Logical (Deductive) Inference

Let KB = { S1, S2,..., SM } be the set of all sentences in our Knowledge Base, where each Si is a sentence in Propositional Logic. Let { X1, X2, ..., XN } be the set of all the symbols (i.e., variables) that are contained in all of the M sentences in KB. Say we want to know if a goal (aka query, conclusion, or theorem) sentence G follows from KB.

Since we don't know the interpretation of these symbols or sentences in the world, we don't know whether the associated facts in the world are true or false. So, instead, consider all possible combinations of truth values for all the symbols, hence enumerating all worlds' truth values:

X1 X2 ... XN | S1 S2 ... SM | S1 ^ S2 ^...^ SM | (S1 ^...^ SM) => G
-------------|--------------|------------------|-----------------
F  F  ... F  |              |                  |
F  F  ... T  |              |                  |
...          |              |                  |
T  T  ... T  |              |                  |

There are 2^N rows in the table.
Each row corresponds to an equivalence class of worlds that, under a given interpretation, have the truth values for the N symbols assigned in that row.
The models of KB are the rows where the next to last column is true, i.e., where all of the sentences in KB are true.
A sentence R is valid if and only if it is true under all possible interpretations, i.e., if the entire column associated with R contains all true values.
To determine if a sentence G is entailed by KB, we must determine if all models of KB are also models of G. That is, whenever KB is true, G is true too. Or, the sentence (KB => G) is valid (by definition of the implication connective). In other words, if the last column of the table above contains only true, then KB entails G; or conclusion G logically follows from the premises in KB, no matter what the interpretations (i.e., semantics) associated with all of the sentences!
The truth table method of inference is complete for PL (Propositional Logic), but not FOL (First-Order Logic). Checking for satisfiability in PL is NP-complete.

Instead of an exponential length proof by truth table construction, is there a faster way to implement the inference process? Yes, using a proof procedure that uses sound rules of inference to deduce (i.e., derive) new sentences that are true in all cases where the premises are true. For example, consider the following:

    P   Q | P   P => Q | P ^ (P => Q) | (P ^ (P => Q)) => Q
    ------|------------|--------------|--------------------
    F   F | F      T   |       F      |       T
    F   T | F      T   |       F      |       T
    T   F | T      F   |       F      |       T
    T   T | T      T   |       T      |       T

Since whenever P and P => Q are both true (last row only), Q is true too, Q is said to be derived from these two premise sentences. This local pattern referencing only two of the M sentences in KB is called the Modus Ponens inference rule. The truth table shows that this inference rule is sound. It specifies how to make one kind of step in deriving a conclusion sentence from a KB.

Therefore, given the sentences in KB, construct a proof that a given conclusion sentence can be derived from KB by applying a sequence of sound inferences using either sentences in KB or sentences derived earlier in the proof, until the conclusion sentence is derived. (Note: This step-by-step, local proof process also relies on the monotonicity property of PL and FOL. I.e., adding a new sentence to KB does not affect what can be entailed from the original KB and does not invalidate old sentences.)