Propositional (Sentential Symbolic) Logic
The Logic of Sentence Structure:
In Aristotelian Standard Form Categorical Logic we used the power of 'distribution' between categorical terms to make valid inferences in 15 Standard Form Categorical syllogisms. But there is yet another way in which humans reason validly using syllogisms. This other level of logic is sometimes called 'propositional' logic or 'sentential' logic since it systematizes the NECESSARY implications that take place between complete sentences, or propositional truth claims themselves.
For example, here is a sequence of ordinary language statements that are not in standard form categorical logic yet the logically necessary inferences are clear to even a mere school child.
"If you give me your lunch cookies, then I will give you my new yo-yo. You give me your lunch cookies. So, here's new my yo-yo."
Of course, with a great deal of effort, this sequence of sentences could be successfully translated into a standard form propositions and, hence, a familiar standard form categorical syllogism of the form:
All things that you give me are things that I will swap for my new yo-yo. Some of your lunch cookie things are things that you give me. Therefore, some of your lunch cookie things are things that you I will swap for my new yo-yo.
It would be quite odd, of course, to hear school children negotiating in this rather stilted categorical language. Moreover, this valid DARII translation fails to capture the CONDITIONAL content of the original ordinary language passage. The syntactic structure of "If...then..." sets up a condition that is SUFFICIENT for an inference to be drawn provided the condition is fulfilled. In the original ordinary language rendition, the first SENTENCE acts as the condition to be fulfilled. The second SENTENCE functions to fulfill that condition, thus generating the valid inference of the last SENTENCE. This gives sufficient insight into the very nature of SENTENTIAL LOGIC, or PROPOSITIONAL LOGIC, or SYMBOLIC LOGIC as case may be.
Understanding this, modern logicians created a system of symbols that adequately express these syntactic logical relations BETWEEN SENTENCES.
We now introduce five special symbols, or statement connectives, that facilitate understanding and testing this propositional logic. They are:
~ (the 'tilde' to symbolize 'not')
• (the 'dot' to symbolize 'and')
ν (the 'wedge' to symbolize 'or')
⊃ (the 'horseshoe to symbolize 'if...then...')
≡ (the 'triple bar' to symbolize 'if, and only if')
This syntax of using statement connectives (the 5 symbols) and statement variables (lower case letters of the alphabet) to form new, compound statements can be stated as a simple rule:
For any statements, p and q,
~ p "It is not the case that 'p' is the case."
p • q "Both 'p' is the case and 'q' is the case."
p • q "Either 'p" is the case or 'q' is the case."
p • q "If 'p' is the case, then 'q' is the case."
p • q "'p' is the case, if, and only if, 'q' is the case."
are all COMPOUND STATEMENTS.
Unlike Aristotelian Categorical Logic, modern symbolic logic (also called propositional logic or sentential logic) is not concerned with the 'distribution' of terms within truth claim sentences. Rather, symbolic logic is only concerned with the necessary logical relationships created by these statement connectives in compound statements. Compound statements are two or more propositional truth claims joined by logical connectives.
Example: Othello is black, and Othello is not black.
There are two complete truth claim propositions here joined by the statement (logical) connective 'and'. This compound statement asserts as factually the case that someone or thing, named Othello, is black AND that this same someone or thing is NOT black. Since each complete statement has its own truth value, i.e. is either factually true or factually false, then this compound statement is a logical contradiction. Additionally, and this is most important to grasp, the ENTIRE COMPOUND STATEMENT has a truth-function. Namely, it is either factually true or factually false that the CONJUNCTION of these two claims TAKEN TOGETHER is factually true or factually false. Assuming that the example refers only to a Shakespearian character, then the CONJUNCTION is a factually FALSE truth-function.
Five Rules of Valid Inference:
Rule of NEGATION:
Since the tilde " ~ " signifies logical negation in symbolic logic, then the factual truth of any simple or compound proposition entails the falsity of its negation as in the Othello example above. A simple truth-table to the right that exhausts all the logical possibilities, graphically demonstrates the rule of NEGATION. Thus it DEFINES the tilde " ~ " symbol.
Rule of CONJUNCTION:
|p||q||p • q|
Since the dot " • " symbolizes logical conjunction in any compound statement formed with this connective. The conjunct is true only in the event that both of the component statements is true. In the event that EITHER of the component statements is false, then the entire CONJUNCT IS FALSE. This is DEFINED by the simple TRUTH TABLE to the right that exhausts every permutation and combination of truth values for each component statement. So, the symbolic logic rule of CONJUNCTION states that; if either, or both, of the conjuncts is factually false, the entire conjunction is factually false, i.e. the compound statement is a FALSE conjunct.
Rule of DISJUNCTION:
|p||q||p ν q|
Since the " ν " signifies inclusive disjunction in symbolic logic then a ν statement is true whenever either (or both) of its component statements is true; it is false only when both of them are false. Thus, the truth table at the right adequately DEFINES the wedge " ν " symbol. Although this roughly corresponds to the English expression "Either . . . or . . . ," notice that in ordinary usage we often exclude the possibility that both of the disjuncts are true—"Either he is here or he is not" doesn't leave open the chance that he is both here and not here. Remember that our logical symbol, ν, is always inclusive by its truth-table definition. If we want to express the more limited sense conveyed by the English expression, we'll have to use a statement of the form "(p • q) • ~ (p • q)."
Rule of IMPLICATION:
|p||q||p ⊃ q|
Since the " ⊃ " symbol is used to symbolize a relationship called material implication; a compound statement formed with this connective is true unless the component on the left (the antecedent) is true and the component on the right (the consequent) is false, as shown in the truth-table at the right that adequately DEFINES the horeshoe " ⊃ " symbol. In this case, there is a reliable correspondence with the conditional statements that are commonly expressed in the English expression "If . . . , then . . . ." Although conditionals have many other uses in ordinary language (to assert the presence of a causal connection, for example), virtually all of them exemplify the basic sense of material implication symbolized by the ⊃.
Rule of EQUIVALENCE:
|p||q||p ≡ q|
Since the " ≡ " is used to symbolize, in which the compound statement is true only when its component statements have the same truth-value—either both are true or both are false. Thus, the truth-table at the right adequately DEFINES the triple bar " ≡ " symbol. This corresponds to a minimal interpretation of biconditional statements expressed in English with the connective phrase " . . . if and only if . . . ."
In compound statements formed with the five truth-functional connectives, one important logical feature remains the same. No matter how long a compound statement is, the truth or falsity of the whole depends solely upon the truth-value of its component statements and the truth-table meaning of the connectives it employs. Thus, if A and B are true while X and Y are false, then the compound statement (A • ~ B) ⊃ ( ~ X ν Y) must be true: since B is true, ~ B must be false, making A • ~ B false; since X is false, ~ X must be true, making ~ X ν Y true; but then the whole ⊃ statement is F ⊃ T , which is true.
And, since the logical relations BETWEEN truth claim propositions is the focus of symbolic logic, then the truth claim sentences themselves may also be symbolized using upper case letters of the alphabet that merely 'stand-in' for the complete sentences themselves. In symbolic logic these upper case letters of the alphabet are known as statement constants. Thus, the ordinary language passage above concerning swapping the yo-yo for some cookies in symbolic logic becomes:
premise 1: P ⊃ Q)
premise 2: P
This symbolic rendition is read aloud as; "If P is the case, then it follows that Q is the case. P is the case. Therefore, it follows that Q is the case."
So, symbolic logic is the logic that obtains BETWEEN truth claim sentences called 'propositions'.
Statement connectives are the syntactic symbols for words that logically join truth claim sentences.
Statement constants are the upper case letters of the alphabet that are 'place-holders' for truth claim sentences.
Statement variables are the lower case letters of the alphabet that are 'place-holders' for truth claim forms.
This rule is recursive in the sense that it can be applied to its own results in order to form compounds of compounds of compounds . . . , etc. As these compound statements become more complex, we'll use parentheses and brackets, just as we do in algebra, in order to keep track of the order of operations. Thus, since A , B , and C are all statements, so are all of the following compound statements:
A • B
A ν ~ C
C ⊃ (B ν A)
~ ( ~ B ≡ C)
(A ν ~ B) ≡ (C ⊃ A)
[A ν ~ (C ν B)]