# Categorical Logic

Categorical Logic Categorical logic is an approach to logic, first developed by Aristotle, that focuses on evaluating arguments whose validity depends on relationships between classes. A categorical statement is a statement that relates two classes. A categorical statement has four parts: • Quantifier (“all,” “no,” or “some”) • Subject term • Copula (“are,” “are not,” “is,” “is not”) • Predicate term A categorical statement also has a quantity (universal, particular) and a quality (affirmative, negative). Combining these quantities and qualities, we get the four standard forms of categorical statement: A All S are P (universal, affirmative) E No S are P (universal, negative) I Some S are P (particular, affirmative) O Some S are not P (particular, negative) The logical relationships between standard form categorical statements with the same subject and predicate terms are captured by the traditional square of opposition: A and E statements are contraries: they cannot both be true, but they can both be false. I and O statements are subcontraries: they cannot both be false, but they can both be true. A and O statements and E and I statements are contradictories: exactly one of each pair must be true. The relationship between A and I and E and O statements is subalternation: A statements imply I statements and E statements imply O statements. There are three kinds of immediate inference: 1 • Conversion: switch the subject and predicate term – Valid for E and I statements – Conversion by limitation (switch subject and predicate, change quantity) is valid for A statements • Obversion: change the quality and replace the predicate term with its term-complement – The term-complement (non-P) of a term (P) denotes the class of things that are not in P – Always valid • Contraposition: replace both terms with their term-complements, then switch them – Valid for A and O statements – Contraposition by limitation (contrapose and change quantity) is valid for E statements There are 256 ways to combine categorical statements into syllogisms. The first premise of a standard form syllogism contains the major term, which is the predicate term of the conclusion. The second premise contains the minor term, which is the subject term of the conclusion. The term contained in both premises but not the conclusion is the middle term. The mood of a syllogism is determined by the kinds of categorical statements of the major and minor premises and the conclusion. So, for example, a syllogism composed of all universal affirmative statements has the mood AAA. The figure of a syllogism is determined by the placement of the middle term in the premises. There are four figures: First Figure M-P S-M ∴ S-P Second Figure P-M S-M ∴ S-P Third Figure M-P M-S ∴ S-P Fourth Figure P-M M-S ∴ S-P The validity of syllogisms can be evaluated with Venn diagrams. Draw three intersecting circles representing the S, P, and M classes. To indicate that a subclass is empty, color it in. For example, to represent “All M are P,” color in the parts of M that don’t overlap with P. To indicate that there is at least one member of a class, draw an “x.” For example, to represent “Some M are P,” draw an “x” where the M and P circles overlap. If a syllogism is valid, the conclusion will be unambiguously represented on the diagram once the premises are represented. The validity of syllogisms can also be evaluated with the following three rules. • Rule 1: The middle term must be distributed in at least one premise. 2 • Rule 2: If a term is distributed in the conclusion, it must be distributed in a premise. • Rule 3: The number of negative premises must equal the number of negative conclusions. A term is distributed in a statement when the statement says something about every member of the class that the term denotes. Statement A E I O Terms Distributed Subject Subject and predicate None Predicate Sometimes an argument will have an unstated or implicit premise or conclusion. Such an argument is called an enthymeme. In order to evaluate an enthymeme, you need to fill in the missing steps before applying one of the methods for evaluating validity. When filling in missing steps, you should be guided by the principle of charity: try to find missing steps that make the argument valid and the premises plausible (this is not always possible). Other arguments consist of categorical syllogisms linked in a chain with subconclusions left implicit. Such an argument is called a sorities. In order to evaluate a sorities, you need to identify the subconclusions, then apply one of the methods for evaluating validity to each of the syllogisms in the chain. According to modern logic, which was developed in the 19th century by figures like George Boole, Augustus De Morgan, Charles Sanders Peirce, and Gottlob Frege, most of the logical relationships on the Aristotelian Square of Opposition do not hold. In modern logic, the four kinds of categorical statements are symbolized as follows: A ∀x(Sx → P x) or ¬∃x(Sx&¬P x) E ¬∃x(Sx&P x) or ∀x(Sx → ¬P x) I ∃x(Sx&P x) O ∃x(Sx&¬P x) In modern logic, A and E statements lack existential import. This means that subalternation fails. For example, “All unicorns have horns” does not imply “There is a unicorn.” A and E statements aren’t contraries: they are both true when the subject term is empty. I and O statements aren’t subcontraries: they are both false when the subject term is empty. The only relationship from the traditional square that is preserved on the modern square is that A and E and I and O statements are contradictories. 3 Paradoxes: Sorites, Liar, Curry 1 Sorites The phenomenon of vagueness has been taken to motivate a number of nonclassical, multi-valued logics. Consider the paradox of the heap. ¬P 1 One grain of sand is not a heap. ¬Pn → ¬P n+1 If n grains of sand don’t make a heap, neither do n+1 grains. P 1,000,000 A million grains of sand is a heap. The problem is that these statements seem true but are jointly inconsistent: the first two imply the denial of the third. One approach to resolving this problem is to hold that some instances of the second statement have an indeterminate truth-value. If these borderline cases are neither true nor false, but instead are truth-value gaps, the argument to the paradoxical conclusion is unsound. Strong Kleene Logic (K3 ) is a 3-valued where the third truth value # is interpreted as “neither true nor false.” This makes K3 paracomplete. The truth conditions for the connectives in K3 are given by the following truth tables, which are due to Stephen Kleene. ¬ 1 0 # 0 1 # ∧ 1 0 # 1 1 0 # 0 0 0 0 # # 0 # ∨ 1 0 # 1 1 1 1 0 1 0 # # 1 # # → 1 0 # 1 1 1 1 0 0 1 # # # 1 # The idea behind these truth tables is that a formula takes the value # when there is not enough classical information to determine a classical truth value. Validity in K3 is defined as an absence of counterexamples, but now there are two ways for a formula to be untrue. As expected, the Law of Excluded Middle (LEM) is not a tautology in K3 . In fact, there are no tautologies in K3 . To see why, notice that an interpretation on which all atomics take the value # will make all formulas #. K3 has trouble accounting for higher-order vagueness. Even if it denies that there’s a determinate threshold between when a vague term does and doesn’t apply, it assumes that there’s a determinate threshold between clear cases and borderline cases. The fuzzy logic Ł (for Jan Łukasiewicz) is a continuumvalued logic, i.e., a logic with a truth value corresponding to every real number in the closed interval between 0 and 1. The meanings of the connectives are defined as follows: v(¬ϕ) = 1 − v(ϕ) 1 v(ϕ ∧ ψ) = M in(ϕ, ψ) v(ϕ ∨ ψ) = M ax(ϕ, ψ) ( 1 if v(ϕ) ≤ v(ψ) v(ϕ → ψ) = 1 − (v(ϕ) − v(ψ)) otherwise Validity in Ł is then defined as follows: where Σ = (α1 , …, αn ), Σ ϕ iff for all v, M in(v(α1 ), …, v(αn )) ≤ v(ϕ). This is a generalization of the “no counterexamples” definition of validity, where a counterexample is a case where the least true premise is more true than the conclusion. Modus ponens is invalid in Ł. To see this, let v(ϕ) = 0.9 and v(ψ) = 0.6. Then v(ϕ → ψ) = 1 − (0.9 − 0.6) = 0.7. But then M in(v(ϕ → ψ), v(ϕ)) > v(ψ). This is to be expected, though, given that Ł is designed to avoid the Sorites Paradox: in Ł there can be a sorites chain with all true premises and a false conclusion. Note also that MP holds when all formulas take classical values. 2 Liar Consider the liar sentence. (L) This sentence is false. Is (L) true or false? Suppose it’s true. Then what it says is true, and it says it’s false, so it’s false. Suppose it’s false. Since it says it’s false, it’s true. So it’s true if and only if it’s false. Contradiction! Might (L) be neither true nor false? This paracomplete response gives rise to a revenge paradox. (G) This sentence is either false or gappy. Is (G) true, false, or gappy? Suppose it’s true. Then what it says is true, and it says it’s either false or gappy, so it’s either false or gappy. Suppose it’s either false or gappy. If it’s false, then it’s either false or gappy, so it’s true. If it’s gappy, then it’s either false or gappy, so it’s true. So it’s true if and only if it’s either false or gappy. Contradiction! One response to this is to bite the bullet and accept that (L) is a true contradiction. Some who accepts that there are true contradictions accepts dialetheism. Since contradictions explode in classical logic, dialetheists must be non-classical logicians to avoid triviality. A logic that rejects the principle of explosion and thus can tolerate inconsistency is paraconsistent. The Logic of Paradox (LP) is a 3-valued logic where # is interpreted as “both true and false” and the truth conditions for the connectives are given by the Kleene tables. The “no counterexamples” definition of validity can be 2 generalized as follows: validity is preservation of designated values. In SL and K3 , the set of designated values is {1}. In LP it is {1, #}. Modus ponens is invalid in LP. Let v(Q) = 0 and v(P ) = #. Then v(P → Q) = #, in which case both premises are designated but the conclusion is not. Every LP-tautology is an SL-tautology, since if ϕ is an LP-tautology there is no case that makes it false. Moreover, every SL-tautology is an LP-tautology. To see this, consider the contrapositive: if 2LP ϕ, then 2SL ϕ. By checking the truth tables, we can see that if an LP-interpretation makes ϕ false, it will still be false if we change all the #s to classical values. 3 Curry Consider the following sentence and argument. (C) If (C) is true, then everything is true. 1. (C) is true. (assume for CP) 2. “If (C) is true, then everything is true” is true. (substitution of identicals) 3. If (C) is true, then everything is true. (transparency of truth) 4. Everything is true. (MP) 5. If (C) is true, then everything is true. (CP) 6. “If (C) is true, then everything is true” is true. (transparency of truth) 7. (C) is true. (substitution of identicals) 8. Everything is true. (MP) This is a troubling argument, as it relies only on conditional proof, modus ponens, and some uncontroversial assumptions about truth. If (C) is wellformed, something has to go! 3 1 True/False Determine whether each statement is true or false. Explain your answer. (4 pts. each) 1. Dialetheists must be paraconsistent logicians. 2. Peirce’s Law ((P +Q) → P) + P is a tautology in K3. 3. Peirce’s Law is a tautology in LP. 4. “All S are P” and “No S are P” are contraries according to modern logic. 5. Every argument that is valid according to traditional categorical logic is valid in QL. 2 Short Answer 6. Briefly explain a problem for either the (a) paracomplete solution to the Sorites paradox or (b) the paracomplete solution to the Liar paradox. (10 pts.)

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