3. FREGEAN LOGIC

1. 3. FREGEAN LOGIC The rules of proof were stated already by Aristotle (384-322 BC). His proposals were later developed and improved on in the Middle Ages, thus forming what is now referred to as Aristotelian-Mediaeval logic. However, it was only at the end of the XIXth century that classical logic attained the form usually accepted nowadays. This was particularly due to the work of Gottlob Frege (1848-1925). Both Aristotelian-Mediaeval logic and Fregean logic share two important features: they are - extensional = the truth-value of a composite statement depends on the truth-value of the composing statements - bivalent = there are two truth-values (true, false)

2. LOGIC OF STATEMENTES Let us start from a series of statements: p, q, r, s… called atomic statements, because they are not further decomposable into simpler statements. Let us see how we can correctly obtain a set of compound statements. We will consider only FIVE ways to obtain a composite statement (connectives): - negation: ¬ (not) - conjunction: Λ (and) - disjunction: V (or) - material implication:  (if…then…) - double material implication(or material equivalence):  (if and only if)

3. note 1. whereas the negation applies to a single statement at a time, the others apply to two statements 2. we consider statements from a formal point of view, disregarding their actual meaning (or content) Each statement can be either TRUE (T) or FALSE (F). The logic of statements considers all possible combinations of these truth-values in both atomic and compound statements.

4. NEGATION: ¬ (not) Given a statement p, it can be either true or false. The negation of p, ¬p, would then be false or true, respectively. To repeat: we have not “opened” p, that is, we are not interested in the actual meaning, or content, of p. We do not even know whether p is true or false. We merely say that p is either true or false, and that its negation, ¬p,is a new statement with an opposite truth-value.

5. The conclusion we have just reached can be synthetically displayed by a truth-table, that is, a table that shows the truth-values (T or F) of the compound statement as a function of the truth-values of the statements (atomic or not) they consist of. truth-tabledefining ¬ (not) A truth–table characterizes the connective it refers to, that is, it both describes and defines it. Analogously, the other four connectives are characterized by their own truth-tables…

6. truth-table defining Λ (and)

7. truth-table defining V (or) (p or q) is to be understood as(p vel q), not as(p aut q). In the former case, (p or q) is true if at least one of its components is true. In the latter case, by contrast, (porq) is true only when either p or q is true, but not both.

8. truth-tabledefining  (if… then…) p is the “antecedent” q is the “consequent” It can be shown that p q is logically equivalent to ¬(p Λ¬q) and ¬p V q material implication is not the same as logical implication, that is, an argument leading to a (proven) conclusion.

9. truth-tabledefining (if and only if, or iff) It can be shown that p q is logically equivalent to two material implications: (p  q)Λ(q  p)

10. It may happen that the last column of a truth-table presents only one kind of truth-values. If it presents only T values, it is a tautology. If it presents only F values, it is a contradiction. tautology contradiction

11. As can be easily verified, also (p p) and ¬(pΛ¬p) are tautologies. These three tautologies formally express the three principles of Aristotelian logic: (p p) is the principle of identity ¬(pΛ¬p) is the principle of non-contradiction (p V ¬p) is the principle of excluded middle (or third) PRINCIPLE OF IDENTITY: A is A, that is, in a correct reasoning, the meaning of terms must be constant PRINCIPLE OF NON-CONTRADICTION: in a given statement, we cannot both state and deny a predicate of the subject, at the same time and within the same range of meaning PRINCIPLE OF EXCLUDED MIDDLE: in a two-value logical system, like ours, a statement is either T or F (used in reductio ad absurdum)

12. We saw that (p q) is logically equivalent to¬(p Λ¬q). Now, if we connect these two compound statements by the material implication, , we obtain a tautology: The two statements are said to be logically equivalent.

13. If any two compound statements, connected by the double material implication, lead to a tautology, the two statements are logically equivalent. Furthermore, there are 4 importantrules that need to be borne in mind while arguing: 1. from false statements both true and false statements can follow (ex falso sequitur quodlibet). 2. from true statements only true statements can follow (ex vero numquam sequitur falsum). 3. from contradictory statements both true and false statements can follow (ex absurdis sequitur verum vel falsum). 4. true statements may follow from false statements (1), from true statements (2), and from a contradiction (3) (verum sequitur a quodlibet).

14. LOGIC OF PREDICATES So far, we have dealt with the logic of statements, that is, with the possible ways statements are connected. We will now “open” the statements. However, we will no longer consider statements as composed by subject, copulative conjunction and predicate (as it was standard in Aristotelian-mediaeval logic). Whereas for Aristotle “Socrates is mortal” can be seen as having the logical form S is P within Fregean logic such statement is seen as composed by a propositional function (or open formula) and an argument: “x is mortal” and “x is Socrates”

15. From this point of view, a propositional function is “incomplete”, so to say, and needs to be saturated. What completes or “saturates” it is the argument. Only when it is saturated, a propositional function becomes a proper statement, that is, something that can be either true or false. Predicate calculus is the part of Fregean logic that deals with the “opening” of statements, by analyzing their saturating argument and what is being saturated.

16. The logic of predicates considers, among others, statements such as: “All men are mortal” (or “Each man is mortal”, or else “For every thing, if it is a man, it is mortal”) “Some men are bald” (or “There are some bald men”) Such statements are said to be quantified. In other words, they present either a universal (Â: “all…”) or an existential (Ê: “some…”) quantifier. Therefore, we can distinguish between: –universally quantified statements, such as “all x are p”: (Âx)px; they are true iff all x are p – existentially quantified statements, such “there is an x that is p”, or “some x are p”: (Êx)px; they are true iff there is at least one x that is p

17. Not all expressions of predicate calculus are statements. Only if each and every variable (x) is bounded,that is, it is quantified either universally or existentially, the expression is a proper statement. In that case, variables are said to be bounded. Otherwise, variables are said to be free, or unbounded. REMARK Propositional functions are expressions containing free variables: therefore, they cannot be said to be true or false. By contrast, expressions containing only bound variables are saturated, and can thus be regarded as proper statements: therefore, they can be true or false.

18. Quantifiers should be handled with great care, especially when dealing with their negation:

19. Therefore, denying the quantifier ≠ denying the predicate. Also, we can say the same thing is different ways: ¬(Âx)px is logically equivalent to (Êx)¬px and therefore (since ¬¬●=●) (Âx)pxis logically equivalent to¬(Êx)¬px and ¬(Êx)pxis logically equivalent to (Âx)¬px and therefore (since ¬¬●=●) (Êx)pxis logically equivalent to ¬(Âx)¬px RULE Â● = ¬Ê¬● and Ê● = ¬Â¬●

20. Furthermore, given the predicate p (“… is from Lucca”), there are 4 possible statements: 1. “All IMT students are from Lucca” 2. “No IMT student is from Lucca” 3. “There is an IMT student who is from Lucca” 4. “There is an IMT student who is not from Lucca” These statements can be equivalently expressed with either the universal or the existential quantifier. And they can be opposed to one another in any of 4 different ways: they can be contradictory to one another, contrary to one another, subcontrary to one another and subalternate.

21. (Âx)px¬(Êx)¬px “all IMT students are from Lucca” = “there are no IMT students who are not from Lucca” (Âx)¬px¬(Êx)px “all IMT students are not from Lucca” = “there are no IMT students who are from Lucca” contradictory statements contrary statements subalternate statements (Êx)px¬(Âx)¬px “there is an IMT student who is from Lucca” = “not all IMT studies are not from Lucca” (Êx)¬px¬(Âx)px “there is an IMT student who is not from Lucca” = “not all IMT students are from Lucca” subcontrary statements

22. contradictory statements cannot be both true or both false contrary statements cannot be both true, but they can be both false subcontrary statements cannot be both false, but they can be both true subalternate statements are statements that are either both true or both false, and one describes a situation that can be deduced from the one described by the other

23. Finally, we can rephrase Aristotelian-Mediaeval logic into Fregean terms, rewriting syllogisms with connectives: All fishes live in water [P1] All sharks are fishes [P2] (P1 Λ P2)  C All sharks live in water [C] Analogously, modus ponens and modus tollens can be formalized as follows: [(pq) Λ¬q]  ¬p modus tollens [(pq) Λp]  q modus ponens Both are tautologies.

24. A FEW REMARKS ON CLASSICAL AND MODERN LOGIC • Aristotelian logic is the logic of terms, whereas modern logic is the logic of statements.As such, Fregean logic is not part of classical logic.Furthermore, Aristotelian logic is concerned only with predicates, not with relations among various predicates. Thanks to its new interpretation of statements, Fregean logic can deal with both the logic of individual terms and the logic of relations within a single framework. • According to Aristotle, the dichotomy subject/predicate is but one side of the coin, whose other side is ontological: the subject corresponds to substance and the predicate is what is said to pertain either essentially or accidentally to that substance.Doing away with the dichotomy subject/predicate equals to freeing modern logic from Aristotelian metaphysics.

25. Quantification: whereas for Aristotle “all” refers to a given realm of individuals, that of Fregean logic has a much wider import.For example:for Aristotle, “All men…” refer to all and only those belonging to the human kind; analogously, “All trouts…” refer to all and only the fishes with this name, and so on;by contrast, for Frege “All men” translates into “All x that are men…”, or better “For every x, when x is a man…” – that is, (Âx)px.Â, Frege’s “all”, refers not only to men, but to anything (any x) that may fall under the function p.

26. Analogously, for the universal categorical statement:“All men are mortal”translates into“For every x, if x is a man, then x is mortal”that is, (Âx) [fx  gx]Whereas Aristotle’s categorical statement assumes the actual existence of men, Frege’s formulation does not.On the one hand, that is, the universal statement A states the truth of the implication, but not the truth of the antecedent.On the other, it does not guarantee the existence of what it assumes.

27. As to the particular categorical statement:“Some men are bald”translates into“There is an x, x is a man, and x is bald”that is, (Êx) [fx Λ gx]In other words, Aristotle’s particular categorical statement is, in fact, the conjunction of two existential statements.Both classical particular statements and modern particular statements have an existential import: they both assume the existence of the subject of the statement (in Aristotle’s terms) or the existence of an argument that saturates the two functions f and g.