gentzen

7/6/12

Indicative sentences in a natural language, English, for instance, are either true or false. For example, 'There are 35 State Governors in the U.S.A.' is an indicative sentence (which happens to be false). Such sentences express statements or propositions. Not all pieces of language express propositions. For example, the question 'What day is it today?' is not either true or false (although reasonable answers to it will be either true or false); again, the greeting 'Have a nice day!' is not either true or false.

7/5/12

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.

11/27/11

So that we can show certain arguments to be valid.

The focus of the course lies with the validity and invalidity of arguments. Now, invalidity can be established by counter-example (by producing an interpretation under which all the premises are true and the conclusion false, at the same time). But validity is a different matter. And the usual approach is to have rules of inference and to do derivations.

8/29/06

Tutorial 2 Example: How Experts Symbolize

[This is a film-- press the 'play' symbol.]

12/19/09

Extending the notation

At first site, having an equivalence between types (lower case letters a..v) and monadic predicates (upper case A..V) might seem limiting. But it can be extending to a degree by introducing definitions for more complex single variable predicates. For example, say you have

(Mx∧Dx)⊃Nx  (*notice the free variable x*)

as a fancier category (those things which if they are M and D are also N), then you could introduce a monadic predicate definition for this, via an equivalence eg

11/4/2020

Under construction

You'd want to check out the respective formation rules, to see what is a well formed formula, and what a variable, etc.

Then the tree rules are:-

Formation Rules for System IV

10/8/09 Under construction

Terms, constants, variables, propositions, and predicates

Terms

<constant> ::= (['0'-'9']) |'∅'| 'U'|'{ }'
<subscript> ::=  ['₁'-'₉']
<functor> ::=  ['a'-'v'](< subscript >)*
<variable> ::=  ['w'-'z'](< subscript >)*
<term> ::=

12/13/09

Sorted Logic

(Many) sorted logic, with sorts or sort labels, is very similar in concept and execution to the types and type labels just discussed (in fact, many texts use the two terms interchangeably). There is, though, a conceptual difference. It is that whereas both ordinary logic and logic with type labels use one (homogenous) domain or universe, sorted logic uses a (usually stratified, heterogenous) domain which consists of two or more sub-domains.

An example will help.

2013

Type labels are one way to approach what seem to be different domains in use in the same reasoning. What we are going to do here is to tag the variables with a type to indicate that they are subject to a particular monadic (one-place) predicate applying to what they pick out. An example will help, say 'All men die' is symbolized

(∀x)(Mx⊃Dx)

5/2/09 10 Software