Propositional Logic - Semantics

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:

  1. 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.
  2. 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 $
  3. Every propositional variable can be put in the interpretation function
  4. For n distinct propositional letters, there are $ 2^n $ interpretations.

The last point can be expressed in the context of truth tables.

Truth tables present every possible combination of interpretations and the truth values of compound propositions derived from the simpler ones. Each row or interpretation is called a possible world.

Below we see that there are two possible worlds in table 2 due to the fact that $ 2^1 = 2 $ and table 3 has 8 possible worlds since $ 2^3 = 8 $. Deriving any other wffs from these atomic wffs will not add possible worlds.

I

P

$ I_1 $

T

$ I_2 $

F

Table 2

I

P

Q

R

$ I_1 $

 T T T

$ I_2 $

 T T F

$ I_3 $

 T F T

$ I_4 $

 T F F

$ I_5 $

 F T T

$ I_6 $

 F T F

$ I_7 $

 F F T

$ I_8 $

 F F F

Table 3

Given certain interpretations of propositional letters, we can solve for the truth values of complex wffs.

A valuation is defined as followed, for any interpretation I, a valuation v of a wff is a function that assigns one truth value to each wff.

In the following scenarios we will use P and Q will be variables for a wff and R will be a variable for any propositional letter. Thus, we can state that:

$ v(R)=I(R) $

Figure 1

Figure 1 shows that whatever interpretation a proposition is given, will also be equivalent to the valuation of the same proposition.

Comments

Post a Comment

Popular posts from this blog

National Accounting - Consumption and Investment

Image
Over the course of a year, a quarter, or a month, we can look at the debits and credits that build up the entirety of an economy. National accounts looks at these debits and credits and record the transactions  While it can be the case we make more than we spend as individuals, this cannot be the case for entire economies. This is the old adage that every sale is something bought. In the context of macroeconomics we say that income over a period of time should be equal to expenditures over that period of time: Expenditures = Income For the case of this post, we will assume a hypothetical economy which only consists of domestic investment and consumption as transactions. We can define the two as follows: Consumption : Transactions that are associated with the destruction of asset values. Investment : Transactions that are associated with increasing assets values. To understand these definitions, we will look at two sets of debit and credit charts to show how these transactions are ...
Read more

Conceptual Analysis

Conceptual analysis is kind of like giving definitions: Conceptual Analysis : A description of constituent properties of a complex property or relation. Some complex property is usually built up of simpler properties. An example of a conceptual analysis is: X is a triangle = df. X is a shape with exactly three angles and straight lines in Euclidean space. This is more than a definition is picking out some properties of a property. Also, = df. means by definition. Another example: X is a bachelor = df. X is an unmarried man. Here a conceptual analysis is: Non-circular, the analysis refers to properties that aren’t the same thing as itself. It tries to include everything it intends to include and exclude what it intends to not exclude. It tries to pick out the properties that do this. (It’s not possible to have one condition without the other). Conceptual analysis helps you: Understand reality : reality contains things with properties, and conceptual analysis picks out properties...
Read more