_logicomix

Boolean logic

Logical propositions expressed in purely symbolic language which provides the means to manipulate them by operations similar to the arithmetic operations. For example, logical AND corresponds faithfully to multiplication, a ∧ b ≅ ab; logical OR corresponds to saturated addition, a ∨ b ≅ a+b (but 1+1 is still 1 in upper-bounded saturated addition).

The central was propositional calculus. What this logical formalism is lacking is the ability (predicate calculus had) to express semantic connections between propositions.

Frege departed from the earlier logicians and Aristotle by explicitly introducing the notion of a variable in logic statements. He introduced statements like "x is a man", propositions that can either be true or false according to the value given to x. Frege also invented the notion of quantifiers. "Grundgesetze der Arithmetic" (The basic laws of arithmetic) is the first work out of school of logicism (math as branch of logic); vol1 pub in 1893, vol2 in 1903 with addendum about Russell's paradox. His symbolic system was too cumberstone, so improved/replaced, but his ideas formed the backbone ofmodern logic.

Godel

his Completness theorem proves all valid stat in Frege's 1st order logic can be proven from a set of simple axioms. in 1931, he proves the incomp theorem for 2ord logic i.e. for a logic powerfull enough to support arithmetic and equally more complex math theories.

Hilbert

"hilbert's problem": 23 problems of math. 11 solved, 7 partly, 8 still unsolved (riemann hyp inluded). The problem no.2 was demanding proof of consistency of arith

Second-order logic

a system that can also accept sets as values of vars. 1st (of 2) Incompleteness theorems says that in a logical axiomatic system rich enough to desc prop of the whole numbers and ordinary arith ops, there will always be propos that are gram correct by the rules of the sys, and moreover true, but cannot be proven within that system. 2nd Incompleteness theorems says that if such a sys were to prove its own consistency it'd be inconsistent.

Predicate calculus

Frege's extension of prop logic dev by Boole. in PC elementary proposiotions (predicates) are composite obj of form P(a,b,c..), where P is a symbol in the lang, and a,b,c are constants or vars. e.g. "older" is propos symbol, "plato" is const and x is a var, then "older(Plato, x)" is a wff. Propositions of this type can be combinaed with Boole's connectives (and, or, not, imply) and prefixed with Frege's quantifiers. So by employing symbols from math, one can create predicates expressing math statements in this formal, log ically exact lang. The arith theorem stating that every int is odd or even: forall x exist y (x=y+y or x=y+y+1). Rigorously defined, the version of the predicate calculus called 1o logic employs simple math objects as vars, whereas in 2ol vars can also be sets, making possible st such as "there is a set S". This lang can exp all math. Whether the sentence in fol or sol is true or false depends on the model whereby the sent is interpreted. However, some sentences, called valid, are true indep of interp cuz they embody basic props of Boolean connectives and Frege's quantifiers.

Russell "history of western phylosophy" 1945