We have been considering the logic of categorical propositions. Now we turn to a more flexible and powerful logic, the logic of propositions. Any proposition, whether it has the subject-predicate form or not, can be assigned a truth-value and placed in a logical relationship with other propositions. Of course, we pay a price in allowing any proposition into this system. We will not have a way to analyze the proposition itself, to examine its internal structure. We must take it as a whole and assign it a truth-value, that is, assert whether the complete proposition is true or false.
We can, however, break compound sentences into their component statements. Propositions, it is important to note, are not the same as sentences. We can analyse a sentence into its component propositions. Thus "Tom and Mary went to the movies" is one sentence but two simple, or atomic, propositions:
Tom went to the movies.
Mary went to the movies.
Thus atomic propositions can be joined together, or compounded. In propositional logic we will be able to determine the truth-value of such compound statements--provided we know the truth-value of the components. Thus, if Tom went to the movies and Mary went to the movies, the compound proposition, expressed in the sentence, "Tom and Mary went to the movies", is true.
This logic of propositions relies on three key notions--truth-value, logical operators and variables, each of which can be symbolized. This symbolization makes this logic look like math, but it also makes it a very useful way of representing arguments.
Every statement is considered to be either true or false. In ordinary language of course, we allow for gradations of truth, even for indeterminancy. Something may be neither true or false. But in propositional logic we do not have room for "maybes" and "perhapses" or "I don't knows". We will insist that every statement be assigned a truth-value; it will be designated true or false. If its truth-value is undetermined, then we will not permit it the status of an atomic proposition. Thus propositional logic is a two-valued and not a multi-valued logic. Multi-valued logics are more like ordinary language, but they are also more complex (and difficult!). Even though propositional logic is limited to two values, one should not underestimate its potency. It is, after all, the logic of computer science. With just two values (plus the logical operators we will discuss next and super-fast computational ability), one can do all the wonderful things that computers can do.
Propositional logic makes extensive use of logical operators, that is, symbols which indicate the precise, syntactical relationships of propositions to one another. It is possible to get by with less, but standardly five operators are used. They are the relationships of conjunction, disjunction, conditionality, and biconditionality and the operation of negation. I will say a bit about each and then give you a table displaying all five.
Conjunction. One way to link propositions together is by asserting both of them. Thus, we sometimes say, "It is raining, and the sun is shining" or even "It is raining, but the sun is shining". These compound sentences bring together two propositions which we wish to assert simultaneously. We will use the ampersand, "&", to symbolize this logical relationship.
Disjunction. A conjuncted proposition is true, if and only if both atomic statements are true. But sometimes a compound proposition will be true, even if one of the atomic statements is false or possibly false. Thus, if I were to say, "Tom is either at the movies or in his room", then the sentence would be true if he is one or the other place. We use a Ú to indicate this disjunctive relationship and mean by it "either . . . or . . . or possibly both". Of course, Tom could not both be at the movies and in his room, so we would not ordinarily use "either-or" to refer to a situation where we thought both atomic propositions were true. But, one could say, "Jane will be in the library or in her room this evening, or possibly both", indicating not that she would be both places at the same time, but that she might be both places within a certain time period. Or, that she would be in one or the other place, but not necessarily both.
Conditional. This is the oddest of the logical operators, because it can be used to compound statements that seemingly have no meaningful relationship to one another. Just why we can do this will be explained in a later lesson. Normally, however, the antecedent (=that which goes before) proposition and the consequent (=that which follows) proposition will be meaningfully, that is, semantically, related. But, in propositional logic, all that is required is a syntactical relationship. The conditional is symbolized by a horseshoe (É) or an arrow (®) and is translated as "if . . . , then". (Online and in the classroom I will use the horseshoe; Shaw, Logic and Its Limits, uses the arrow.) An example of a conditional statement is "If the sun is shining, then Terry is at the beach".
Biconditional. Sometimes we will want to assert two conditional statements simultaneously, hence the need for the biconditional, symbolized by three parallel bars (º) or a double arrow («). (Shaw uses the double arrow.) We could dispense with it entirely, relying on two conditional sentences conjoined together. Thus, we could say, "If the sun is shining, then Terry is at the beach, and if Terry is at the beach, then the sun is shining". Instead, we will rely on the biconditional to assert, "Terry is at the beach, if, and only if, the sun is shining". Note that this sentence means the same as the former, longer one. (In a later lesson, we shall show that this is the case.)
Negation reverses the truth-value of an asserted proposition. In a two-valued logic, as noted above, every proposition is either true or false. If I say, "The sun is shining", I can symbolize it as "S". S, the symbolized proposition, will then be either true or false. If the sun is shining, S is true. If the sun is not shining (~S), then S is false. Negation, symbolized by the tilde (or a short, straight line) preceding a variable, indicates that the proposition's truth-value is reversed. Note that negation is not equivalent to falsification. For, if the sun is not shining, then ~S is true.
These logical relationships/functions can now be displayed in a table:
Operator Name Function Sense Translation & ampersand conjunction and p & q: "p and q" Ú wedge disjunction or p Ú q: "p or q [or possibly both]" É horseshoe conditional if . . . then p É q: "if p then q" º triple bar biconditional if and only if p º q: "p if and only if q" ~ tilde negation not ~p: "not-p" or "it is not the case that p is true" For completeness sake, I should mention that we will use parentheses and brackets to group statements. For instance, (P É Q) Ú R and (P & Q) É [R Ú (S & T)]. Note that the first level of grouping uses parentheses, and the second, more inclusive, level uses brackets. Parentheses and brackets, then, are forms of punctuation in symbolic, propositional logic.
Note also that there is a tendency to use capital letters to symbolize atomic statements.
With the basic notions of truth-value, logical operators and variables in place we can now develop more fully the syntax of propositional logic. In the next lesson we will continue our exposition of this powerful symbolic logic, defining the logical operators by means of truth tables. Then we will have a completely syntactical, symbolic calculus.
Copyright © 1999, Michael Eldridge