natural language exposition", the sense of implication is strict: there should be no exceptions.