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In logic, as in math and all other walks of life, there are degrees of formality. In logic and math, formality usually takes the form of symbols. Ao it might be seen to alienate itself from ordinary language, even though its aim is to clarify. While it deletes specific meaning, it establishes a storehouse of meaning, or what I call a repository of information.

The product of formalization is the deductive system. Its most prestigious mode has a set of special forms from which deduction begins and which take on special importance as axioms. The remaining, unlimited forms of the system are derived (or calculated -- hence the word, calculus) from the axioms in accordance with a set of transformation rules -- i.e., demonstrations of consequences from postulates. Logical inference.

Let me say that again. If you have a whole bunch of statements (statement forms) that you identify as a system, you can select a few of them from which to derive all the rest. The system is then a deductive system.

In Deductive Sentential Logic, the method of deduction proceeds from compound sentences known as conjunctions, disjunctions, conditionals, and negations, which are discussed in Ordinary Logic Makes a Statement or Two.  The deductive arguments in this category are analogous to, but different in character from those based on the syllogism. The difference is due to the fact that the syllogism contains words like 'all' and 'no' as a critical part of its logical structure, whereas the sentential logic doesn't.

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Arguments containing compound sentences depend for their validity on truth functions of the four basic compound sentences. So the key to understanding the validity of these arguments is to recognize that the compound sentences have their own logical identity and their own truth-value. The truth condition of each compound sentence is dependent on, but separate from, the truth condition of its parts. This relationship defines the truth value function.

In either case, the truth function of a negation is the negative of the truth-value of the original sentence. Therefore, if a sentence, P, is true, the compound sentence, -P, is false. And if P is false, then -P is true. This format adopts the principle of non-contradiction, a principle accepted in ordinary logic, but not accepted in Fuzzy Logic.

The negation argument can be extended validly to include any level of negation. A double negation, for instance, yields the original statement:

The conjunction is a compound sentence containing two sentences linked by the key word 'and.' If P and Q are individual sentences, we can form the compound sentence, 'P and Q'. Intuitively, you would expect that P and Q is true only if both P and Q are individually true. So, for example, the sentence:

Today is Tuesday and I am on my way to the shopping center.

is a true statement if today is actually Tuesday and I'm actually heading for the shopping center.

But if today is really Thursday, and even though I'm going shopping, the conjunction is false, as you'd expect. In general, the conjunction is true if both its parts are true; otherwise it's false.

Two valid conclusions that can be drawn form the conjunction, P and Q are as follows:

and

That is, if P and Q is true, then P is true, and Q is true, by virtue of the truth function.

Being a linkage of two sentences with the key word 'or,' the disjunction is true only if at least one of the alternative sentences is true, but otherwise it is false.

An example of a valid argument from the disjunction is therefore:

More controversial that the other compound sentences, the conditional is generally understood to be false if and only if the antecedent is true and the consequent is false. Otherwise it is true. Familiar valid arguments drawing on the conditional are as follows:

and

Denial of the consequent is to deny the premise.

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Truth tables can be used to determine the validity of arguments based on compound sentences.  But they can also get clumsy for arguments with many variables. In this respect, a more formal deductive approach is less complicated, though not necessarily less direct. Deductive arguments have central importance in both logic and math. 

For examples, see The Elements of Logic, by Stephen F Barker.

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