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Unit 6: Formalized Symbolic Logics

a disjunction as antecedent. However, this makes for a problem for those cases when we want to Notes
express a disjunction with an implication as a disjunctive. In such cases, we must retain at least
one pair of parentheses.

Semantics

The semantics of logic is similar to the semantics of algebra. Algebra is unconcerned with the
real-world meaning of variables like x. What is interesting is the relationship between the
variables expressed in the equations we write; and algebraic methods are designed to respect
these relationships, no matter what meanings or values are assigned to the constituent variables.
In a similar way, logic itself is unconcerned with what sentences say about the world being
described. What is interesting is the relationship between the truth of simple sentences and the
truth of compound sentences within which the simple sentences are contained. Also, logical
reasoning methods are designed to work no matter what meanings or values are assigned to the
logical “variables” used in sentences. Although the values assigned to variables are not crucial
in the sense just described, in talking about logic itself, it is sometimes useful to make these
assignments explicit and to consider various assignments or all assignments and so forth. Such
an assignment is called an interpretation. Formally, an interpretation for propositional logic is
a mapping assigning a truth value to each of the simple sentences of the language. In what
follows, we refer to the meaning of a constant or expression under an interpretation i by
superscripting the constant or expression with i as the superscript.
The assignment shown below is an example for the case of a logical language with just three
propositional constants, viz. p, q, and r.
i
p = true
q = false
i
i
r = true
The following assignment is another interpretation for the same language.
i
p = false
i
q = false
r = true
i
Note that the expressions above are not themselves sentences in Propositional Logic. Propositional
Logic does not allow superscripts and does not use the = symbol. Rather, these are informal,
Meta level statements about particular interpretations. Although talking about propositional
logic using a notation similar to that propositional logic can sometimes be confusing, it allows
us to convey meta-information precisely and efficiently. To minimize problems, in this book
we use such meta-notation infrequently and only when there is little chance of confusion.
Looking at the preceding interpretations, it is important to bear in mind that, as far as logic is
concerned, any interpretation is as good as any other. It does not directly fix the interpretation
of individual logical constants. On the other hand, given an interpretation for the logical constants
of a language, logic does fix the interpretation for all compound sentences in that language. In
fact, it is possible to determine the truth value of a compound sentence by repeatedly applying
the following rules.
1. If the truth value of a sentence is true in an interpretation, the truth value of its negation is
false. If the truth value of a sentence is false, the truth value of its negation is true.

2. The truth value of a conjunction is true under an interpretation if and only if the truth
value of its conjuncts are both true; otherwise, the truth value is false.
3. The truth value of a disjunction is true if and only if the truth value of at least one its
conjuncts is true; otherwise, the truth value is false. Note that this is the inclusive or

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