the organon and the logic perspective of computation

THE ORGANON AND THE LOGIC PERSPECTIVE OF COMPUTATION Panagiotis Katsaros - Nick Bassiliades - Ioannis Vlahavas Aristotle University of Thessaloniki - Greece

Computation I Calculation (from the Greek "κάχληξ" - gravel) is a deliberate process that transforms one or more inputs into one or more results, with variable change. An effective procedure to solve a problem (logic, mathematics, theory of computation) was not a well-defined (i.e. mathematically) notion until 1936: 1. Finite number of exact instructions. 2. When it is applied to a problem of its class: A. It always terminates after a finite number of steps. B. It always produces a correct answer. 3. It requires no ingenuity to succeed.

Computation II General mathematical models of computation: Turing Machine (1936): a mathematical model of a device that manipulates symbols on a strip of tape based on a table of rules.

λ-calculus (1936): a formal system for expressing computation based on function abstraction and application using variable binding and substitution.

Logic system

Purpose

Language

Proof/decision

Syllogistic

validity of arguments

categorical statements

deduction

Truth-functional logic truth of sentences

natural language

deduction

Boolean algebra

truth of propositions

algebraic expressions

calculation

Predicate logic

predicate formulas (truth) symbolic language

deduction + decision

λ-calculus

function calculation

symbolic language

computation

Turing machine

function calculation

symbolic language

computation

Syllogism P1: P2:

No birds are mammals. Some birds are pets.

Syllogism P1: No birds are mammals. P2: Some birds are pets. Concl: Some pets are not mammals.

Syllogism P1: No birds are mammals. P2: Some birds are pets. Concl: Some pets are not mammals. The syllogism is defined as "a logos in which, certain things having been supposed, something different from the things supposed results of necessity because of their being so".

Syllogism P1: No birds are mammals. P2: Some birds are pets. Concl: Some pets are not mammals.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

Middle term.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

Major term.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

Minor term.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y. Categorical propositions: A All X are Y. (Universal affirmative) E No X are Y. (Universal negative) I Some X are Y. (Particular affirmative) O Some X are not Y. (Particular negative) The arrangement of the middle term determines the figure.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

E I O figure 3

Categorical propositions: A All X are Y. (Universal affirmative) E No X are Y. (Universal negative) I Some X are Y. (Particular affirmative) O Some X are not Y. (Particular negative) The arrangement of the middle term determines the figure.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

E I O figure 3

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

E I O figure 3

• Aristotle was the first who introduced variables in reasoning!

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

E I O figure 3

• Aristotle was the first who introduced variables in reasoning!

• Valid syllogisms are those in which no false conclusion can be inferred from true premises.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

E I O figure 3

• Aristotle was the first who introduced variables in reasoning!

• Valid syllogisms are those in which no false conclusion can be inferred from true premises.

• The placement of the terms and not

their contents determine the validity of the syllogisms.

Logical form P1: No X are Y. P2: Some X are Z. Concl: Some Z are not Y.

E I O figure 3

• Aristotle was the first who introduced variables in reasoning!

• Valid syllogisms are those in which no false conclusion can be inferred from true premises.

• The placement of the terms and not

their contents determine the validity of the syllogisms.

• 256 logical forms - 24 unconditionally valid

Syllogistic proofs I Aristotle defines the proof (in Posterior Analytics) as: ". . . a syllogism which creates scientific knowledge, that is such a syllogism owing to which, if we only have the proof, we possess that knowledge. Hence, if knowledge is such as we have stated, then the premises of demonstrative knowledge must be true, primitive, direct, better known (than the conclusion) and must be its cause".

Syllogistic proofs II • Perfect (complete) deductions (six •

forms): no external term is needed to show the necessary result. Imperfect (incomplete) deductions: additional terms are needed, because of the terms supposed, but not assumed through the premises.

Syllogistic proofs II • Perfect (complete) deductions (six •

forms): no external term is needed to show the necessary result. Imperfect (incomplete) deductions: additional terms are needed, because of the terms supposed, but not assumed through the premises.

• Imperfect are "reduced" to perfect forms:

Syllogistic proofs II • Perfect (complete) deductions (six •

forms): no external term is needed to show the necessary result. Imperfect (incomplete) deductions: additional terms are needed, because of the terms supposed, but not assumed through the premises.

• Imperfect are "reduced" to perfect forms: • direct deductions that include perfect first-figure deductions & conversions (subject to predicate interchanges)

Syllogistic proofs II • Perfect (complete) deductions (six •

forms): no external term is needed to show the necessary result. Imperfect (incomplete) deductions: additional terms are needed, because of the terms supposed, but not assumed through the premises.

• Imperfect are "reduced" to perfect forms: • direct deductions that include perfect •

first-figure deductions & conversions (subject to predicate interchanges) proof through the impossible, where the conclusion is inferred from a pair of premises by assuming the denial of the conclusion

Syllogistic proofs III • Premise - conclusion combinations that cannot be valid syllogisms are proved by counterexample (a method that is still used).

Syllogistic proofs III • Premise - conclusion combinations that cannot be valid syllogisms are proved by counterexample (a method that is still used). Direct deduction of AEE figure 2 Proof: Step 1 - All C are B. (proposition A) Step 2 - No A are B. (proposition E)

Syllogistic proofs III • Premise - conclusion combinations that cannot be valid syllogisms are proved by counterexample (a method that is still used). Direct deduction of AEE figure 2 Proof: Step 1 - All C are B. (proposition A) Step 2 - No A are B. (proposition E) Step 3 - No B are A. (E from step 2 by conversion of E)

Syllogistic proofs III • Premise - conclusion combinations that cannot be valid syllogisms are proved by counterexample (a method that is still used). Direct deduction of AEE figure 2 Proof: Step 1 - All C are B. (proposition A) Step 2 - No A are B. (proposition E) Step 3 - No B are A. (E from step 2 by conversion of E) Step 4 - No C are A. (from steps 3, 1 by perfect fig 1 EAE)

Syllogistic proofs III • Premise - conclusion combinations that cannot be valid syllogisms are proved by counterexample (a method that is still used). Direct deduction of AEE figure 2 Proof: Step 1 - All C are B. (proposition A) Step 2 - No A are B. (proposition E) Step 3 - No B are A. (E from step 2 by conversion of E) Step 4 - No C are A. (from steps 3, 1 by perfect fig 1 EAE) Step 5 - No A are C. (concl. from step 4 by conversion of E)

Syllogistic proof system Axioms (true propositions): law of identity - "All A are A." "Some A are A." law of excluded middle - for every proposition, either its positive or negative form is true. law of non-contradiction - contradictory propositions are not true simultaneously. The square of opposition tells us when a contradiction arises by two propositions that are asserted true together. Inference rules: three rules of conversion (simple implications) obversion: the predicate is negated and replaced by the opposite quality contraposition: both terms are negated and their order is reversed

Deductive systems in (mathematical) logic I They are used as syntactic devices for establishing semantic properties. We start with an (artificial) language and a semantics assigning meaning to expressions in the language (logic formulas). Deductive systems are still defined in the same way as Aristotle's syllogistic proof system: axioms, i.e. true formulas rules of inference, i.e. patterns with constant and variable parts, which yield true formulas, when given true formulas.

Deductive systems in (mathematical) logic II

A deduction is viewed as a tree labelled with formulas, where the axions are leaves, the inference rules are interior nodes and the label of the root is the formula whose semantic property is established. Important questions for a deductive system: Soundness - Are the axions true and is the truth preserved by the inference rules? Completeness - Can every true formula be deduced?

From the standpoint of mathematical logic I Syllogistic has been criticized (e.g. by B. Russell). Most criticisms are due to an extensional interpretation of the categorical terms, which is dominant in mathematical logic. Extensional meaning consists of the members of the class denoted. Intensional meaning refers to the qualities or the attributes connoted by the terms. Syllogistic is a theory of logos (speech) and the terms are words with meaning of both intensional and extensional kinds.

From the standpoint of mathematical logic II Symbolic formalizations of the syllogistic during the second half of the last century opened new dimensions of research (e.g. soundness and completeness). Lukasiewicz states: "The syllogistic of Aristotle is a system the exactness of which surpasses even the exactness of a mathematical theory, and this is its everlasting merit. But it is a narrow system and cannot be applied to all kinds of reasoning, for instance to mathematical arguments". Syllogistic is a formal system, which should be modeled independently of predicate logic and set theory and should not be considered as an incomplete forerunner of them.

Logic system

Purpose

Language

Proof/decision

Syllogistic

validity of arguments

categorical statements

deduction

Truth-functional logic truth of sentences

natural language

deduction

Boolean algebra

truth of propositions

algebraic expressions

calculation

Predicate logic

predicate formulas (truth) symbolic language

deduction + decision

λ-calculus

function calculation

symbolic language

computation

Turing machine

function calculation

symbolic language

computation

Truth-functional logic of the Stoics It is a logic of propositions (the so-called assertibles), not a logic of terms. Every simple assertible is classified in one out of three affirmative and three negative types and its type is determined by the form of the sentence through which it is expressed. Non-simple assertible can be composed by constituent assertible, which may be themselves nonsimple and the used connectives (conjunctive and disjunctive), which are conceived as two-or-moreplace functors. The Stoics were the first who introduced logic reasoning over the natural language by using rules for reasoning with the logical connectives “not”, “and”, “or” and “if . . . then”.

Boolean algebra I • George Boole introduced an abstract algebra

of propositions, where the logical connectives (and, or, not) behave like algebraic operators.

• He considered logic reasoning by dealing with subsets of some given set:

X ∪ Y for the proposition p or q (p ∨ q) where p is true for elements in X and q is true for elements in Y X ∩ Y for proposition p and q (p ⋀ q) X’ (complement of X) for proposition not-p (¬p)

• The conditional “if p then q” is written as p −> q that is equivalent with ¬p ∨ q

• If p and q represent propositional variables,

they can be either true or false and they are assigned one of the values 0 (false) or 1 (true).

Boolean algebra II • To compute the truth-value of a compound

proposition, when given the truth values for the variables in the formulas, we only need to know how the connectives determine truth-values:

• Such a formalized representation is the key to

mechanize the computation of truth values for complex propositional formulas.

• Categorical propositions in Boolean algebra:

All X are Y. X∙Y=X Some X are Y. X∙Y≠0 where ∙ denotes set intersection and 0 denotes the empty set.

Boolean algebra III • Boole is considered the father of mathematical logic.

• Boole explicitly accepted Aristotle’s logic as “a collection of scientific truths” and he regarded himself as following in Aristotle’s footsteps.

• He thought that he was supplying a unifying

foundation for Aristotle’s logic and that he was at the same time expanding the ranges of propositions and of deductions that were formally treatable in logic.

• “. . . using mathematical methods . . . has led to

more knowledge about logic in one century than had been obtained from the death of Aristotle up to . . . when Boole’s masterpiece was published” by Rosenbloom.

Predicate logic I • Predicate logic has a precisely defined • • •

•

vocabulary and syntax. We refer to objects using constants, for example a, b, c, . . . and variables, such as x, y, z, . . ., when the object is indefinite. It is also possible to use function symbols for complex objects. We can use unary, binary or ternary predicates (denoted with capital letters) depending on the number of objects they relate. The language also incorporates the propositional connectives ⋏, ⋎, ¬ and has a very expressive way to specify quantification.

Predicate logic II “Every student is younger than some instructor.” We define the following predicates: S x: x is a student I x: x is an instructor Y x y: x is younger than y ∀x (Sx −> (∃y (Iy ⋀ Yxy)))

Predicate logic II “Every student is younger than some instructor.” We define the following predicates: S x: x is a student I x: x is an instructor Y x y: x is younger than y ∀x (Sx −> (∃y (Iy ⋀ Yxy)))

Categorical propositions: A All A are B. ∀x (Ax −> Bx) E No A are B. ¬∃x (Ax ⋀ Bx) I Some A are B. ∃x (Ax ⋀ Bx) O Some X are not Y. ¬∀x (Ax −> Bx)

Proof systems for predicate logic I David Hilbert and Wilhelm Ackermann introduced a formal system with axioms and inference rules for proving predicate logic well-formed formulas. For the completeness of Hilbert’s proof system it was essential to prove that any well-formed formula, which is true, can be proved. In this case, formal proofs of well-formed formulas would be a mechanical manipulation of symbols. Kurt Gödel showed in his doctoral thesis that Hilbert’s system is complete. However, Hilbert envisioned a formal system with appropriate axioms, in which ALL the theorems of mathematics would be possible to be proved. This system would also have to provide axioms for the mathematical theory of interest.

Proof systems for predicate logic II Kurt Gödel proved with his incompleteness theorems that no such system that will be both consistent and complete exists.

Hilbert then posed the “decision problem”: is there an effective decision procedure, which could decide for any well-formed logic formula with natural numbers, if it is provable in his formal system?

The birth of computation • In 1936, Alonzo Church and Allan Turing by working independently introduced their mathematical models which can manipulate all wellformed formulas and include all effective procedures that would be compliant with Hilbert’s decision problem. • With their models of computation they eventually proved that no such decision procedure exists. • A new science was born, the Theory of Computation.

CONCLUSION Today, we are interested to develop formal systems that are not only sound and complete, but they are also computable. This is the field of computational logic with important results for: automated theorem proving logic programming and many other applications There are also new computational formulations of Aristotle’s logic. The theory of syllogisms continues to drive the developments, in a way that Aristotle certainly would not have been imagined.