Induction in colloquial English means 'educated guess', Mathematical induction however, is a kind of deductive reasoning, unlike plain 'induction'.
An argument is valid if it is impossible both for its premises to be true and its conclusion to be false. An argument can be valid even though the premises are false.
Deductive arguments are generally evaluated in terms of their validity and soundness, For a deductive argument to be considered sound the argument must not only be valid, but the premises must be true as well.
Many writers draw a technical distinction between the form ``p implies q " and the form ``if p then q ". In this view, writing ``p implies q " asserts the existence of a certain relation between the logical value of p and the logical value of q while writing ``if p then q " simply forms a compound sentence whose logical value is a function of the logical values of p and q . Notice that a relation is a mathematical object while a sentence, whether open or closed, is a syntactic form that exists in the domain of signs.
How young students understand this: Students' understandings of logical implication