Ungenerated Being

There are various types of logical connectives which are used in complex wffs. The ones we shall focus on consist of the conjunction, disjunction, and negation. In order to define these operators, we will have to introduce interpretation and valuation. The following topics will be categorized under the topic of logical semantics. Semantics are rules for assigning truth values to wffs. An Interpretation is a function that takes propositional letters as an input and assigns a single truth value. An interpretation: To interpret proposition P, put it in the interpretation function so that I(P)=T . This is read as “P is true under interpretation I. A proposition can be interpreted in two possible ways, but one interpretation cannot give both truth values. IE: $ I_1 (P)=T, I_2 (P)=F $ Every propositional variable can be put in the interpretation function For n distinct propositional letters, there are $ 2^n $ interpretations. The last point can be expressed in the context of truth tables....

 The following are my posts on propositional logic: Propositional Logic - Preface Propositional Logic - Syntax Propositional Logic - Semantics Propositional Logic - Conjunction Propositional Logic - Disjunction Propositional Logic - Negation Propositional Logic - Conditionals Propositional Logic - Biconditional Propositional Logic - Deductive Hypotheses Propositional Logic - Rules of Inference Propositional Logic - Transformations of the Conditional Propositional Logic - Other Topics The following are my posts on Predicate Logic: Predicate Logic - Introduction Predicate Logic - Inference Rules The following are my posts on Alethic Logic (Modal Logic): Modal/Alethic Logic - Introduction The following are my posts on Discrete Math: Discrete Math - Naïve Set Theory Discrete Math - Other Set Theory Basics Discrete Math - Relations Discrete Math - Functions

As we showed above, propositional logic is consistent of symbols which take the place of atomic propositions and logical operators. There are two other building blocks in propositional logic which include its syntax and semantics. Syntax refers to how to set up grammatically meaningful propositions in a symbolic logic system. The following are examples of syntax: ‘The dog is brown.’ Is a sentence. ‘The dog is brown, and the dog is microscopic.’ Is a sentence with two propositions connected by a conjunction. ‘The dog and’ is not a full sentence. In all of these cases we are looking at the grammar of the sentences, not the truth content nor interpretation. It is unlikely that the sentence in point (2) is true, but it can be said to be true or false as a proposition. It is impossible for sentence in point (3) to be converted into a proposition. One aspect to using proper syntax in any symbolic logic language is to be sure that the propositions expressed are well formed formulas. ...

Propositions are statements, statements are propositions, these things are one in the same. Statements are claims which can either be true or false. I am a dog; this is a proposition because it could be true or false. I could be a dog, or I could not be a dog. It is highly unlikely I am a dog, but there is still a possibility that it is true. However, there are statements which cannot be true or false. Take this sentence fragment: An orange plus three. Is this an example of a proposition? No, an orange plus three cannot be valued as true or false.   Propositions are not only limited to sentences used in natural language or un-paraphrased English (or any language really). Propositions can also be mathematical. The following are mathematical statements: 1. $ 7>8 $ 2. $ 1+1=2 $ 3. $ 89*5+9(4-3) > 34 $  All of these statements are either true or false. To be precise, propositions (2) and (3) are true and proposition (1) is false. Let us see some mathematical expres...