Orthologiko Esografia
Above art by Steve McCaffery
This is just meant to be a surmise of formal #logic, via tables of #operators or #quantifiers, rules of #computation, and a glossary of relevant terms.
Logic
Logic is the study of the “correct” way by which some alleged verbally expressed derivation from verbally expressed past knowledge, without reliance on mere intuition or perception, can be determined to constitute new knowledge. Reiterated, it is the study of the “correct” way to derive new verbally expressed knowledge from established verbally-expressed knowledge without reliance on mere intuition or perception.
Truth-Values and Evidence
For logic, a truth-value is an epistemically relevant value that is assigned to a sentence. The epistemic relevance of such a value is that it distinguishes and distributes the status a sentence may have for knowledge, called its epistemic status. A proposition is a verbal expression that can have or take a truth-value. Systems of logic that only have $2$ truth-values with opposite valences, namely true v. false, are characterized as following the principle of bivalence and may thus be called a bivalent logic. The paradigmatic proposition is a meaningful declarative sentence.
Relationships among or within propositions that are alleged to possess specific truth-values that allow or determine a truth-value to be granted for some other proposition or set of propositions are said to be evidential in relation to that or those other proposition(s), hence constituting an evidential relationship. When a #proposition is claimed or alleged to have a specific truth-value, namely one which would warrant it a favorable epistemic status, it is called an assertion. A set of propositions with an asserted evidential relationship among themselves is known as an argument, whereby the subset of propositions acting as evidence are known as premises and the propositions evidenced by them are known as conclusions. The process of going from a #premise to a #conclusion in an #argument is known as an inference.
Formal Logic
Formal logic is logic which focuses on studying evidential relationships among or within sentences independent of what those sentences are about or regarding, what utility they serve, or what they refer to. The latter may be called the content of the sentence. Without the content, what is left of a sentence is its form, i.e. the relationships among its components—that is, the structure of the sentence. Types of formal logic include propositional calculus, predicate calculus, etc.
Validity
Validity refers to a property of a formal aspect of an evidential relationship such that the form of that relationship is truth-preserving. Truth preservation is the process by which the (intrinsically?) epistemically favored truth-value, or “positively valent” truth-value, found in the premises of the argument, or the components of the premises of the argument, persist in the conclusion(s) and/or is “transferred” to the conclusion(s) itself/themselves as a whole. Namely, under a principle of bivalence, if the premises of an argument being true guarantee the truth of its conclusion, or the conclusion of an argument cannot be false provided its premises are true, then that argument can be described as valid. Otherwise, the argument would be described as invalid. An argument which is valid while in fact the premises are also true is described as sound. For testing if an argument is valid or invalid, skip to Validity Testing.
Informal Logic
Informal logic is logic which assesses the derivation of verbally-expressed knowledge in terms of the context that determines the scope of those verbal expressions, in addition to the forms which determine them, such that they can be regarded as knowledge. Unsurprisingly, informal logic renders the content of the sentence, i.e. its utility, reference, or object, relevant, as it has baring on questions of scope.
Propositional Calculus
Propositional calculus is simply another term for sentential logic or propositional logic. All mean a formal logic that takes as its smallest unit of analysis the simple sentence that can have or take a truth-value (henceforth “truth-apt” for short). To know what a “simple sentence” is, jump to Sentences and Notation.
Sentences and Notation
A sentence is simple, or is a simple sentence, insofar as it contains no more than one clause or it contains no component sentences. A component sentence is anything in a given sentence that can be replaced by some other sentence while that given sentence remains grammatical. In propositional calculus, (usually-declarative) truth-apt simple sentences are taken as primitives which can be combined to thus comprise compound sentences. In propositional calculus, compound sentences appear as juxtaposed propositions jointly “affixed” in some way.
Sentential Constant
The convention in propositional calculus is for any (usually-declarative) truth-apt simple sentence to be represented as the upper-case of the letter closest to its beginning, or the first letter of its subject, when trying to keep the sentence bound to its discursive environment or trying to hint at the sentence's content-specificity. A (usually-declarative) truth-apt simple sentence that, symbolized, nonetheless hints at its content-specificity or is otherwise bound to its discursive environment is called a constant.
$$ S = \text{ The sun is rising.} $$
Sentential Variable
When a (usually-declarative) truth-apt simple sentence is represented as a lower-case of the letter nearest the beginning of its sentence or the first letter of its subject, this is so that it can then be taken as any possible sentence provided it excludes whatever other sentences may surround it. A (usually-declarative) truth-apt simple sentence symbolized in this way, for this purpose, is called a variable.
$$ s = \text{ The sun is setting} $$
Sentential Operators and Notation
A sentential operator is a word in a sentence that meaningfully joins two sentences or more, thereby producing a compound sentence. An operator is said to be truth-functional if:
- Modeled as function, it outputs one option from the set of possible truth-values
- The truth-value the function model outputs is invariant provided an input with a constant truth-value or a constant set of true-values
- The outputted truth-value applies to a structure that encompasses the elements for which the truth-value input(s) of the function apply
A system of logic is truth-functional provided all its available operators are truth-functional. A sentential operator is for the propositions of propositional calculus simply an operator. The minimalist set of truth-functional operators for propositional calculus can be perused below.
| symbol | operator name | operand name | meaning |
|---|---|---|---|
| $\land\text{, }\cap$ | #conjunction | #conjunct | and |
| $\lor\text{, }\cup$ | #disjunction | #disjunct | or |
| $\to\text{,}\implies\text{,}\rightarrow$ | #conditional / implication | #antecedent / implicant, consequent | if ___ then ___ |
| $\iff\text{,}\leftrightarrow$ | #biconditional | implicants/autocedents, parasequents | if and only if |
| $\neg\text{,}\sim$ | #negation | negata | not |
Formulae and Notation
A formula in propositional calculus is simply an operation. The operator joining the largest component sentences for the largest compound sentence in a set of interdependent operations is known as the major operator, and that largest compound sentence may be called a major operation. Component sentences containing an operator are known as subformulae while compound sentences containing an operator are known as superformulae. Operative compound sentences that also act as operative component sentences are set apart via parentheses. Example below, wherein subscripted $m$ indicates the major operator, subscripted $f$ indicates the subformulae and subscripted $s$ indicates the superformulae.
$$ ((p\land q)_{f_0} \to_m (p \lor s)_{f_1})_s $$
Computing Truth in Propositional Calculus
Truth Table Computations
The output of a sentential operator in each case of combined truth-values, known as a rule of computation, can be examined by creating a table structure called a truth-table, consisting of columns each of which is a component or compound sentence and rows each of which is a case comprised of given truth-values for each of the enlisted/columnar propositions. Truth-tables have utility for exploring the possibility space of computable truth-values for a set of propositions.
The number of rows aside from the columnar row is $t^c$ where $c$ is the number of smallest/“simplest” component sentences and $t$ is a constant representing the number of possible truth-values for those sentences. The conventions for constructing the truth-table are as follows:
- Enlist the smallest/“simplest” component sentences into the columnar row in alphabetical order.
- Fill each column's row with alternating truth-values, each alternated truth-value doubled as one moves across each column in a direction opposite the scriptural convention (in English, that direction would be from right to left as normally things are written from left to right). That is, provided $t$ is a constant representing the number of possible truth-values, while $i$ is the index of the column in the reversed column sequence, the number of times a specific truth-value consecutively repeats down each column prior or posterior to alternation is: $c_i = t^i \text{, for } i \geq 0$.
- Optionally append the largest compound sentence, the superformulae in which is found the major operator, or otherwise the ordered premises of an argument, to the columnar row, with its formulae or those premises acting as columns, each row of those columns being the resulting truth-value of the operator for that formula or that premise's major operator.
Rules of Computation in Propositional Calculus
Some examples, that show the rules of computation for the different sentential operators (see Sentential Operators and Notation), assuming “T” stands for true and “F” stands for false under a principle of bivalence:
| $p$ | $\neg{p}$ |
|---|---|
| F | T |
| T | F |
negation true as its negata is false and false as its negata is true
| $p$ | $q$ | $p \land q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
conjunction true if and only if its conjuncts are both true
| $p$ | $q$ | $p \lor q$ |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
inclusive disjunction true if and only if ≥1 disjunct is true
| $p$ | $q$ | $p \veebar q$ |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
exclusive disjunction true if and only if only 1 disjunct is true
| $p$ | $q$ | $p \to q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
conditional false if and only if while consequent is false antecedent is true[^1]
| $p$ | $q$ | $p \iff q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
biconditional true if and only if both implicants/parasequents have the same truth-value[^2]
Truth Tree Computations
The output of a sentential operator in each case of combined truth-values can be examined by creating a tree structure called a truth tree, representing the truth-values of the smallest subformulae and the leftover truth-values of previous operations at each level of the tree as sibling nodes. Truth trees have utility for evaluating the truth-value of the largest compound sentence or the truth-value output of the major operator of a formula when and if the truth-value of “simplest” component sentences or “smallest” subformulae is known, assumed or conjectured; otherwise, it can be useful for testing the validity of an argument. The conventions for constructing a truth-tree are the following:
- Specify the truth-values of each smallest/“simplest” component sentence, or the primitive operands of the formulae, above each of them.
- Starting with the smallest subformulae or operations, draw solid arrows from each subformulae operands towards a shared point wherein the truth-value of their respective subformulae operations are written then drawing dashed arrows for and to sibling nodes of and for leftover operands' truth-values. These leftovers are usually from and due to the resolved subformulae themselves having acted as operand of a larger operation that also contains that leftover.
- Repeat #2 for each subsequent higher-order formulaic operation until hitting a root node with a resolved truth-value for the major operator.
Validity Testing
Validity testing is the use of a computational test to either demonstrate that an argument is valid or demonstrate that an argument is not invalid. The former is a method of verification for validity while the latter a method of falsification for invalidity. There are two formal methods of testing for validity:
- The falsity-constraint test: Simplifies the truth table for an argument so that only rows or instances wherein a conclusion is false are included, such that one then computes or goes through the truth-values of the premises to detect whether they are all true for any such instance. If there is no instance in which they are all true, then the argument is valid. Otherwise, the argument is invalid.
- The counterfactual test: Simplifies the truth table for an argument by simply assuming it is invalid, i.e. by forcing the conclusion to be false while the premises are true, only to then “work backwards” to demonstrate what would have to be the truth-value combination of the component sentences for this to occur. If, in attempting to do so, one encounters a contradiction, i.e. one has produced a situation in which at least one component sentence would have to be several “contravalent” truth-values at once (in the case of bivalence, would need to be simultaneously true and false) for the argument to be invalid, then the argument is valid. Otherwise, the argument is invalid.
An informal test would be to simply find an argument with the same form but premises known to be true and a conclusion known either to be false or nonsensical as a counterexample to the presumed validity of a given argument.
Predicate Calculus
// TODO: Complete the Predicate Logic Section
[^1]: This truth table represents the material conditional, a specific interpretation of the conditional based purely on constraining its scope, i.e. reducing its computational overlap with other sentential operations. Other interpretations of the conditional may lead to different rules of computation. [^2]: A bit meta-, isn't it?
#bivalence #counterexample #contravalence #variable #constant #validity #preservation #soundness #assertion