Logic 2

Primer on Logic

E. M. Segal:
page 2
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II. Categorical Logic:

Categorical propositions: Propositions of a subject-predicate type connected by a copula (is, are). The subject is an individual (e.g. Socrates, John, this book) or a category (eg. men, elephants, green things), and the predicate is a category. Propositions define a relationship between the subject and predicate.
Types of categorical propositions (Aristotelian): a) Universal affirmative, (e.g. All men are mortal; All water is wet)
b) Universal negative, (e.g. No men are mortal; No apple is sweet)
c) Particular affirmative, (e.g. Some men are mortal; Some paperclips are plastic), and
d) Particular negative, (e.g. Some men are not mortal; Some talk show hosts are not racists) In studying Categorical Logic, one can substitute variables (e.g. A, B, S, P) for the categories.
In a syllogism the two premises each contain a common category (the middle term) and a unique one. The conclusion contains the unique categories from the premises. (e.g., Some A are B, No B are C, Therefore No A are C). Some syllogisms are valid and some (such as this example) are invalid.

Syllogisms

III. Relational terms and linear reasoning. Relations: There are relations in addition to a copula. Aristotle did not consider these in his (categorical) logic. One can specify various relations between the subject and predicate terms (e.g. greater than, equal to, not equal to, hit). The relationship has content; different relations imply different logical properties.

Linear Problems
If a relation is between two objects, it is called a binary relation, and the objects are referred to as arguments of the relation. In the proposition A is greater than B, (Formally this can be written aGb, or Ga,b) 'a' and 'b' are the 'arguments' of the binary relation greater than. [There are higher order relations, e.g., give, sell; A gave B to C, John gave the book to Mary, which have more than two arguments.]
Linear or ordered relations. These are relations that can be applied to syllogism-like arguments. These syllogisms contain relations that are binary, transitive, asymmetrical, and nonreflexive (e.g. greater than, larger than, faster than).
Transitive (def.): if aRb, and bRc, then aRc.
Asymmetrical (def.): if aRb, it is not the case that bRa.
Nonreflexive (def.): it is not the case that aRa.
Intransitive (e.g. next to, sees): aRb and bRc does not imply aRc.
Symmetrical (e.g. similar to, near, equal to, not equal to), (def.): If aRb, then bRa.
Reflexive (e.g. identical to, equal to) (def.): aRa.

IV. Conditional Reasoning and Symbolic Logic. Symbolic logic refers to formal reasoning systems.
Reasoning principles are rules of inference based on the form of the representations.
Well-formed formulas: A notational system will define symbol strings that represent propositions. Other strings cannot formally represent propositions.
In Elementary propositional or sentential logic,
Simple propositions are not formally analyzed. They are often represented by a single letter, (e.g., A, B). Compound propositions are formed by combining simple propositions using "logical terms" to connect propositions with one another.
Logical terms include not, which relates to a single proposition (e.g., ~A. If A is The sky is blue, ~A (not A) is It is not the case that the sky is blue , or The sky is not blue). Some other logical connectives are: and, or, if then, and if and only if. These relate two propositions (e.g., A&B, AÚ B, AÉ B, Aº B, read A and B, A or B, If A then B, and A if and only if B, respectively.
In many logics these connectives are truth functional; that is, they assign truth values to the compound propositions only as a function of the truth values of the simpler propositions they contain.
Law of the excluded middle --All propositions are either true or false. This means we must assign a truth value to any sentence if we are to analyze an argument using this logic. Most (if not all) logics of this type obey this law.
Truth functional logic --Truth of composite propositions is determined by the truth of its component propositions
Rules of inference: Principles by which one can verify the validity of conclusions from premises. Some rules of inference in propositional logic include Modus Ponens (If A É B, and A, conclude B), Modus Tollens (If AÉ B, and ~B, conclude ~A).
Hypothetical Syllogism
(If AÉ B, and B É C, conclude AÉ C)
Tautology: a logically true sentence--A sentence which has to be true by form alone, i.e. whatever the truth of its component propositions, it is true. E.g. The Yankees will either win the World Series this year or they will not win the World Series this year.
Empirical sentence--A sentence representing a proposition that may be true or false. E.g. It is raining in Beijing.
Logically false sentence --A sentence that must be false as determined by its form alone. E.g. It is raining and it is not raining in Beijing. (Joe gave 50% of the pie he baked to each of three friends. is a false sentence and is necessarily false, but its falseness depends on the meanings of the terms 50% and three, not on the sentence form, therefore it is not false because of its form.)
Truth table— The truth of the composite propositions in certain logics are determined entirely by the truth of its component propositions. A way to represent truth of composite propositions as a function of the truth of its component propositions. These may represent whether a composite proposition is a tautology, an empirical sentence, or a logically false one. Truth tables may also be used to evaluate whether an argument is valid or not. In a truth table the top row contains the elementary propositions and the set of composite propositions that comprise the proposition under analysis. For each of the remaining rows of the table the truth of the elementary propositions are assigned systematically such that each row has a different combination of truth values. The truth of the composite propositions are then determined by the rules of assignment, that is what their truth value would be if the elementary propositions had the truth of their assignment for that row. One might envision a given row as a possible world where the truths af the elementary propositions are as assigned. In that world, in a truth functional logic, each composite proposition has the truth of its rule-governed assignment. Some elementary examples: P, Q, and R stand for elementary propositions; ~ means 'not'; & means 'and,' Ú means 'or,' É means 'if…then…', and º means 'if and only if'. In the tables T is the semantic assignment of 'true' and F the assignment of 'false.'
Table 1

P	~P
T	F
F	T

Table 1 asserts if P is true, not P is false (row 1); and if P is false, not P is true (row 2).
Table 2

P	Q	P&Q	PÚ Q	P É Q	Pº Q
T	T	T	T	T	T
T	F	F	T	F	F
F	T	F	T	T	F
F	F	F	F	T	T

Table 2 represents several of the logical connectives. In it you can see that according to this simple logic a. P&Q is true if P is true and Q is true, and is false otherwise (From columns 1, 2, and 3).
b. PÚ Q is true if P is true, if Q is true, or if both P and Q are true. It is false only when both arguments, P and Q are false.
c. P É Q is false only when P is true and Q is false. It is true whenever P is false. These latter cases are called "counterfactual conditionals and are the basis for much discussion and controversy.
d. Pº Q is true if the truth values of P and Q are the same and false if they are different.
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