In this thread I will be sharing my notes and general thoughts on Kleenes introductory text on Mathematical Logic. I am currently studying Kleene and other logicians and analytical philosophers to gain a further perspective in language construction, type theory, recursive theory, and computability in general.
My reasoning for starting with Kleenes text on Mathematical Logic which is treated classically is because I desire to read his text on Metamathematics afterwards which is a very difficult text to read. I have never read Kleene so this introductory text will allow me to familiarize myself with his views and conventions.
Chapter 1 – Propositional Calculus
Section 1: Linguistic Considerations
Mathematical logic in the context of our study has a double meaning given our study of logic will be using logic, as such it is necessary to separate the logic we our investigating from the logic we use in our investigation in order to avoid paradoxical and circular logic. To achieve this, it is appropriate and valid to split up our logics into separate languages with their own unique identities – one language serving as the object of study and the other as the means to study that object. From this necessity arises the notion of convention from construction, both of which lay the reality of logic and all of mathematics at the feet of man. It was the role of the logician, a philosophical school of thought Kleene was acquainted with, to make their primary concern the language we use to convey the epistemic condition of our statements. This is especially the case for a logical intuitionist such as Kleene who views convention as the way to meaning and Man as the way to convention.
Language as it is used day to day is imprecise and riddled with ambiguity. It is by this Kleene justifies the Propositional Calculus which in his opinion is a means of confronting ambiguity. A logician such as Kleene makes it his primary goal to remove ambiguity from language through the construction of symbolic systems, the propositional calculus laying at the foundation, with clear and concise conventions of meaning. Symbolism is then the bridge between ambiguity and mathematical precision.
Propositional calculus is the calculus of declarative sentences, that is sentences that can be said to be statements that are either true or false but not both. The reality of ‘true or false but not both’ is a matter of convention within the language we call Classical logic which will be the specific language Kleene concerns himself most with in this introductory treatise on mathematical logic. There are other languages, that is constructions with opposing conventions, in which a declarative sentence may be only true or false. Once such example would be Intuitionist Logic that Kleene played such a big role validating, in which the value of declarative sentences is not that of truth but rather justification.
The nature of a declarative sentence, that being its identity, is found in what it expresses and not how it is expressed. As such a proposition or declarative sentence can be said to be equivalent if they have the same meaning even though they may be expressed differently – hence meaning is identity. Kleene states the following example, “thus ‘John loves Jane’ and ‘Jane is loved by John’ express the same proposition, but ‘John loves Mary’ expresses a different proposition. Notice that in this primitive example, that which changes the meaning of the proposition is the subject of the predicate. In this sense, Kleenes example is a Aristotelian one which does not necessarily grow the same fruit as that of the logicians of Kleenes time. Kleene rectifies this by stating that what the value of the proposition hinders on is the outcome and by this he creates the possibility for variability of the subject while maintaining consistency without the Aristotelian subject-predicate constraint.
The power of the propositional calculus is its capacity to construct complexity out of primitive statements. Within the language Kleene constructs for instance, he justifies the creation of two separate categories for declarative statements which he now calls formula, one being the prime formula and the other being composite formula. The prime formula being defined as all primitive propositions that cannot be constructed by other propositions and the composite formula being defined as those propositions constructed by the combination prime and composite propositions. As such, the combination of two or more prime formula is the construction of a composite formula and the combination of two or more composite formula is the construction of a composite formula. The question that comes naturally is what is our means of combination?
Proposition connectives allow for the combination of primary or composite formula that then manifest other composite formula of greater complexity. We obtain these connectives once again through the process of reducing ambiguity via symbolic convention by doing away with language such as “if….then…”, “and”, “or”, etc. As can be seen all a propositional connective is are those words we use to connect statements. In the practice of logic we do away with the spelled constructs and replace them with symbolic gestures that convey concrete meaning. There are five major connectives: equivalence(‘equivalent to’, ‘if and only if’), implication(‘if….then…’), conjunction(‘and’), disjunction(‘or’), and negation(‘not). There symbolic representations are those listed respectively: ∽, ⊃ , ∧ , ∨ , ¬. By these symbolic propositional connectives and their use in the combination of symbolic formula the Logician believes ambiguity can be done away with and a axiomatization of mathematics and even reality may be achieved.