First of all, we have to consider that the propositional connectives are a (very simple) mathematical model of natural language, suited for modelling formal arguments.

In the context of classical logic, their definition is through truth-table; having defined them, we may check how they are "proxing" natural language mechanism.

Someone do their job in a good way (negation and conjunction), someone with some arbitrariness (disjunction, inclusive : vel instead of aut); someone with a big approximation : the conditional.

We may find a useful the discussion in Stephen Cole Kleene, Mathematical Logic (1967), pag.9 and pag.58-on.

As Kleene says, a lot of controversies aroused around the truth-functional definition of the conditional connective.

The very relevant point to notice is that the mathematical model of "if A, then B" represented by truth-tables does not require any sort of "causal link" between them.

Having said that, we must take into account the key role played by the truth-functional connective "if ... then" and the inference rule of modus ponens that allows us to infer from the premises A and "if A, then B", the conclusion B.

We must read it as Gottlob Frege did in his Begriffsschrift (1879):

assuming as true both the premises, the assumption that "if A, then B" is True, rule-out the row T-F in the truth-table for implies, while the assumption that also A is True rule out two other rows (F-F and F-T, respectively). Then, the conclusion that B is True is licensed.

So, assuming the truth-functional definition of "if A, then B", we have that the truth of A is a sufficient condition for that of B.

This "mechanism" is what is used again and again in mathematical proofs: either having assumed some axiom A as true or having available some already proven theorem A, we may prove a new theorem B through a deductive argument showing "if A, then B".

And that is the reason why we:

use the material implicaton over alternatives.

On "alternative" (i.e. non truth-functional) conditionals, see e.g. Indicative Conditionals, Counterfactuals, Strict conditional.

why do we stick with the material implication?

First of all, we have to consider that the propositional connectives are a (very simple) mathematical model of natural language, suited for modelling formal arguments.

In the context of classical logic, their definition is through truth-table; having defined them, we may check how they are "proxing" natural language mechanism.

Someone do their job in a good way (negation and conjunction), someone with some arbitrariness (disjunction, inclusive : vel instead of aut); someone with a big approximation : the conditional.

We may find a useful the discussion in Stephen Cole Kleene, Mathematical Logic (1967), pag.9 and pag.58-on.

As Kleene says, a lot of controversies aroused around the truth-functional definition of the conditional connective.

The very relevant point to notice is that the mathematical model of "if A, then B" represented by truth-tables does not require any sort of "causal link" between them.

Having said that, we must take into account the key role played by the truth-functional connective "if ... then" and the inference rule of modus ponens that allows us to infer from the premises A and "if A, then B", the conclusion B.

We must read it as Gottlob Frege did in his Begriffsschrift (1879):

assuming as true both the premises, the assumption that "if A, then B" is True, rule-out the row T-F in the truth-table for implies, while the assumption that also A is True rule out two other rows (F-F and F-T, respectively). Then, the conclusion that B is True is licensed.

So, assuming the truth-functional definition of "if A, then B", we have that the truth of A is a sufficient condition for that of B.

This "mechanism" is what is used again and again in mathematical proofs: either having assumed some axiom A as true or having available some already proven theorem A, we may prove a new theorem B through a deductive argument showing "if A, then B".

And that is the reason why we:

use the material implicaton over alternatives.

On "alternative" (i.e. non truth-functional) conditionals, see e.g. Indicative Conditionals, Counterfactuals, Strict conditional.

First of all, we have to consider that the propositional connectives are a (very simple) mathematical model of natural language, suited for modelling formal arguments.

In the context of classical logic, their definition is through truth-table; having defined them, we may check how they are "proxing" natural language mechanism.

Someone do their job in a good way (negation and conjunction), someone with some arbitrariness (disjunction, inclusive : vel instead of aut); someone with a big approximation : the conditional.

We may find a useful the discussion in Stephen Cole Kleene, Mathematical Logic (1967), pag.9 and pag.58-on.

As Kleene says, a lot of controversies aroused around the truth-functional definition of the conditional connective.

The very relevant point to notice is that the mathematical model of "if A, then B" represented by truth-tables does not require any sort of "causal link" between them.

Having said that, we must take into account the key role played by the truth-functional connective "if ... then" and the inference rule of modus ponens that allows us to infer from the premises A and "if A, then B", the conclusion B.

We must read it as Gottlob Frege did in his Begriffsschrift (1879):

assuming as true both the premises, the assumption that "if A, then B" is True, rule-out the row T-F in the truth-table for implies, while the assumption that also A is True rule out two other rows (F-F and F-T, respectively). Then, the conclusion that B is True is licensed.

So, assuming the truth-functional definition of "if A, then B", we have that the truth of A is a sufficient condition for that of B.

This "mechanism" is what is used again and again in mathematical proofs: either having assumed some axiom A as true or having available some already proven theorem A, we may prove a new theorem B through a deductive argument showing "if A, then B".

First of all, we have to consider that the propositional connectives are a (very simple) mathematical model of natural language, suited for modelling formal arguments.

In the context of classical logic, their definition is through truth-table; having defined them, we may check how they are "proxing" natural language mechanism.

Someone do their job in a good way (negation and conjunction), someone with some arbitrariness (disjunction, inclusive : vel instead of aut); someone with a big approximation : the conditional.

We may find a useful the discussion in Stephen Cole Kleene, Mathematical Logic (1967), pag.9 and pag.58-on.

As Kleene says, a lot of controversies aroused around the truth-functional definition of the conditional connective.

The very relevant point to notice is that the mathematical model of "if A, then B" represented by truth-tables does not require any sort of "causal link" between them.

Having said that, we must take into account the key role played by the truth-functional connective "if ... then" and the inference rule of modus ponens that allows us to infer from the premises A and "if A, then B", the conclusion B.

We must read it as Gottlob Frege did in his Begriffsschrift (1879):

assuming as true both the premises, the assumption that "if A, then B" is True, rule-out the row T-F in the truth-table for implies, while the assumption that also A is True rule out two other rows (F-F and F-T, respectively). Then, the conclusion that B is True is licensed.

So, assuming the truth-functional definition of "if A, then B", we have that the truth of A is a sufficient condition for that of B.

This "mechanism" is what is used again and again in mathematical proofs: either having assumed some axiom A as true or having available some already proven theorem A, we may prove a new theorem B through a deductive argument showing "if A, then B".

And that is the reason why we:

use the material implicaton over alternatives.

On "alternative" (i.e. non truth-functional) conditionals, see e.g. Indicative Conditionals, Counterfactuals, Strict conditional.