Forays into propositional logic
I figured I would post this here for "posterity's sake", and to wrap my head around what is going on. Because none of this makes any sense to me. If someone knows propositional calculus maybe they could help explain it to me!
Formal logic is the abstract study of propositions, statements, or assertively used sentences and of deductive arguments. From these things, formal logic is concerned with abstracting the logical forms.
Formal logic uses symbolic notation in order to test for validity. Formal logic can be seen as wholesomely complimentary, or indistinguishable from that of pure mathematics. Formal logic is not to be confused with that of the dirty and empirical process of studying reasoning like unto that of psychology. *gross* Let us also distinguish formal logic from the art of correct reasoning, nor the art of persuasion.
Abstraction: the process of considering something independently of its associations, attributes, or concrete accompaniments.
- abstractio: draw away
A priori: denoting knowledge that is independent of all particular experiences. "From what is before"
A posteriori: knowledge derived from experience. "From what is after"
A priori and a posteriori are used philosophically in order to distinguish between arguments from causes and arguments from effects. First seen as used by Albert of Saxony.
Two broad methods of reasoning:
Deduction: Work from the more general to the specific.
Induction: Works inversely to deduction, reasoning from the specific, or the observable, to the general.
The premise of a logical statement is the starting point from which we reason to the proposition (the conclusion). It must follow with strict neccesity. We must show that it is inconsisent to deny the proposition if the premises are true.
In order to establish truth in an argument there are two distinctions to be met:
1. The conclusion must really follow from the premises.
2. The premises must be true.
An argument which meets this criteria may be called "sound." Firstly, we are not entirely concerned with the validity of the premise, but mostly concerned with the logical cogency and strict adherence of the proposition being born out from the premise.
We may call a deduction "valid" regardless of the veracity of the premise.
Inference: a conclusion reached on the basis of evidence and reasoning
"Every X is a Y. Some Z's are X's. Some Z's are Y's"
- This is an inference form.
- Expressions may be inserted in X Y Z.
- X Y Z are "variables", similar to that of algebra.
"Every DOG is a MAMMAL. Some QUADRUPEDS are DOGS. Some QUADRUPEDS are MAMMALS."
- An example of a valid inference form.
"Every X is a Y. Some Z's are Y's. Some Z's are X's."
- An example of an invalid inference form.
Formal logic is concerned with inference forms in abstraction, not in any specific instance of them. Formal logic seeks to discriminate between valid and invalid inference forms and to explore and systematize the relations that hold among the valid ones.
Proposition form: An expression of which the instances are not inferences from several propositions to a conclusion but rather propositions taken individually. A valid proposition form is one for which all of the instances are true propositions.
"Nothing is both an X and a non-X"
System of Logic:
- Requires a "symbolic aparatus."
- A symbolic aparatus consists of a set of symbols (rules for stringing variables together to create formulas)
- Rules for manipulating these formulas - Attaching meaning to these symbols and formulas.
If a symbolic aparatus has no attached meaning to symbols it is said to be uninterpreted (purely formal)
If a symbolic aparatus has meaning attached to manipulation it is said to be (interpreted)
This is important because systems of logic have properties independent of interpretation.
Axiom: an unproved formula from which other things may be proved
Theorem: The conclusion of certain axioms.
Questions for the philosopher:
- What is the correct analysis of the notion of truth?
- What is a proposition, and how is it related to the sentence by which it is expressed?
- Are there some kinds of sound reasoning that are neither deductive nor inductive?
The simplest and most basic branch of logic.
It deals exclusively with complete, unanalzyed propositions and certain combinations into which they enter.
The symbols of "PC" are comprised of:
Variables: p q r (with or without numerical subscripts)
- Variables are to take propositions as input.
- We are to assume that every propositional variable is either true or false and is never true and false
Operators: ~ · v ⊃ ≡
- These function as to form new propositions from one or more given propositions. These produced propositions are "arguments".
Correspondance of operators to English is as follows:
~ not (negation)
⊃ if...then (implies)
≡ Forces a supposed equivalence "is equvalent to"
Brackets or parentheses
Truth values of propositions are "truth" and "falsity"
Adjudication of operations:
if p is true, then ~p is false
if p is false, then ~p is true
p and q is true if both p and q are true in all other cases if both p and q are not true then this argument is false
At least one variable must be true. If p is false, q must be true, or else the argument is false
Consists of an antecedant (the first variable) and a consequent (the second variable).
p ⊃ q is only a false argument if p is true and q is false
This argument solves as true when both variables have the same truth value, whether true or false.
A monadic (unary) operator requires only one variable to argue. A dyadic (binary) operator requires two arguments.
A truth-functional operator has a truth value determined in every case.
A truth function is composed of truth-functional operators.
PC Formation Rules:
1. A variable standing alone is a wff (well formed formula)
2. If "a" is a wff, so is "~a"
3,. If a and b are wff's (a and b), (a b), (a v b), (if a then b), and (a is equivalent to b) are wff's.
alpha and beta are known as "metalogical variables", not used in PC but used to explain PC.
A wff in which all instances are false is said to be "unsatisfiable."
A wff in which some instances are true and some are false is said to be "contingent."
A decision problem is the problem of finding an effective procedure for testing the validity of any wff of the system. The procedure thus created is called the "decision procedure."
In the event that a decision procedure is found the system is then said to be "decidable." Some systems may be proved to be unsolvable, in this case these systems are said to be undecidable.
PC is known to be a decidable system. The method of truth tables are said to be the most important theoretically.
The following are some valid formulas of PC:
Law of Identity
(p is equivalent to p) is true
Law of Double Negation
(p is equivalent to not not p) is true
Law of Excluded Middle
(p or not p) is true
Law of Noncontradiction
not(p and not p) is true
De Morgan laws
(p or q) is equivalent to (q or p)
(p and q) is equivalent to (q and p)
[(p or q) or r] is equivalent to [p or (q or r)]
[(p and q) and r] is equivalent to [p and (q and r)]
Law of Transposition
(if p then q) is equivalent to (if not q then not p)
[p and (q or r)] is equivalent to [(p and q)or(p and r)]
[p or (q and r)] is equivalent to [(p or q)and (p or r)]
Law of Permutation
[if p then (if q then r)] is equiv to [if q then (if p then r)]
Law of Syllogism
if (if p then q)then[if(if q then r)then (if p then r)]
Law of Importation
Law of Exportation
The following are notes from Principia Mathematica:
Let us begin with "atomic propositions." These are perhaps irreducible. And to reduce them further would be outside the scope of our present inquiry.
We can posit a negative definition for "atomic propositions" Propositions containing no parts which are propositions, and not containing the notions of "all" or "some."
Examples of atomic propositions: "This is red." "This is earlier than that."
"Atomic propositions" may also be defined positively. These are positive atomic propositions.
R1(x) - meaning "x has the predicate R1"
R2(x,y) - meaning "x has the relation R2 (in intension) to y" -note: "intension" is the internal content of a concept
R3(x,y,z) - meaning "x,y,z have the triadic relation R3 (in intension)"
R4(x,y,z,w) - meaning "x,y,z,w have the tetradic relation R4 (in intension)"
We may assume that this relation may go on ad infinitum.
Logic is concerns solely with the hypothesis of there being propositions of such-and-such a form.
First; let "p,q,r,s,t" denote atomic propositions.
Now our primitive idea "p | q" this is to say that p is incompatible with q. It may be said that this is true when either or both are false.
This may also be read "p is false or q is false" or "p implies not q"
However, if p is true, q may not be
p | q is to be read "p stroke q"
The "principal stroke" is the stroke which belongs to the first "atomic proposition".
The secondary stroke belongs to the molecular propositions. And so on for the tertiary. A finite number of steps may reduce us to the first stroke.
Atomic and molecular propositions are both known as "elementary propositions." Therefore an elementary proposition includes all those things which are atomic and molecular and therefore may be finitely reduced to a principle stroke.
If we know p then we may infer r.
If we are not given p, but p is used suitably, we may know p to be true.
A function is that which is true given it's constituents