# Logic. A Primer with Addendum

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1 Logic A Primer with Addendum

2 The Currency of Philosophy Philosophy trades in arguments. An argument is a set of propositions some one of which is intended to be warranted or entailed by the others. The one supported is the conclusion. Those offered in support are the premisses One can look for ordinary conclusion markers: Therefore, p ; Consequently, p ; It follows that p ; So, p ; Ergo, p.

3 Propositions I The units of arguments are propositions. A proposition is an assertion (typically) expressed by a declarative sentence. A proposition may provisionally be thought of as the meaning of a declarative sentence. It is also, for these reasons, a mind- and language-independent entity which has its truth conditions essentially. Generally speaking, declarative sentences express propositions; they are truth-evaluable; they typically report beliefs. More generally, where s is a declarative sentence, it is always possible to ask, sensibly: Is it true that s? So, some examples of sentences which are not declarative: Commands Invitations Questions

4 Propositions II We will assume bi-valence: proposition (or a declarative sentence presenting it) is either true or false. So, every proposition has a truth value. Further, no proposition is ever both true and false. Tricky for: Vacuous reference: The first female president of Notre Dame was born in Quebec. Complex sentences: Thankfully, he no longer beats his wife.

5 An Extra-logical Feature of Propositions Propositions are made true by truth-makers, like facts. (The expression real facts is pleonastic; the expression alternative fact is oxymoronic.) A proposition is true when what it claims about the world is so; it is false otherwise. Beliefs are true when they have as their contents true propositions. So, truth-makers make beliefs true. Thus, the world makes true beliefs true; true beliefs do not make the world the way the world is. N.b.: these are extra-logical features of propositions and can be (indeed, must be) be set aside in the study of logic. Correct rules of logic are indeed useful. It staggers the imagination to picture a world in which they have no authority. But their utility derives from their correctness, not the other way around. Joel Feinberg

6 Relations between Propositions Logic focuses on the relations between propositions. The relations of immediate concern to us are: Consistency Entailment Warrant

7 Consistency A set of propositions is consistent if and only if (iff) there exists some possible situation in which they can be true together. Otherwise they form an inconsistent set, or are inconsistent. Two propositions are contradictory iff it is the case that if one is true, the other is false; or, equivalently, if one is false, the other is true. So, e.g.: (i) The ball is red; and (ii) It is not the case that the ball is red. Two propositions are contraries iff they can be false together but cannot be true together. So, e.g.: (i) The ball is (altogether) red; and (ii) The ball is (altogether) green.

8 Entailment and Warrant Consider again our definition of argument: an argument is a set of propositions some one of which is intended to be warranted or entailed by the others. Two kinds of arguments: deductive and inductive a deductive argument is an argument where one proposition is represented as being entailed by some other propositions. an inductive argument is an argument where one proposition is represented as being warranted by some other propositions.

9 Entailment and Validity A set of premisses entails its conclusion iff their being true requires the truth of the conclusion. A valid argument is an argument such that its premisses entail its conclusion. An argument is valid if it has the following feature: if its premisses are true, then its conclusion cannot fail to be true. N.b. This does not say that a argument is valid only if it has true premisses. In fact, it says nothing at all about whether the premisses are or are not true. Validity is a matter of form or structure, rather than of content.

10 Some Valid Arguments If it is raining, then the field is wet; it is raining; so, the field is wet. If Mickey is a mouse, then he eats cheese; Mickey is a mouse; consequently, Mickey eats cheese. If the Republicans win the next election, then the problem of global warming will be ignored; unfortunately, they will win; so, the problem of global warming will be ignored. If at least some Buddhists are Republicans, then at least some Republicans wear saffron robes; some Buddhists are definitely Republicans; so, at least some Republicans wear saffron robes. If Lassie is a fish, then some fish bark like dogs; Lassie is a fish; so, some fish bark like dogs.

11 Some Invalid Arguments If you are a status-conscious bourgeois dog, then you own a Jaguar. You do own a Jaguar. So, I guess you are a status-conscious bourgeois dog. Some birds are animals with webbed feet. Some animals with webbed feet can swim beneath the surface of the sea. So, at least some birds can swim beneath the surface of the sea. If it s raining, then the sidewalks are wet. They re very wet; so, it must be raining.

12 Moving from Natural Language to Canonical Form Researchers have shown that relaxing activities promote health. Many people find smoking both enjoyable and relaxing. Same again with wine, at least in moderation. So, in its own humble way, smoking, contrary to what some have claimed, actually promotes health. Conclusion? Premisses? Valid or Invalid?

13 Some Canonical Forms of Deductive Arguments Modus Ponens: if p, then q; p; therefore q. Modus Tollens: if p, then q; not-q; therefore not p. N.b. these both derive from the same contention, viz. that p is sufficient for q. N.b. that this cuts two ways: whenever p is sufficient for q, then q is necessary for p. e.g. If there is fire, then oxygen is present. Or, equivalently, there is fire only if oxygen is present. Thus, one can conclude on the basis of the same conditional if p then q (if there is fire, then oxygen is present): Since there is fire, oxygen is present. (This is the basis of MP) Since there is no oxygen present, there is no fire. (This is the basis of MT)

14 Argument Chains 1. If the Democrats win the next election, then the economy will do well. 2. If the economy does well, then the environment will suffer. 3. If the environment suffers, then the poor will suffer inordinately. 4. If the poor suffer inordinately, there will be a revolution. 5. So, if the Democrats win the next election, there will be a revolution.

15 Again 1. If God exists, then an omnipotent, omnibenevolent, omniscient being exists. 2. If an omnipotent being exists, then she can rid the universe of all the evil of which she is aware. 3. If an omnibenevolent being exists, then she will want to rid the world of all evil of which she is aware. 4. If an omniscient being exists, then she is aware of all the evil that there is in the universe. 5. It follows that if there is a being who can rid the world of all the evil of which she is aware, and she is aware of all the evil there is, and who wants to rid the universe of all the evil she can, then there is no evil. 6. So, if God exists, there is no evil. 7. There is evil. 8. So, there is no God.

16 The Gold Standard A sound argument is a valid argument with all true premisses. We can test for validity without knowing the truth values of the premisses of an argument. To judge an argument for soundness, we must first determine validity and then assess for truth.

17 Three Common Fallacies Petitio Principii (Begging the Question): implicitly tandem arguments where the conclusion of the first is called to support a conclusion of the second, when the conclusion of the second was a premiss in the first. (i) The Bible is the word of God; obviously, whatever God says is true; so, whatever the Bible says is true. You ask: How do I know that the Bible is the word of God? (ii) Well, I ll tell you: we ve just seen that whatever the Bible says is true; and the Bible itself tells us that Bible is the word of God; so, it follows that it is true that the Bible is the word of God. Circular Reasoning: an argument whose conclusion is also one of its own premisses. Ad hominem In business, it sometimes pays to maximize profits by skirting the laws when possible. Of course, sometimes one is caught and sometimes not and when one is caught, one is required to pay huge fines. Still, as long as one is careful and not too flagrant, the probabilities that one will be caught are acceptably low. So, you see, in business under capitalism, it sometimes really does pay to maximize profits by skirting the law now and again. Professor Smedley claims that if the Republicans win the next election, the poor will suffer inordinately. You know what though? He s an idiot. You know what else he said? He said that capitalism is doomed to suffocate under its own weight within the next fifty years. You know what else? He s a hypocrite, too: he drives a Jaguar. A big, fat bourgeois Jaguar. There s no reason to believe that the poor will suffer at all if the Republicans win. Maybe they ll all get rich and be able to afford Jaguars, just like Professor Smedley Idiot. Hypocrite.

18 Addendum: Two Important Distinctions The Necessary/Contingent Distinction The A Priori/A Posteriori Distinction

19 Necessary/Contingent The Character of this Distinction This is a metaphysical distinction, in the domain of entities and, derivatively, propositions which characterize them. The Distinction A proposition is necessarily true/false iff it is true/false and could not possibly have been false/true (or as Leibniz suggests, a proposition is necessary iff it is true in all possible worlds). A proposition is contingent iff it is true in some possible worlds and false in others.

20 The A Priori/A Posteriori Distinction The Character of this Distinction This is an epistemological distinction, in the domain of knowledge. The Distinction One has a priori knowledge that p iff one knows p by reason or conceptual resources alone (that is, the extra-mental world makes no contribution to the justification of p). A posteriori knowledge is knowledge that is not a priori. N.b. this is a point about justification, not genesis.

21 A Co-extensivity Hypothesis Although drawn from different domains, these distinctions are co-extensive: p is known a priori iff p is necessary p is known a posteriori iff p is contingent

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