10/9/26
Welcome!
These web pages provide an introduction to logic to the level of Propositional and Predicate Calculus.
The focus of the program is on arguments and the question of whether they are valid. Arguments have the form <list of premises> ∴<conclusion>. An argument is valid if and only if it is not possible for all its premises to be true and its conclusion false at one and the same time; an argument which is not valid is invalid.
An argument can be proved to be invalid by displaying a scenario (truth table, a possible world, an interpretation ...) under which all its premises are true and its conclusion false. EasyDeriver has the tools to allow you to do this.
An argument can be proved to be valid by displaying a suitable derivation or proof of it, and EasyDeriver's also has the tools to do this.
These web pages, or Notes, should be accompanied by a suitable textbook, such as:
M.Bergmann, J.Moor, J.Nelson, The Logic Book
A.Hausman, H.Kahane, P.Tidman, Logic and Philosophy
W.Hodges, Logic
C.Howson, Logic with Trees
R.C.Jeffrey, Formal Logic: Its Scope and Limits
H.Leblanc and W.Wisdom, Deductive Logic
B.Mates, Elementary Logic
M.D.Resnick, Elementary Logic
Unfortunately these textbooks use slightly different choices of rules and symbols one from another. To adjust to this the Notes are in different major sections, with the sections tailored to particular texts. This section is particularly suitable for
M.Bergmann, J.Moor, J.Nelson, The Logic Book
You are invited to review
Notation
Not all logicians, and logical texts, use the same symbols for the so-called 'logical connectives'. Nor do they use the same sequences of symbols for 'well formed formulas'.
Here are typical possibilities for symbols
'not' : ∼ (the 'tilde'), ¬ (looks like the top right corner of a box)
'and': ∧, & (the ampersand), . (just a period)
'or': ∨ (usually just this, vel)
'implication': ⊃ , →
'equivalence': ≡, ↔
'existential quantifier': ∃, ∑
'universal quantifier':∀, ∏
So, in a logic book, you might see (A&B)→C and that is just the same as (A∧B)⊃C.
And you might see (∀x)(Fx ⊃ Gxy) and that might be just the same as ∀x(F(x)→G(x,y)).
The software running here can easily manage or render any of these. But we should explain what we favor, and help you find what you prefer.
The 'bergmann' system
Bergmann uses ∼, &, ∨, ⊃, ≡ as its sentential connectives. Sentence letters are upper-case A, B, C, etc. possibly with subscripts. Parentheses can be used (primarily for disambiguation). So, for example, ~(A&B) is a sentential formula.
Predicates are upper-case A, B, C, etc. possibly with subscripts. Constants are lower case [a-v], possibly with subscripts. Variables are lower case [w-z], possibly with subscripts. So, a simple unary predicate formula might be Aa; a binary one Bgf, a ternary one Hwxa etc.
Quantifiers are written in parentheses, as is the scope of a quantified (sub) expression. So the argument. Socrates is a man, All men are mortal, therefore, Socrates is mortal might be symbolized
Ms, (∀x)(Mx⊃M₂x) ∴ M₂s