Course info

GODFREY OKOYE UNIVERSITY, UGWUOMU-NIKE ENUGU.

SECOND SEMESTER COURSE FOR 2019/2020 SESSION

DEPARTMENT OF PHILOSOPHY IN THE FACULTY OF ARTS

PHI 424: FURTHER LOGIC

CONTRIBUTIONS OF GOTTLOB FREGE TO LOGIC

Friedrich Ludwig Gottlob Frege was born in 08/11/1848 in Wismar of Northern Germany. His parents were Karl Alexander Frege and Auguste Frege who both worked at a girl’s private school as principals. Gottlob Frege was a German logician, mathematician and philosopher who played a crucial role in the emergence of modern logic and analytic philosophy. Frege’s logical works were revolutionary, and are often taken to represent the fundamental break between contemporary approaches and the older, Aristotelian tradition. He invented

1.      modern quantificational logic

2.     created the first fully axiomatic system for logic, which was complete in its treatment of propositional and first-order logic

3.     represented the first treatment of higher-order logic.

4.     proponents of logicism

5.     influential in philosophy of language.

6.     Influenced thinkers as Russell, Carnap, Wittgenstein.

7.     founder of modern logic

Frege was trained as a mathematician but had strong interest in logic following particular interest in the foundations of arithmetics. Quite unlike Kant but like Leibniz, he believed that the truths of arithmetic are logical, analytic truths. Kant thought that arithmetical knowledge was grounded in “pure intuition”. Frege subscribed to logicism though his logicism was limited to arithmetic; unlike other important historical logicists, such as Russell, Frege did not think that geometry was a branch of logic. However, Frege’s logicism was very ambitious in another regard, as he believed that one could prove all of the truths of arithmetic deductively from a limited number of logical axioms. Indeed, Frege himself set out to demonstrate all of the basic laws of arithmetic within his own system of logic.

Though Frege had common course with Leibniz and Boole to develop logically perspicuous language in which logical relations and possible inferences would be clear and unambiguous, he turned down their methods as imprecise and antiquated. Hence for Frege, the formulae of mathematics is the paradigm towards such perfect language that will give way to clarity and  unambiguous writing. This is the language he addressed as the “Begriffsschrift” where “Begriff” refers to “concept”. He patterned this logical language in line with the language of arithematic but replaced the subject/predicate style of logical analysis with the notions of ‘LOGICAL FUNCTION’ represented as following:

“f(x) = x² + 1″

Here:

1.      f  is a function that assumes x as an argument

2.     yields as value the result of multiplying x by itself (x²) and adding 1.

Since Frege intended to expand this logical language beyond the scope of arithmetic, he wildened the notion of function to allow arguments and values other than numbers. For instance, the concept of “ATTENDANCE” in class becomes a function that has the True as value for any argument that shows positive attendance and the False as value for anything else. Hence A() where “j” is a constant for Joseph and “h” for Hellen. Then “A(j)” stands for the True that Joseph is present, while “A” stands for the False that Hellen is absent. 

 

The values of such concepts could then be used as arguments to other functions. In his own logical systems, Frege introduced signs standing for the negation and conditional functions. His own logical notation was two-dimensional. However, let us instead replace Frege’s own notation with more contemporary notation.

1.      Conditional function, “→” is understood as a function the value of which is the False if its first argument is the True and the second argument is anything other than the True. Therefore, “H → H(j)” stands for the True, while “H(j) → H” stands for the False.

2.     The negation sign “~” stands for a function whose value is the True for every argument except the True.

3.      Conjunction and disjunction signs could then be defined from the negation and conditional signs.

4.     Frege also introduced an identity sign, standing for a function whose value is the True if the two arguments are the same object, and the False otherwise, and a sign, which he called “the horizontal,” namely “—”, that stands for a function that has the True as value for the True as argument, and has the False as value for any other argument.

 

Frege understands quantifiers as “second-level concepts”. The distinction between levels of functions involves what kind of arguments the functions take. In Frege’s view, unlike objects, all functions are “unsaturated” insofar as they require arguments to yield values. But different sorts of functions require different sorts of arguments. Functions that take objects as argument, such as those referred to by “ + ( )” or “H”, are called first-level functions. Functions that take first-level functions as argument are called second-level functions. The quantifier, “∀x(…x…)”, is understood as standing for a function that takes a first-level function as argument, and yields the True as value if the argument-function has the True as value for all values of x, and has the False as value otherwise. Thus, “∀xH(x)” stands for the False, since the concept H( ) does not have the True as value for all arguments. However, “∀x[H(x) → H(x)]” stands for True, since the complex concept H → H does have the True as value for all arguments. The existential quantifier, now written “∃x(…x…)” is defined as “~∀x~(…x…)”.

 

 

LOGICAL OPERATORS

Operators or connectives are also called logical constants or functions; they are called constants because they have a fixed specific meaning; apart from operating on propositions to yield other propositions, they give our arguments their form or structure.

           Propositional logic deals with propositions and because a is either definitely true or definitely false, it follows that it allows only operators, which have the quality of being truth-functional. By a truth functional operators we mean one which enables us to know the truth value of the proposition(s) resulting from applying it just by knowing the definition of the operation(s) and the truth-value of the operation(s) to which it is applied. That is, the operators are such that the truth or falsity of the propositions which they yield is wholly determined by the truth value of the proposition which they operate. In order to know the truth value of the operation which results from applying an operator to operations used.

           We have five such logical operators (or connectives) each of which though roughly corresponds to some fairly common English expression is defined in precise logical terms; these in English expression are “AND” “OR”, “NOT”, “IF”------ “THEN”, and “IF AND ONLY IF”. These respectively are what we know as Conjunction, Disjunction (or alteration), Denial (or Negation, Implication ( or Condition) and Bi-conditional or (Material Equivalence).

      i.          Conjunction

Conjunction is a truth-functional connective similar to “and” in English and is represented in symbolic logic with the dot “.” The ordinary language definition of the dot is a connective forming compound propositions which are true only in the case when both of the propositions joined by it are true. One way of expressing this definition is by way of a table that gives the truth-value of the components of a proposition and the truth-value, which they determine of the compound; this is called truth table. The truth table offers a schematic devise for correlating the truth-values of the component propositions, called truth conditions with the truth-value they determine the compound propositions to have. A truth table is thus a graphic representation of an operator and its definition in terms of truth-value.  In procedure, the table displays the condition under which a proposition formed by a logical operator is to be regarded as true or otherwise false.

Consider the following example:

“Amarachi left and Uchechi arrived”

Can be symbolized as “A .U” 

(i.e (without quotation marks), so long as we remember that the statement does not mean “Uchechi arrived after Amarachi left” which is a simple proposition).

           There are four possible states of affairs which might have occurred with respect to Amarachi leaving and Uchechi arriving. In the context of truth-functional logic, given the notion of truth-value, for any possible combination of two independent propositions, as in above instance, there are four possible combinations of truth-value, namely; both are true; or the first is true and the second false; or the first is false and the second is true; or both are false. With this possibility of true vale combination, if we use the letters “T” and “F” to designate a true truth-value and “0” for false) any compound proposition (using the variables “p” and “q” to represent this) can be clearly shown to have the following possibilities:

p	q	p . q
T	T	T
T	F	F
F	F	F

 

Other ordinary language conjoiners besides “and” include some uses of “yet”, “but”, “also”, “still” “nevertheless” , “although”, “however”, “moreover”, even “comma” (,) and “semi colon” (;); all of them can be indifferently symbolized with the dot “.”

The dot as a truth functional connective doesn’t do everything that the “and” does in English. It might be thought of in terms of a “minimum common logical meaning” to conjoined statements. It is notable that, “And” in English sometimes expresses temporal succession, not just conjunction. “She cursed like a sailor and hung up.” Here the “and” should be translated as “.” Because it does express conjunction; but our translation will no longer make it clear which act was performed first. One function of the “and” in English is to do so, but that function is not truth-functional and cannot be captured by our operators. “And” sometimes also functions as a slovenly substitute for the infinitive, as in “Try and make me.” Here the meaning is even further from truth-functional conjunction.

           The point on prominent relief is that the truth functional connectives are more limited than their corresponding English connectives: the whole meaning of the truth functional connective is given in its truth table. It is nevertheless instructive that so long as we do not expect more from truth-functional connectives, there should be few difficulties in translation.

    ii.          Disjunction

Disjunction (or as it is sometimes called, alternation) is a connective which forms compound propositions which are false only if both statements (disjuncts) are false. The connective “or” in English is quite different from disjunction.  “Or” in English has distinctly different senses. The first is the exclusive sense of “or” is “Either A or B (both not both)” as in “You may go to the left or to the right”. In Latin, the word is “aut”. Then, the second which is the inclusive sense of “or”: “Either A or B (or both).” as in “Ugonma is at the library or Ugonma is studying.” In Latin, the word is vel.” It is this second sense that we adopt its use, accordingly we use the “vel” or “wedge” symbol: “v” The truth table definition of the vel is    

p	q	p v q
T	T	T
T	F	F
F	F	F

 

 

 

 

iii. Conditional Statements and Material Implication

     The word “implies” has several different meanings in English, and most of these senses of the word can be conveyed in the ordinary language connection of statements with “If … then …” In symbolic logic, implication is present for “If … then …” propositions which assert some logical or causal or other relationship. Implication is a relation that holds for conditional statements-there are many types of conditionals:

1.     Logical: For example, “If all philosophers are great thinkers and Agu is a philosopher, then Agu is a great thinker.”

2.     Definitional: For example, “If Nwokoye is anemic, the Nwokoye has a low concentration of erythrocytes in his blood.”

3.     Causal: For example, “If you strike the match, it will light.”

4.     Decisional: For example, “If you contribute to the scholarship scheme, then the company you work for will match the amount.” Material implication is the weakest common meaning for all types of “If … then …” statements. By convention the first part of the conditional is termed the antecedent (also less often called the “implicans” or the “protasis”), and the second part of the conditional is the consequent (less often termed the “implicate” or “apodosis”). For example, in the conditional statement “If you study diligently, then you might see positive results”, the antecedent is “You study might see positive results.”

In general, the weakest common meaning is that (1) if the antecedent and consequent of a conditional statement are true, then the conditional as a whole is true, but (2) if the antecedent is true, then the consequent is false, then the conditional as a whole as a whole is false. Thus, we can display these vales in the following truth table:

p	q	p É q
T	T	T
T	F	?
F	F	?

 

If we assume completeness for our truth functionality, then lines (3) and (4) of the truth table for “p É q” must have truth values unique to the substitution instances for implication. Let’s try out various combinations of truth values. If the resultant truth values for “p É q” on lines (3) and (4) of the truth table, were both false, then this truth table would be the same truth table for conjunction (or the dot “.”). Consequently, these two lines cannot both result in false because conditionals mean something different from conjunctions. If the resultant truth values were a T and a F respectively, for lines (3) and (4) of the truth table for “p É q”, then the truth of the conditional would depend on the truth of the consequent regardless of the first statement.

     However, “If p then q” does not mean “q whether or not p.” If the resultant truth values were respectively a F and a T for lines (3) and (4) of the truth table, then a similar objection would apply. This objection can be explained with the help of the following tentative truth table:

p	q	p É q
T	T	T
T	F	F?
F	F	F?

          

Suppose we have the conditional statement, “If the match is struck, the match lights”. By the above truth table, if we do not strike the match and the match lights, then the conditional would be false. But surely the match could light in many other ways than the method of striking. i.e, the tentative truth table implies the match lights only in case the match is struck; we want to allow that the match could light in other ways. The final suggestion for the truth table for “É” is this:

p	q	p É q
T	T	T
T	F	F
F	F	T