All people are over 10 ft tall.
Trump is a person.
Therefore, Trump is over 10 ft tall.
Valid—because conclusion follows from premises.
But one of the premises is false, so it is not sound.
All deductive reasoning is based on one principle, namely:
Whatever follows from a truth is a truth.
In other words,
If P, and P entails Q, then Q.
In some cases, a case of deductive-reasoning can be shown to be a substitution-instance of this logical schema. When this is the case, the reasoning in question is said to be ‘formally’ valid, and the validity of the argument can therefore be established strictly computationally. (A computation is simply a case of formal reasoning.)
Here is an example.
Premise 1: For any three numbers x, y, and z, if x>y and y>z, then x>z
Premise 2: a2>a1
Premise 3: a3>a2
Premise 4: a4>az3
Premise 1000: a1000>a999
Give or take a few technicalities, this argument is a substitution instance of the previously mentioned argument-schema and its validity can therefore be computationally established.
When reasoning is valid but not formally so, validity can only be known intuitively. In other words, there is no way, apart from intuition, to know that a case of informally valid reasoning is in fact valid.
Here is an example:
Premise: John is literate (meaning that he can read).
Conclusion: John is sapient (meaning that he has at least some intelligence).
This last argument is valid, since the conclusion follows from the premise, but it is not formally so, since there are invalid arguments that have the same form (e.g. John is green; therefore, John is a giraffe). The validity of such an argument can obviously be seen, but it cannot, tautologously, be formally established
This last point is subject to one qualification. Informally valid reasoning is capable of ‘relative’ proof. If someone rejects an informally valid argument, it can be shown that, in so doing, he is committed to accepting some position that he himself rejects.
For example, suppose you reject the contention that John must be sapient, given that he is literate. I can respond by pointing out that, if you are right, then no intelligence is involved in mapping meanings onto ink-marks and the like, and also that no intelligence in any of the other operations involved in absorbing a text—which, though implied by your position, you are unlikely to accept and which, consequently, therefore constitutes a proof of sorts, albeit only a relative and informal one, of the position in question.