Inference - Logical Truth - Empiricism - Positivism

Basics of logical argument:

Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement.

Logical truths (including tautologies) are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. However, it is not universally agreed that there are any statements which are necessarily true.

A logical truth was considered by Ludwig Wittgenstein to be a statement which is true in all possible worlds^[1]. This is contrasted with synthetic claim (or fact) which is true inthis world, as it has historically unfolded, but which is not true in at least one possible world, as it might have unfolded. The proposition “If p and q, then p” and the proposition “All husbands are married” are logical truths because they are true due to their inherent meanings and not because of any facts of the world. Later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations.

The existence of logical truths is sometimes put forward as an objection to empiricism because it is impossible to account for our knowledge of logical truths on empiricist grounds.

Rules of inference must be distinguished from axioms of a theory. In terms of semantics, axioms are valid assertions. Axioms are usually regarded as starting points for applying rules of inference and generating a set of conclusions. Or, in less technical terms:

Rules are statements ABOUT the system, axioms are statements IN the system

(Assumes closed System?) (see Godel Esher Bach: Hofstadter)

Overview

In formal logic (and many related areas), rules of inference are usually given in the following standard form:

  Premise#1
  Premise#2
        ...
  Premise#n   
  Conclusion

This expression states, that whenever in the course of some logical derivation the given premises have been obtained, the specified conclusion can be taken for granted as well. The exact formal language that is used to describe both premises and conclusions depends on the actual context of the derivations. In a simple case, one may use logical formulae, such as in:

  A→B
  A        
  ∴B

This is just the modus ponens rule of propositional logic. Rules of inference are usually formulated as rule schemata by the use of universal variables. In the rule (schema) above, A and B can be instantiated to any element of the universe (or sometimes, by convention, some restricted subset such as propositions) to form an infinite set of inference rules.

A proof system is formed from a set of rules chained together to form proofs, or derivations. Any derivation has only one final conclusion, which is the statement proved or derived. If premises are left unsatisfied in the derivation, then the derivation is a proof of a hypothetical statement: "if the premises hold, then the conclusion holds."

Different Types of Logic

Non-classical logics

Main article: Non-classical logic

Non-classical logic is the name given to formal systems which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.^[3]

• Computability logic is a semantically constructed formal theory of computability, as opposed to classical logic, which is a formal theory of truth; integrates and extends classical, linear and intuitionistic logics.
• Fuzzy logic rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1.
• Intuitionistic logic rejects the law of the excluded middle, double negative elimination, and the De Morgan's laws;
• Linear logic rejects idempotency of entailment as well;
• Modal logic extends classical logic with non-truth-functional ("modal") operators.
• Paraconsistent logic (e.g., dialetheism and relevance logic) rejects the law of noncontradiction;
• Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment;

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