Preparatory Background: Logic for Metaphysics

Introduction to Arguments in Metaphysics

• Metaphysical Inquiry and the Search for Truth:

  • In metaphysics, as in other branches of philosophy and the natural sciences, the central goal is finding the truth regarding specific topics.

  • Truth cannot be attained through random guessing, blind speculation, or stabs in the dark.

  • Relying uncritically on unexamined personal beliefs or inherited views is inadequate for discovering truth (though common sense deserves a degree of respect).

  • Seeking out an authoritative group of elders to accept what they have said is not a valid method of discovery.

  • Instead, philosophy relies on arguments: series of statements rationally supporting a specific position so that individuals can evaluate for themselves why a position is correct.

  • Philosophers dedicate significant effort to seeking out and evaluating good arguments to ensure a trustworthy methodology for arriving at truth.

• Definitions and Core Components of an Argument:

  • Argument: A series of statements in which someone is presenting reasons in defense of some claim.

  • Distinction from ordinary usage: In philosophy, an argument is not two people yelling at each other, nor is it merely a single person's position or view.

  • Premise: A statement offered as part of an argument as a reason for accepting a certain claim.

  • Conclusion: The part of an argument that is being argued for, for which reasons are being offered.

• Fundamental Metaphysical Positions Defined:

  • Theism: The thesis that God exists.

  • Atheism: The thesis that God does not exist.

• Standard Introductory Metaphysical Arguments (Prose Form):

  • The Argument from Design (for Theism): The complexity and organization of the universe shows that it must have been designed. But there cannot be something which is designed without there being a designer. So, the universe must have a designer. Therefore, God exists.

  • The Problem of Evil (for Atheism): If there were a God, he would not allow evil to exist in this world. But there is evil in this world. Therefore, God does not exist.

• Numbered Premise Form:

  • Numbered Premise Form: A way of stating arguments so that each premise as well as the conclusion are given a number and presented each on their own line.

  • Function: Formally displaying arguments in numbered premise form allows easy reference back to specific premises, making it straightforward to analyze, criticize, or single out questionable premises needing further defense.

  • The Argument from Design in Numbered Premise Form:

  1. The complexity and organization of the universe shows that it must have been designed.

  2. But there cannot be something designed without there being a designer.

  3. So, the universe must have a designer.     Therefore,

  4. God exists.

  • The Problem of Evil in Numbered Premise Form:

  1. If there were a God, he would not allow evil to exist in this world.

  2. But there is evil in this world.     Therefore,

  3. God does not exist.

• Indicator Words for Premise and Conclusion Identification:

  • Premise Indicators: Words and phrases such as since, for, because, due to the fact that, etc.

  • Conclusion Indicators: Words and phrases such as hence, thus, so, therefore, it must be the case that, etc.

• Exercise 0.1: Recognizing Premises and Conclusions:

  • Passage A: "Americans should reject nationalized health care. This is because a system with nationalized health care is one in which someone's parents or baby will have to stand in front of the government's death panel for bureaucrats to decide whether they are worthy of health care. Any system like that is downright evil."

  • Reconstructed Numbered Premise Form:

    1. A system with nationalized health care is one in which someone's parents or baby will have to stand in front of the government's death panel for bureaucrats to decide whether they are worthy of health care.

    2. Any system like that is downright evil.       Therefore,

    3. Americans should reject nationalized health care.

  • Passage B: "If we don't nationalize health care there may be those, especially the young and healthy, who will take the risk and go without coverage. And if we don't nationalize health care, there will be companies that refuse to give their workers coverage. When people go without coverage, the rest of the country pays for them. So, if the young and the healthy or employees go without coverage, then the rest of the country will have to pay more in taxes. No one should have to pay more taxes. Therefore, we should nationalize health care."

  • Reconstructed Numbered Premise Form:

    1. If we don't nationalize health care, there may be those, especially the young and healthy, who will take the risk and go without coverage.

    2. If we don't nationalize health care, there will be companies that refuse to give their workers coverage.

    3. When people go without coverage, the rest of the country pays for them.

    4. So, if the young and the healthy or employees go without coverage, then the rest of the country will have to pay more in taxes.

    5. No one should have to pay more taxes.       Therefore,

    6. We should nationalize health care.

Concept of Validity in Deductive Arguments

• Technical Definition of Deductive Validity:

  • Deductively Valid: An argument is deductively valid when there is no possible way for the premises of the argument to all be true while its conclusion is false. The premises of the argument logically imply its conclusion.

  • Deductively Invalid: An argument is deductively invalid when it is possible for the premises of the argument to all be true while its conclusion is false.

• Core Attributes of Validity:

  • Validity is strictly a logical feature of an argument concerning the logical relation between premises and conclusion.

  • It does not depend on whether the premises are actually true in reality.

  • Validity asks: If the premises were true, would the conclusion have to be true?

  • Evaluating validity involves attempting to construct a coherent possible scenario (free of contradiction) where all premises are true and the conclusion is false.

  • Contradiction: Any sentence or statement of the form $P \text{ and } \text{not-}P$ (or $P \land \neg P$).

  • Counterexample: An example that shows an argument is invalid by providing a way in which the premises of the argument could be true while its conclusion is false; or an example that shows a statement is false by providing a way in which it could be false.

• Detailed Case Analyses of Validity:

  • Argument 1 (Universe & Aliens):

  1. If the universe were to end tomorrow, we would never know if there exist alien life forms.

  2. The universe will not end tomorrow.     Therefore,

  3. We will get to know if there exist alien life forms.

  • Assessment: Deductively Invalid.

  • Counterexample Scenario: Premise 1 is true (we currently have not discovered aliens). Premise 2 is true (the universe survives past tomorrow). Yet the conclusion can be false: we still might never learn if alien life forms exist because the universe ends next week instead of tomorrow. Since a coherent scenario exists where premises are true and conclusion is false, the argument is invalid.

  • Argument 2 (The Big Bang):

  1. All events have a cause.

  2. The Big Bang is an event.     Therefore,

  3. The Big Bang has a cause.

  • Assessment: Deductively Valid.

  • Scenario Analysis: If premises 1 and 2 are assumed to be true, the conclusion must be true. Denying the conclusion requires asserting that the Big Bang does not have a cause, which creates a direct contradiction (that the Big Bang both is an event requiring a cause and does not have a cause).

• Independence of Truth and Validity (Table 0.1 Breakdown):

  • Validity depends entirely on logical connection, not actual factual truth.

  • Theoretical combinations of premise truth and conclusion truth:

  1. Premises All Actually True, Conclusion Actually True:

     • Valid Example: 1. If Paris is in France, then it is in Europe. 2. Paris is in France. Therefore, 3. Paris is in Europe.

     • Invalid Example: 1. If Paris is in France, then it is in Europe. 2. Paris is in Europe. Therefore, 3. Paris is in France.

  2. Premises All Actually True, Conclusion Actually False:

     • Valid Example: Impossible. A valid argument can never have all true premises and a false conclusion.

     • Invalid Example: 1. If Paris is in Spain, then it is in Europe. 2. Paris is in Europe. Therefore, 3. Paris is in Spain.

  3. Premises At Least One False, Conclusion Actually True:

     • Valid Example: 1. If Paris is in China, then it is in Europe. 2. Paris is in China. Therefore, 3. Paris is in Europe.

     • Invalid Example: 1. If Paris is in France, then it is in Asia. 2. Paris is in Asia. Therefore, 3. Paris is in France.

  4. Premises At Least One False, Conclusion Actually False:

     • Valid Example: 1. If Paris is in Spain, then it is in Asia. 2. Paris is in Spain. Therefore, 3. Paris is in Asia.

     • Invalid Example: 1. If Paris is in Spain, then it is in Asia. 2. Paris is in Asia. Therefore, 3. Paris is in Spain.

• Exercise 0.2: Testing Arguments for Validity:

  • Argument A: "All lawyers like basketball. Barack Obama is a lawyer. Therefore, Barack Obama likes basketball." -> Valid (If all lawyers like basketball and Obama is a lawyer, he must like basketball).

  • Argument B: "Some snakes eat mice. Mice are mammals. Therefore, some snakes eat some mammals." -> Valid (If snakes eat mice and mice are mammals, those snakes eat mammals).

  • Argument C: "If the Pope is a bachelor, then the Pope lives in an apartment. The Dalai Lama is a bachelor. So, the Dalai Lama lives in an apartment." -> Invalid (Premise 1 makes a conditional claim about the Pope, not the Dalai Lama; the Dalai Lama being a bachelor does not satisfy the premise's condition).

  • Argument D: "All birds can fly. Penguins are birds. But penguins cannot fly. Therefore some birds can't fly." -> Valid (Premises assert penguins are birds that cannot fly, directly implying some birds cannot fly).

Concept of Soundness in Deductive Arguments

• Definition of Soundness:

  • Sound: An argument is sound just in case it has all true premises and is deductively valid.

  • Soundness provides a compelling reason to believe its conclusion because it establishes:

  1. The conclusion logically follows from the premises (validity).

  2. The premises are, as a matter of fact, actually true in reality.

• Evaluation of Previous Arguments for Soundness:

  • Argument 1 (Aliens): Unsound because it is deductively invalid.

  • Argument 2 (Big Bang): Valid, but unsound if one doubts or denies the actual factual truth of Premise 1 ("All events have a cause").

  • Argument 3 (Greece in the European Union):

  1. Greece is a member of the European Union. (Actually True)

  2. All members of the European Union lie north of the Equator. (Actually True)     Therefore,

  3. Greece lies north of the Equator. (Actually True)

  • Assessment: Sound (Satisfies validity and has all actually true premises).

• Standard Method for Evaluating Arguments in Metaphysics:

  1. Are all of the premises of this argument true? (If not, identify which are false and why).

  2. Does the conclusion follow from the premises? (Is the argument valid?).

  3. If the answers to both questions are 'yes,' then the argument is sound, giving compelling reason to accept the conclusion.

• Exercise 0.3: Assessing Exercise 0.2 Arguments for Soundness:

  • Argument A: Unsound (Premise 1 "All lawyers like basketball" is false).

  • Argument B: Sound (Deductively valid and both premises "Some snakes eat mice" and "Mice are mammals" are actually true).

  • Argument C: Unsound (Deductively invalid).

  • Argument D: Unsound (Premise 1 "All birds can fly" is false).

Methods for Criticizing Arguments

• Two Options for Rational Critique:

  1. Challenge one or more of the argument's premises on factual grounds.

  2. Challenge the logical validity of the argument's inferences.

• Structural Terminology for Complex Arguments:

  • Major Conclusion: The final conclusion of an argument.

  • Minor Conclusion: A statement that is argued for on the way to arguing for an argument's major conclusion.

• Worked Example: Criticizing the Argument from Design:

  1. The complexity and organization of the universe shows that it must have been designed.

  2. But there cannot be something which is designed without there being a designer.

  3. So, the universe must have a designer. (Minor conclusion)   Therefore,

  4. God exists. (Major conclusion)

  • Criticizing Premise 1: Argue that complexity and organization have no bearing on design, or argue that high complexity implies a lack of a designer (e.g., a designer might prefer a simpler universe).

  • Criticizing Premise 2: Argue that being designed does not imply a designer (debating the definition of design).

  • Evaluating Inference 1 (1 & 2 to 3): Valid step ($1$ and $2$ logically imply $3$).

  • Evaluating Inference 2 (3 to 4): Invalid step! Assuming step 3 is true (the universe has a designer) does not contradict step 4 being false (God does not exist). The universe could have been designed by a non-divine entity (e.g., an alien or committee), providing a counterexample to Inference 2.

• Exercise 0.4: Criticizing the Cosmological Argument:

  • Argument:

  1. Everything that happens in the universe must have a cause.

  2. Nothing can be a cause of itself.

  3. So, there must exist a first cause.

  4. If there is a first cause, then this first cause is God.

  5. Therefore, God exists.

  • Analysis:

  • Independent Premises: 1, 2, and 4.

  • Minor Conclusion: 3 (derived from 1 and 2).

  • Major Conclusion: 5 (derived from 3 and 4).

  • Premise Skepticism: Premise 1 can be challenged via quantum mechanics (uncaused events). Premise 4 assumes without proof that a first cause must be God.

  • Validity Analysis: The move from 1 and 2 to 3 is invalid unless an additional premise ruling out infinite causal regresses is added. The move from 3 and 4 to 5 is valid ($A \land (A \implies B) \implies B$).

  • Soundness: Unsound due to unstated assumptions and questionable independent premises.

The Principle of Charity and Enthymemes

• Principle of Charity:

  • Principle of Charity: A convention of philosophical debate to, when reasonable, try to interpret one's opponent's claims as true and her arguments as valid.

  • When an opponent's claim can be interpreted in multiple ways, choose the interpretation that makes the claim true rather than obviously false.

• Enthymemes:

  • Enthymeme: An argument that is incomplete as stated and invalid, although it is easy to supply the missing premises that the argument would need to be valid.

  • Authors omit premises because they are considered too obvious to state, or to avoid boring or insulting the reader's intelligence.

  • The principle of charity compels readers to supply obvious intended premises to reconstruct the argument into its valid form.

• Worked Example: Argument against Abortion:

  • Prose: "Anytime one ends the life of a person, it is murder. Abortion ends the life of a fetus. So, abortion is murder. Therefore, abortion is wrong."

  • Initial Form:

  1. Anytime one ends the life of a person, it is murder.

  2. Abortion ends the life of a fetus.

  3. So, abortion is murder. (Minor conclusion)     Therefore,

  4. Abortion is wrong. (Major conclusion)

  • Critique of Validity as Stated:

  • Inference 1 (1 & 2 to 3): Invalid, because 1 refers to a person, while 2 refers to a fetus. Missing premise: "A fetus is a person."

  • Inference 2 (3 to 4): Invalid, because 3 claims abortion is murder, while 4 claims it is wrong. Missing premise: "Murder is wrong."

  • Charitable Reconstruction:

  1. Anytime one ends the life of a person, it is murder.

  2. Abortion ends the life of a fetus.     *2.5 A fetus is a person. (Fixes Inference 1 validity)

  3. So, abortion is murder.     *3.5 Murder is wrong. (Fixes Inference 2 validity)     Therefore,

  4. Abortion is wrong.

  • Note: Supplying premises to restore validity does not force acceptance of the argument; premises 2.5 and 3.5 are now explicit targets for evaluation and rational disagreement.

• Exercise 0.5: Supplying Missing Premises for Anti-Thales Arguments:

  • Context: Thales argued "Everything is water."

  • Argument A: "There is no water on Saturn. Therefore, not everything is water."

  • Missing Premise: Saturn exists (or Saturn is something).

  • Argument B: "There were things that existed in the first seconds immediately after the Big Bang. Water did not come into being until hundreds of thousands of years after the Big Bang. So, not everything is water."

  • Missing Premise: If something existed before water came into being, that thing was not water.

Propositional Logic

• Need for Formal Logic:

  • When arguments have complex premises or inferences, formal logic systems provide rigorous syntactic methods for evaluating validity independent of specific subject matter.

• Argument Form:

  • Argument Form: The structural shape or pattern an argument has, independent of its specific subject matter.

• Propositional Logic Mechanics:

  • Uses single letters to represent basic or atomic propositions (e.g., $H$: Sally is human, $M$: Sally is mortal, $D$: Determinism is true, $N$: No one has free will).

  • Logical Connectives: Symbols used to build complex propositions out of simpler ones.

• Table of Logical Connectives (Table 0.2):

  • And (Conjunction): Symbolized as $\land$ or & (e.g., $H \land M$).

  • Or (Inclusive Disjunction): Symbolized as $\lor$ (e.g., $H \lor M$).

  • If… then (Conditional): Symbolized as $\implies$ or $\supset$ (e.g., $H \implies M$).

  • Not (Negation): Symbolized as $\neg$ or $\sim$ (e.g., $\neg H$).

  • If and only if (Biconditional): Symbolized as $\equiv$ or $\iff$ (e.g., $H \equiv M$).

• Symbolic Representation of Arguments:

  • Argument 4:

  1. $H \implies M$

  2. $H$     Therefore,

  3. $M$

  • Argument 5:

  1. $D \implies N$

  2. $D$     Therefore,

  3. $N$

• Standard Valid Argument Forms in Propositional Logic:

  • Modus Ponens:

  1. $A \implies B$

  2. $A$     Therefore,

  3. $B$

  • Modus Tollens:

  1. $A \implies B$

  2. $\neg B$     Therefore,

  3. $\neg A$

  • Simplification:

  • Form 1: 1. $A \land B$ Therefore, 2. $A$

  • Form 2: 1. $A \land B$ Therefore, 2. $B$

  • Disjunctive Syllogism:

  • Form 1: 1. $A \lor B$, 2. $\neg A$, Therefore, 3. $B$

  • Form 2: 1. $A \lor B$, 2. $\neg B$, Therefore, 3. $A$

• Exercise 0.6: Translations in Propositional Logic:

  • Key: $I$: The universe is infinite; $U$: The future is unknown; $O$: The future is open; $F$: Humans have free will.

  • A. "Either the universe is infinite or the universe is not infinite." -> $I \lor \neg I$

  • B. "If humans have free will and the future is open, then the future is unknown." -> $(F \land O) \implies U$

  • C. "Humans have free will if and only if the future is open." -> $F \equiv O$

  • D. "It is not the case that either the universe is infinite or the future is open." -> $\neg (I \lor O)$

• Exercise 0.7: Recognizing Valid Argument Forms:

  • A. "Either the future is open or the universe is not infinite. The future is not open. Therefore, the universe is not infinite."

  • Symbolization: $O \lor \neg I$, $\neg O$, Therefore $\neg I$

  • Form: Disjunctive Syllogism.

  • B. "If humans have free will, then the future is open. The future is not open. Therefore, humans don't have free will."

  • Symbolization: $F \implies O$, $\neg O$, Therefore $\neg F$

  • Form: Modus Tollens.

  • C. "If humans have free will, then the future is open. The future is open. Therefore, humans have free will."

  • Symbolization: $F \implies O$, $O$, Therefore $F$

  • Form: None of the above (Fallacy of Affirming the Consequent).

  • D. "If humans have free will, then the future is open. Humans have free will. Therefore, the future is open."

  • Symbolization: $F \implies O$, $F$, Therefore $O$

  • Form: Modus Ponens.

  • E. "The future is open and it is unknown. So, the future is unknown."

  • Symbolization: $O \land U$, Therefore $U$

  • Form: Simplification.

First-Order Predicate Logic

• Limitations of Propositional Logic:

  • Argument 6: 1. Alex respects everyone who loves the Beatles. 2. Betty loves the Beatles. Therefore, 3. Alex respects Betty.

  • Symbolized in propositional logic: 1. $A$, 2. $B$, Therefore 3. $C$ (invalid form).

  • Requires predicate logic to reveal internal subject-predicate structures and quantification.

• Components of First-Order Predicate Logic:

  • Predicates: Capital letters ($F, G, H, T, P, A, M$) placed before subject terms.

  • 1-place predicate: $T s$ ("Shaq is tall", where $T$ = is tall, $s$ = Shaq).

  • 2-place predicate: $A s l$ ("Shaq admires Ludwig", where $A$ = admires, $s$ = Shaq, $l$ = Ludwig).

  • 4-place predicate: $M p b k c$ ("Professor Plum murdered Mr. Body in the kitchen using the candlestick").

  • Names / Constants: Lowercase letters from the beginning of the alphabet ($a, b, c, \dots$) representing specific named objects.

  • Variables: Lowercase letters from the end of the alphabet ($x, y, z, w, u, v, \dots$) standing in for values in a domain.

  • Existential Quantifier ($\exists$): Combined with a variable to state that something exists that satisfies a condition.

  • Example: $\exists x T x$ ("There exists an $x$ such that $x$ is tall" / "Somebody is tall").

  • Example: $\exists x M x b k c$ ("Somebody murdered Mr. Body in the kitchen using the candlestick").

  • Example: $\exists x (T x \land F x)$ ("There is something that is tall and friendly").

  • Multi-variable quantifier: $\exists x \exists y ((C x \land D y) \land L x y)$ ("Some cats love some dogs").

  • Universal Quantifier ($\forall$): Combined with a variable to state that everything in the domain satisfies a condition.

  • Example: $\forall x H x$ ("Everyone is happy").

  • Categorical statement "All philosophers are happy": $\forall x (P x \implies H x)$.

  • Contrast with