Logic in Philosophy

Presentation to The Philosophy Forum, April 18, 2007

1.1 Logic comes from the classic Greek logos (meaning 'word', 'reason' or 'principle'). It is concerned with the validity of inference and demonstration. It is a key component of philosophy along with epistemology and ontology. Logic establishes the correctness of argumentation and judgement[1], almost invariably with deductive reasoning from given axioms. Its greatest strength is the ability to uncover fallacies and its greatest challenge is paradoxes.

1.2 Shorthand used in this presentation will include the following boolean operatives: '!' shall stand for "not", '&' for "and", '|' for "or", "=" for "equal".

1.3 The two broad divisions of logic are (abstract) formal logic and (contextual) informal logic. Formal logic is commonly used in mathematics, computing and electronics. It is especially good at highlighting contradictions and resolving multiple connected propositions. Informal logic is often used in journalism, law and debates. and is especially useful for highlighting fallacies.

1.4 In informal logic, fallacies include:

1.4 The classic basic laws of thought are usually attributed to Aristotle. They include (in propositional form and with Betrand Russell's explanations); The Law of Identity: A = A ("Whatever is, is."), The Law of Non-Contradiction: !(p | !p)[2] ("Nothing can both be and not be.") and The Law of the Excluded Middle: p & !p (Everything must either be or not be.")

1.5 A syllogism takes the consists of three parts: the major premise, the minor premise, and the conclusion. The major premise includes terms are the syllogism's major term and middle term. The minor premise (or subsumption) includes syllogism's minor and middle term only. Propositions in a syllogism may be of four types only: A-type: universal and affirmative; I-type: Particular and affirmative; E-type: Universal and negative; O-type: Particular and negative

1.6 In modern formal logic, syllogistic propositions have been absorbed by predicate calculus which accounts for problem of multiple generality [2]. In addition there has been the development in the twentieth century of modal and temporal logic (concepts like possibility, existence, and necessity, eventually, formerly, can, could, might, may, must), and their combination with Boolean logic and algebra [3]. Logical notation has been established, including universal and existential quanitification and significant work has been conducted on set theory and the axiomatisation of arithmetic.

1.7 Propositional Calculus [4]: A proposition is a statement which can be classified as either unambigiously true or false. A compound statement applies logical connectives to one or more propositions. Propositions have logical equivalences (also found in the laws the algebra of sets, or the axioms and properties of Boolean algebra). e.g. [5],

Propositions have inference rules. An inference is the method by which one can derive new propositions from previous ones. [6] e.g.,

A hypothetical syllogism, for example, informs us that a conditional is transitive.

Connectives are evaluated with an order of precedence; ! first, | second, & third, -> fourth, fifth.

Propositions and connectives lead to resolution tables. [6]

1.8 In the previous presentation pragmatic means of verifying statements of truth (external verification), justice (mutual agreement) and beauty (subjective assertion) was proposed. Propositional logic can be applied to these statements to statements of truth, justice and beauty when the appropriate means of verification is used. Thus it is quite possible to use logic to evaluate propositions of scientific facts (true, false), social facts (accurate, inaccurate), legal norms (legal, illegal), moral norms (right, wrong), sensual expressions (pleasurable, painful), and aesthetic expressions (beautiful, ugly). However, not that statements with different means of verification are incommensurable.

1.9 A contradiction consists of a logical incompatibility between two or more propositions. This differs from a logical paradox is an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition. Contradictions are often used as a "negative proof" (see reductio ad absurdum). Paradoxes occur when statements cannot be consistently interpreted; often arising from self-reference, infinity, and circular definitions.

1.10 Some famous logical paradoxes and "false paradoxes" include:

The Liar Paradox: This, and variations. derives the philosopher-poet Epimenides of the 6th BCE, himself a Cretan, who reportedly wrote: "The Cretans are always liars." This isn't actually a paradox, but rather erroneously arises from a false dichotomy: that either the speaker always lies, or always tells the truth - when it is possible that the speaker occasionally does both.
Infinite Series: Again, not really a paradox, but rather an illustration of the nature of infinite series which seems to contradict the law of identity. 1/3 = 0.333... 3 * 1/3 = 1. 3 * 0.333... = 0.9999... Therefore 1 = 0.999... .
Petronius' paradox: "Moderation in all things, including moderation."
Grelling-Nelson paradox: Is the word "heterological" (meaning "not applicable to itself), a heterological word?
Russell's Paradox: A more fully developed version of self-referential paradoxes expressed in the terms of set theory. First, conceive of a set that included other sets (e.g., the set of all sets, the universal set). Then consider the set of all sets that do not include themselves. Does this set include itself? It does if it does not, and it does not if it does.

1.11 Issues of debate in logic include:

Bivalence and the law of the excluded middle: Classical logic assumes that statements are either true or not true. In the early 20th century a number of mathematicians rejected such bivalence, and thus also rejected the law of the excluded middle. Ternary logic, for example, includes true, false and possible. Modal logic is not truth conditional either. "Fuzzy logic" incorporates an infinite number of "degrees of truth", represented by a real number between 0 and 1. Bayesian probability, proposes concept of probability can be defined as the degree to which a person believes a proposition.

Paraconsistent Logic: Paraconsistent logics attempts to deal with contradictions in a a controlled and discriminating way, minimising the "principle of explosion" or "ex contradictione sequitur quodlibet" (from a contradiction, anything follows). Paraconsistent logic has significant overlap with many-valued logic. The most prominent contemporary defender of dialetheism is Graham Priest, a philosopher at the University of Melbourne.

Gödel's completeness and incompleteness theorems (1929, 1931): The former established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. The latter however showed that no sufficiently strong formal system can prove its own consistency. They are widely regarded as showing that a complete and consistent set of axioms for all of mathematics is impossible.

Endnotes

[1] The function of logic to establish validity of argumentation is evident in Artistotle's Organon and following that, the Peripatetic school and the Muslim Muta'zilites and all the way up to the medieval Christian Scholastics. In other traditional societies, logic was often used for theological, legal and ethical applications. These include the Hindu schools of Nyaya and Vaisheshika (which also contributed to Buddhism), and the Mohist school in China. Immanual Kant introduced the idea of logic as a means of judgement.

[2] "Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned." (Avicenna, Persian Philosopher and Physician, 980-1037 CE)

[3] Syllogistic logic could not logically develop sentences which used multiple quantifiers (e.g., "Some cat is feared by every mouse"). This was solved by Gottlab Frege in the 1879 book Begriffsschrift by using multiple bindings.

[4] Boolean logic is named after George Boole who developed an algebraic system of logic in the mid 19th century. In 1938, Claude Shannon showed how electric circuits with relays could model for Boolean logic, leading to the development of electronic computers.

[5] For the purpose of this paper I shall be using the following basic connectives: "not" (!), "and" (&), "or" (|), "conditional" (->), and "biconditional" (if and only if) (). "Not" is a unary operator, it takes a single term (!P). The rest are binary operators, taking two terms to make a compound statement (P | Q, P & Q, P -> Q, P Q).

[6]. Note that the Y/N stands for "yes/no" in the above examples, except in the section on logical equivalences, where the algebra sets of universal and null make more intuitive sense. In mathematics this is often expressed as T/F (true, false) or 1/0 (on/off, or high/low in electronics).

[7] These have some delightfully literal names in logic. For example, in P -> Q, if P is false (and Q is true) the compound statement is vacuously true; such as the claim "If I win the lottery, then I'll buy all my friends a present". Others include the "constructive dilemma" and the "destructive dilemma".