Two valued logic is usually called Bivalent. This is because two valued logic has the fundamental thesis that all propositions have two, and only two, values that can possibly be attributed to them. The two values that every proposition will have is True or False. The Bivalent logic will be able to tell you which propositions are True and which propositions are False. The only classes for propositions are Truth class and False class. Truth class is everything and False class is nothing.
Bivalent logic would only work if have only those two classes. From having these two classes it will always give you true propositions. It will be able to answer only two types of questions.
Suppose a student asks Bivalent logic a question, specifically a question on mathematics. The student asks, "What isn't true in mathematics?". The response is "two plus two isn't five". The student asks, "What is true in mathematics?". The response is "two plus two is four". So it can only give answers to the questions of form of "What isn't true" & "What is true".
Bivalue logic deals with propositions, but these propositions must be of a certain sort. The propositions must take on a certain form for Bivalue logic to be applied. The mathematical logic text "Mathematical Logic and Computability", gives some indication of mathematical logic being a bivalent logic.
"A vocabulary for propositional logic is a non-empty set P(o) of symbols; the elements of the set P(o) are called proposition symbols and denoted by lower case letters p, q, r, s, p,q, ... In the standard semantics of propositional logic the proposition symbols will denote propositions such as 2+2 = 4 or 2+2 = 5. Propositional logic is not concerned with any internal structure these propositions may have; indeed, for us the only meaning a proposition symbol may take is a truth value – either true or false in the standard semantics." - (Italics my emphasis)The Bivalent logic machine will only answer questions that have meaning. The only questions that concern the machine are propositions that are either true or false, and nothing more and nothing less. It will always give true answers to questions that meet that requirement of meaningful questions, i.e. meaningful propositions.
As the authors move further on to show of mathematical logic that:
"The primitive symbols of the propositional logic are the following: (1) propositional symbols p,q,r...from P(o); (2) negation sign (~); (3) conjunction sign (&); (4) disjunction sign (V); (5) implication sign (-->); (6) equivalence sign (<-->); (8) left bracket [ ; (9) right bracket ]."We can represent all these primitive symbols of Bivalent logic, or at least mathematical logic, like this: Primitive Symbols = (p, ~, &, V, -->, <-->, [,and ]).
"Any finite sequence of these symbols is called a string. Or first task is to specify the syntax of propositional logic; i.e. which strings are grammatically correct. These strings are called well-formed formulas. The phrase well-formed formula is often abbreviated to wff (or a wff built using the vocabulary P(o) if we wish to be specific about exactly which proposition symbols may appear in the formula). The set of wffs of propositional logic is then inductively defined by the following rules: (W:P ) Any proposition symbol is a wff; (W:~ If A is a wff, then ~A is a wff; (W:∧, ∨, -->, <-->) If A and B are wffs, then [A ∧ B], [A ∨ B], [A --> B], and 0 [A <--> B] are wffs." - (Italics are authors)The primitive symbols will now be able to be joined together. This joining together will be based on some rules, and these rules determine what is a well formed formula. So if well formed formula is presented based on some primitive symbols, then Bivalent logic may deal with them.
We can represent these rules of Bivalent logic, at least mathematical logic, like this:
WFF of Primitive Symbols= [(p, ~, &, V, -->, <-->, [,and ]) P, ~P, P&Q, PvQ, P-->Q, A<-->Q]
Another way to put this would be that (i) all propositional symbols are wff, (ii) all wff that are prefixed with ~ are wff, (iii) all two wff joined by either (&,V,-->, or <-->) and enclosed by brackets are wff. Bivalent logic can only deal with propositions that are either of these criterion and either true or false.
Here is a truth table of two propositions, both of which are wff:
P ~P
T F
F T
This truth table exhausts all logically possible outcomes of either well formed formula P or well formed formula ~P. These are the only possible outcomes for Bivalent logic. The truth table shows that each well formed formula is consistent with each proposition being either true or false.
Unvalent Logic will have some slight alternations. Unvalent logic will only deal with propositions that take on one truth value and only one truth value. This truth value would be truth itself, perhaps might be false value. So Unvalent logic will not deal with any propositions that take on two truth values or more than two truth values.
This alteration leads to an automatic change. It would immediately lead to a well formed formula not having the primitive symbols of (2) through (9). So it will not accept the symbols (~, &,V,-->, <-->). It will also not accept as well formed formula like (ii) and (iii). This also leads to an immediate change in the laws of logic.
For example, the Bivalent logic has at least three laws. It is the Law of Identity, Law of Excluded Middle, and Law of Non-Contradiction. None of these laws would hold within Unvalent logic. All of this becomes obvious from recognition that the logical operator of negation, i.e. ~, is rejected from Unvalent logic.
Here are all three laws represented in symbolic notation:
(LoI): P-->P or ~P-->~P
(LEM): Pv~P or ~PvP
(LNC): ~(P&~P) or ~(~P&P)
Excluded Middle has a negation within it, and Unvalent rejects negation, thus Excluded Middle is rejected. Likewise, Non-Contradiction has two negations in it, so Non-Contradiction is rejected. Because Identity in Bivalent logic equally applies to negation and non-negation, and negation is rejected in Unvalent logic, the identity is also rejected.
With two valued logic, each proposition with a propositional symbol (either p,q, or r) is implicitly saying that "it is true that...". For example, take the propositional symbol P. P is either true or false. P states that "it is true that two plus two is four". The negation operator is implicitly saying that "it isn't true that...". So given symbol P and applying ~ operator to it, then we obtain ~P, which means that "it isn't true that two plus two is four".
So LoI says, If "it is true that two plus two is four" then "it is true that two plus two is four" & If "it isn't true that two plus two is four" then "it isn't true that two plus two is four". LEM says, either "it is true that two plus two is four" or "it isn't true that two plus two is four". LNC says, not both "it is true that two plus two is four" & "it isn't true that two plus two is four". So Unvalue logic drops all propositions with "it isn't true that..." from being well formed formula. It would drop two valued logics well formed formula.
If all golden retrievers are dogs & some beloved household pets are golden retrievers, then some dogs are beloved household pets, is a true proposition in two valued logic, but isn't true in Uncvalue logic. This is because it has the operators of --> and &. Implication is dropped as a logical operator in this single value logic because implication is defined by material implication.
Implication: P-->Q
Material Implication: ~PvQ
As becomes obvious, material implication contains negation operator within it. This means that whenever a conditional statement is made in two value logic, there is a negation contained within it. If P-->Q is true, then ~PvQ is true. If P-->Q is true, then a negation is also true. Substitute the wff X for the wff (P-->Q) and suppose the proposition X is true, then a negation is also true. Single value can't have a negation true, so P-->Q can't be true. Furthermore, suppose that P-->Q, then the contraposition would be true as well, which is ~Q-->~P. So P-->Q being true implies a negation is true as well.
In two value logic, there are two basic logical operators. They are negation and one of the other four operators of -->, &, V, or <-->. Usually, the two basic logical operators are implication and negation, or ~ or -->. Because --> can be replaced with material implication, which is just a combination of negation and disjunction, or ~ and V, the disjunction can't be the second operator.
If we only assume propositional symbols P and Q & logical operators of negation and implication, then we can define the other logical operators.
Conjunction (&): ~(P-->~Q)
Disjunction (V): (~P-->Q)
Equivalence (<-->): (P-->Q)&(Q-->P)
The law of identity is an implication, and implications contain negations within them, so implications aren't valid in single value logic. The law of excluded middle is a disjunction, and we notice that disjunction contains negation within it, so disjunction isn't valid in single value logic. The law of non-contradiction is a conjunction, and we notice that conjunction contains negation within it, so conjunction isn't valid in single value logic. Equivalence contains both implication and conjunction, and so they aren't valid in single value logic.
It becomes obvious that single value logic will not have negations, conjunctions, disjunctions, equivalence, or implications. It won't answer "What isn't true?" questions. It will only answer "What is true?" questions. It will give you nothing but true statements, and only applies to true statements.
The primitive symbols of the Uncal logic are (1) propositional symbols p,q,r...from P(o). The set of well formed formula of single value logic is then inductively defined by the rule (i) (W:P ) that any proposition symbol is a well formed formula.
No comments:
Post a Comment