Paradigms of Logic

Controversies in logic

It is by no means the case that logicians agree on what the 
principles of logic are.

Bivalence and the law of the excluded middle

The logics discussed above are all "bivalent" or "two-valued"; that is, the 
semantics for each of these languages will assign to every sentence either 
the value "True" or the value "False." Systems which do not always make this 
distinction are known as non-classical logics or non-Aristotelian logics.

In the early 20th century Jan Lukasiewicz investigated the extension of the 
traditional true/false values to include a third value, "possible", so 
inventing ternary logic, the first multi-valued logic.

Intuitionistic logic was proposed by L. E. J. Brouwer as the correct logic 
for reasoning about mathematics, based upon his rejection of the law of the 
excluded middle as part of his intuitionism. Brouwer rejected formalisation 
in mathematics, but his student Arend Heyting studied intuitionistic logic 
formally, as did Gerhard Gentzen. Intuitionistic logic has come to be of 
great interest to computer scientists, as it is a constructive logic, and is 
hence a logic of what computers can do.

Modal logic is not truth conditional, and so it has often been proposed as a 
non-classical logic. However modal logic is normally formalised with the 
principle of the excluded middle, and its relational semantics is bivalent, so 
this inclusion is disputable. However, modal logic can be used to encode 
non-classical logics, such as intuitionistic logic.

Logics such as fuzzy logic have since been devised with an infinite number of 
"degrees of truth", represented by a real number between 0 and 1. Bayesian 
probability can be interpreted as a system of logic where probability is the 
subjective truth value.

Implication: strict or material?

It is easy to observe that the notion of implication formalised in classical 
logic does not comfortably translate into natural language by means of "if... 
then...", due to a number of problems called the paradoxes of material 
implication.

The first class of paradoxes are those that involve counterfactuals, such as 
"If the moon is made of green cheese, then 2+2=4", puzzling because natural 
language does not support the principle of explosion. Eliminating these 
classes of paradox led to David Lewis's formulation of strict implication, 
and to a more radically revisionist logics such as relevance logic and 
dialetheism.

The second class of paradox are those that involve redundant premises, falsely 
suggesting that we know the succedent because of the antecedent: thus "if that 
man gets elected, granny will die" is materially true if granny happens to be 
in the last stages of a terminal illness, regardless of the man's election 
prospects. Such sentences violate the Gricean maxim of relevance, and can be 
modelled by logics that reject the principle of monotonicity, such as 
relevance logic.

Is logic empirical?


What is the epistemological status of the laws of logic? What sort of arguments 
are appropriate for criticising purported principles of logic? In an influential 
paper entitled Is logic empirical? Hilary Putnam, building on a suggestion of 
W.V.O. Quine, argued that in general that the facts of propositional logic have 
a similar epistemological status as facts about the physical universe, for 
example as the laws of mechanics or of general relativity, and in particular 
that what physicists have learned about quantum mechanics provides a compelling 
case for abandoning certain familiar principles of classical logic: if we want 
to be realists about the physical phenomena described by quantum theory, then we 
should abandon the principle of distributivity, substituting for classical logic 
the quantum logic proposed by Garrett Birkhoff and John von Neumann.

Another paper by the same name by Sir Michael Dummett argues that Putnam's 
desire for realism mandates the law of distributivity: distributivity of logic 
is essential for the realist's understanding of how propositions are true of 
the world, in just the same way as he has argued the principle of bivalence is. 
In this way, the question Is logic empirical? can be seen to lead naturally 
into the fundamental controversy in metaphysics on the relationship between 
bivalence and realism.