Logical Proofs

A sentence is valid or necessarily true if and only if it is true under all possible interpretations in all possible worlds. If a sentence is valid, it is so by virtue of its logical structure independent of what possible interpretations are like.

A sentence is satisfiable if and only if there is some interpretation in some world for which it is true. A sentence that is not satisfiable is unsatisfiable.

Making Inferences

Reasoning and inference are generally used to describe any process by which conclusions are reached. Inference is often of three types:

  1. Abduction

    Abduction is a process to generate explanations. It aims to give a hypothetical explanation. It is only a plausible inference.

    If there is an axiom E=>F and an axiom F, then E does NOT logically follow. This is called Abduction and is not a sound rule of inference.

  2. Induction

    Hypothesises a general rule from observations.

  3. Deduction or rational inference

    These are inferences which are made by the sound rules of inference. This means that the conclusion is true in all cases where the premise is true. These rules preserve the truth.

Sound rules of inference

Looking at the truth tables you will easily see that the sound rules are:

  1. Modus ponens (or implication-elimination)

    From an implication and the premise of the implication you can infer the conclusion.

    If there is an axiom E F and an axiom E, then F follows logically.

  2. Modus tolens

    If there is an axiom E F and an axiom F, then E follows logically.

  3. Resolution.

    If there is an axiom E F and an axiom F G then E G follows logically.

    In fact, resolution can subsume both modus ponens and modus tolens. It can also be generalized so that there can be any number of disjuncts in either of the two resolving expressions, including just one. (Note, disjunct are expressions connected by , conjuncts are those connected by .) The only requirement is that one expression contains the negation of one disjunct from the other.

    To verify soundness we can construct a truth-table with one line for each possible model of the proposition symbols in the premise, and show that in all models where the premise is true, the conclusion is also true.

    Example of truth table for resolution:
    A
    B
    C
    AB
    BC
    AC
    F
    F
    F
    F
    T
    F
    F
    F
    T
    F
    T
    T
    F
    T
    F
    T
    F
    F
    F
    T
    T
    T
    T
    T
    T
    F
    F
    T
    T
    T
    T
    F
    T
    T
    T
    T
    T
    T
    F
    T
    F
    T
    T
    T
    T
    T
    T
    T

<<<Previous Next>>>