Colin Howson, [1997] Logic with trees Chapter 12 Section 2
The Howson [1997] does not expand on modal logic (and modal trees) so a text like
Rod Girle [2000] Modal Logics and Philosophy
would definitely be a help here.
[Modal logic is a vast area, what is being presented here is the briefest of glimpses through the shop window (a book like the Girle would help you go further).]
2013
The Colin Howson book uses a notation like R(a,b,c) for the application of a predicate R to the arguments or terms a, b, c.
It employs the upper case letters A-Z, perhaps followed by subscripts, to be predicates, so, for example, R, S₁, T₁2 are all predicates.
The software supports this.
But the software makes an extension.
Often, when working informally, authors will write Red(x) to mean that the predicate Red is applied to the variable x.
[These might help, you need only scan them.]
K.H.Blasius et al. eds.[1990] Sorts and Types in Artificial Intelligence
Maria Manzano [1996] Extensions of First-Order Logic
John Sowa [2000] Knowledge Representation
2013
Colin Howson, [1997] Logic with trees Chapter 11
Set theory is an extensive topic introduced elsewhere. It can be written as a first order theory.
There is one axiom schema, Abstraction (or Comprehension), which can generate infinitely many axioms
∀y(yε{x:Φ[x]}↔Φ[y])
Axiom Schema of Abstraction (or Specification or Comprehension). The Set Builder Axiom Schema.
2013
Colin Howson, [1997] Logic with trees Chapter 9 & 11
It is common in this setting (which is arithmetic) to use functional terms like s(x), s(1), s(0) to mean the successor of x, 1, and 0, respectively. Equally common is the notation x', 1', and 0' to mean the same thing. The latter is quicker and shorter (though not semi-nmemonic)-- we will use it here.
2013
Colin Howson, [1997] Logic with trees Chapter 9
Groups can be characterized by three proper symbols {=,+,0} (ie identity, one infix operator, we will use '+', and an identify element '0') and the three proper axioms
∀x∀y∀z((x+y)+z=x+(y+z)), (*associativity*)
∀x(x+0=x∧0+x=x), (*identity element, right and left*)
∀x∃y(x+y=0∧y+x=0) (*inverse*)
Colin Howson, [1997] Logic with trees
The word 'terms' in logic means 'names' and thus far we have met two kinds of terms: constants (or proper names), and variables.
2013
Colin Howson, [1997] Logic with trees Chapter 2
Howson [1997] has a number of exercises. Many of them you will be able to do in the Widget below.
Here are a few hints
