Monday, May 28, 2012

Introductory Logic, Part 1

Home  >  Philosophy  >  Logic

This is part 1 of my series on logic and critical thinking.
  1. Introductory Logic, Part 1—Introducing both logic in general (such as the difference between a deductive and inductive argument) and propositional logic in particular
  2. Introductory Logic, Part 2—More propositional logic
  3. Introductory Logic, Part 3—A defense of the material conditional

Bait

This post begins my series on logic and critical thinking. First a little bait for both atheists and theists. For atheists, here’s an argument from evil:
  1. If God exists, then gratuitous evil does not exist (we can define gratuitous evil as evil from which no greater good results from its existence).
  2. Gratuitous evil does exist.
  3. Therefore, God does not exist.
In logic, an argument uses a series of statements called premises (like lines 1 and 2 above) that attempts to support a conclusion (like line 3 above). Whether the premises in the above argument are true or not, one can use symbolic logic to prove that the conclusion follows from the premises, and this article will show how an atheist can do that. For theists, here’s a moral argument:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
Whether the premises are true or not, one can use symbolic logic to prove that the conclusion follows from the premises, and this article will show how a theist can do that.

And now that I have your attention...

In my previous blog entry I used symbolic logic to help analyze the “Could an omnipotent being create an immovable stone?” paradox, which helps illustrate the importance of logical and analytical thinking. Hopefully if it’s one thing theists and atheists can agree on it is the importance of logic as well as analytical and critical thinking. Part of what inspired this is seeing some atheists claim that a conclusion didn’t follow from premises when anyone with basic training in logic would’ve seen that the conclusion did indeed follow. To be fair, there are no doubt some theists out there who have likewise made embarrassing logical errors.

In this article I’ll explain the difference between deductive and inductive arguments as well as introducing some easy-to-learn symbolic logic. Why symbolic logic? Learning symbolic logic significantly helps one learn to think logically and for getting an intuitive feel for how an argument’s conclusion can follow logically from the premises. Symbolic logic is also great for introducing various rules of logic.

Deductive Arguments

A deductive argument tries to show that it’s logically impossible (i.e. self-contradictory, like a married bachelor) for the argument to have true premises and a false conclusion, and thus that the conclusion follows from the premises by the rules of logic. If it’s logically impossible for an argument to have true premises and a false conclusion the argument is deductively valid or valid. An example of a deductively valid argument:
  1. If it is raining, then my car is wet.
  2. It is raining.
  3. Therefore, my car is wet.
The above example uses a famous rule of logic called modus ponens which has this structure:
  1. If P, then Q
  2. P
  3. Therefore, Q.
Another famous rule of logic is called modus tollens where “not-Q” means “Q is false.”
  1. If P, then Q
  2. Not-Q
  3. Therefore, not-P.
An argument is deductively invalid or invalid if it is not deductively valid. An example of an invalid argument:
  1. If it is raining, then my car is wet.
  2. My car is wet.
  3. Therefore, it is raining.
In logic lingo, a deductively valid argument with all its premises being true is called a sound argument. And since a valid argument having true premises guarantees the truth of its conclusion, a sound deductive argument has a true conclusion.

Inductive Arguments

The symbolic logic I’ll deal with in this article will deal with deductive arguments, but it’s worth talking about another type of argument: the inductive argument. Whereas a deductive argument attempts to have premises that (if true) guarantee the truth of the conclusion, an inductive argument has premises that are intended to make probable the conclusion without guaranteeing it. There are two types of inductive arguments. One type is called an enumerative inductive argument and it’s basically the type of reasoning scientists use to establish physical laws, generalizing a truth about past experiences to future experiences. An example:
  • We’ve seen the law of gravity work in conditions C1.
  • We’ve seen the law of gravity work in conditions C2.
  • We’ve seen the law of gravity work in conditions C3.
  • We’ve seen the law of gravity work in conditions Cn.
  • Therefore, (probably) the law of gravity holds true in all physical situations.
Some enumerative induction arguments take on the following form:
  • All observed A’s are B’s.
  • x is an A.
  • Therefore, probably, x is a B.
Another example of enumerative inductive logic:
  • We’ve sampled a thousand animals of this species in this forest and three-fifths of them have black fur.
  • Therefore, probably three-fifths of the animals of this species in this forest have black fur.
Terminology warning: some people define “induction” to mean only the first sort of enumerative induction rather than including the “population sampling” sort. Here I’m using the broader definition of inductive reasoning as an argument with premises that are intended to make the conclusion more probable.

Another kind of inductive argument is called inference to the best explanation. This is where out of a pool of live options, a certain explanation is selected as the best and most rational one. When comparing competing theories, some factors include but are not limited to the following:
  • Explanatory power refers to making the data probable. A theory having more explanatory power than another means it makes the data more probable (e.g. given our background knowledge that doesn’t include data D, it’s more likely that we’d see data D on theory #1 than on theory #2).
  • Explanatory scope refers to how many things a theory explains. A theory has more explanatory scope than another if it explains more things.
  • Simplicity utilizes a number of factors, but among them is Ockham’s razor (also spelled Occam’s razor) that says to not multiply explanatory entities beyond necessity. All other relevant factors held constant, simpler explanations are preferred over more complex ones.
Inference to the best explanation is also called abductive reasoning or abduction. Some writers define the terms somewhat differently, defining induction to mean just the first case of enumerative induction and use the term “abduction” to distinguish between the more narrowly defined “induction.” A less ambiguous catch-all term for arguments that are not deductive is nondeductive. In this article, I’ll consider “inference to the best explanation” as a sort of inductive reasoning.

Inductive arguments are by nature invalid, but if the premises of an inductive argument make the conclusion highly probable it is said to be an inductively strong argument (another term I’ve seen used for this is nondeductively valid). The strength of an inductive argument comes in varying degrees, from making the conclusion highly probable to not at all probable (the latter case results in the argument being inductively weak). A strong inductive argument with true premises is called a cogent argument.

Symbolic Logic: Some Symbols

In this article I’ll talk about some symbolic with regards to deductive argument. Some symbols:

ConnectiveSymbolic
Logic
English
Meaning
Notes
∧ (conjunction)p ∧ qp and qThe p and q parts are called conjuncts.
∨ (disjunction)p ∨ qp or qThe p and q parts are called disjuncts.
→ (conditional)p → qIf p, then qThe p part is called the antecedent and the q part is called the consequent. Sometimes p → q is read as “p implies q.”
↔ (biconditional)p ↔ qp, if and only if qThis means the same thing as “p → q and q → p.”
¬ (negation)¬pNot-pThe negation of P is ¬P, and ¬P means “not-P” or “P is false.”


As suggested in the above table, the symbols →, ¬, ∨, and ∧ are called connectives. It’s a somewhat misleading name since ¬ doesn’t connect propositions even though the other three connectives do. Still, it’s a popular label used by a lot of logic textbooks. The type of symbolic logic I’ll discuss here uses letters to represent propositions. While the terminology varies among writers, I’ll call a single letter a simple statement and one more or more simple statements with one or more connectives is called a compound statement. For example, “¬P” and “A ∧ B” are compound statements.

Oddly enough, the symbols for the connectives are not as standard as one might think (though in my experience the disjunction operation being symbolized as ∨ is pretty standard). Below are some alternate forms:

OperationExampleAlternate Forms
conjunctionA ∧ BA & B, A • B, AB
implicationA → BA ⊃ B
material equivalenceA ↔ BA ≡ B
negation¬A~A, –A, A


In the system of logic I’m using (called propositional logic), simple statements are true or false. What makes a compound statement true? Here’s a summary of how the connectives work in making a compound statement true/false:

Type of
connective
EnglishSymbolic
Logic
When it’s true/false
Conjunctionp and qp ∧ qTrue if both are true; otherwise false
Disjunctionp or qp ∨ qFalse if both are false; otherwise true
ConditionalIf p, then qp → qFalse if p is true and q is false; otherwise true
Biconditionalp, if and only if qp ↔ qTrue if both have the same truth-value (i.e. both are true or both are false); otherwise false
NegationNot-p¬pTrue if p is false; false if p is true


We can also use truth tables like those below to express what makes a compound statement true/false, where T symbolizes true and F symbolizes false.

Truth Table of Connectives

pq   pq   pq   pq   ¬p
TTT   TTT   TTT   TTT   TF
TFF   TTF   TFF   TFF   FF
FFT   FTT   FTT   FFT     
FFF   FFF   FTF   FFF     


The disjunction (p ∨ q) being used here is an inclusive or, i.e. true if at least one of the disjuncts is true (otherwise false). This is in contrast to an exclusive or, which is true if and only if exactly one disjunct is true.

The type of conditional (p → q) being used here is called a material conditional. A material conditional is equivalent to “It is not the case that the antecedent (p) is true and the consequent (q) is false,” such that the only way for a material conditional to be false is for it to have a true antecedent with a false consequent, as the truth table for it illustrates. When you look at the truth table, a material conditional might seem like a pretty weak claim (in the sense that it doesn’t claim very much), since the antecedent and consequent don’t even have to be related to each other for a material conditional to be true. Thus, “If there is a married bachelor, then Minnesota is awesome” constitutes a true material conditional since it is not the case that we have a true antecedent (there is a married bachelor) with a false consequent (Minnesota is awesome). But it turns out that a material conditional is enough for modus ponens and modus tollens to be valid rules of inference, since in a true material conditional if the antecedent is true, then the consequent is true as well.

It’s worth noting, however briefly, how English can be translated into symbolic logic.

EnglishSymbolic Logic
p is a sufficient condition for qp → q
q is a necessary condition for p
q, if p
p only if q


Notice that the “conversion” from English to the material conditional doesn’t necessarily work so well the other way (material conditional to English), e.g. p → q doesn’t necessarily talk about necessary or sufficient conditions; it just says it isn’t the case that p is true and q is false. On the other hand, if p is a sufficient condition for q, then it follows that p → q.

EnglishSymbolic Logic
p is a necessary and sufficient condition for qp ↔ q
p if and only if q
p just in case q


Odd as it may seem, in the philosophical literature “p just in case q” often means p ↔ q.

Order of Operations and Logical Terminology

Logic uses parentheses in a sort of “order of operations” (if you remember your algebra) to determine what to calculate first. For example, suppose we had this:

(A ∨ B) ∧ C

So if we were going for an “inside to outside” strategy of creating a truth table we’d first determine the truth-value (i.e. whether it’s true/false) of (A ∨ B) as opposed to checking whether B ∧ C is true. If we need more groupings inside other groupings, we first use brackets [] and then braces {} like so:

¬{[(A ∨ B) ∧ C] → D}

Another note: the order of operations is that negation is done to the immediate thing the ¬ is connected to. For example, this:

¬P ∨ C

Should be viewed as this:

(¬P) ∨ C

Rather than this:

¬(P ∨ C)

Two statements are said to be logically equivalent if it is logically impossible for one to be true when the other is false. So in propositional logic, two statements are logically equivalent or truth-functionally equivalent if and only if they yield identical truth-values in their truth-tables, and so p → q is logically equivalent to both ¬(p ∧ ¬q) and ¬p ∨ q. To make it easier to follow I’ve filled in the truth-tables in stages:

Stage 1

¬(p¬q)   ¬pq   pq  
 T  T    T T   T T
 T  F    T F   T F
 F  T    F T   F T
 F  F    F F   F F


Stage 2

¬(p¬q)   ¬pq   pq  
 T FT   FT T   TTT
 T TF   FT F   TFF
 F FT   TF T   FTT
 F TF   TF F   FTF


Stage 3

¬(p¬q)   ¬pq   pq  
 TFFT   FTTT   TTT
 TTTF   FTFF   TFF
 FFFT   TFTT   FTT
 FFTF   TFTF   FTF


Stage 4

¬(p¬q)   ¬pq   pq  
TTFFT   FTTT   TTT
FTTTF   FTFF   TFF
TFFFT   TFTT   FTT
TFFTF   TFTF   FTF


In every line of the truth tables, the truth-value is the same for p → q, ¬(p ∧ ¬q), and ¬p ∨ q; thus making all three statements logically equivalent.

Consider also these truth-tables:

Stage 1

p¬p   p¬p
T FT   T FT
F TF   F TF

Stage 2

p¬p   p¬p
TTFT   TFFT
FTTF   FFTF


It is said that a statement is tautologous if the form of the statement is sufficient to make it true, e.g. P ∨ ¬P, which is true for every truth-value assignment. A statement is self-contradictory if affirms both the truth and falsehood of the same thing, hence the law of noncontradiction which says “For any proposition p, it is impossible for p to be both true and false at the same time and in the same context.” In propositional logic, an example of a self-contradiction is P ∧¬P. We also have “truth-functional” synonyms for tautologies and contradictions, as well as other terms:
  • A statement is truth-functionally true if and only if every truth-value assignment makes it true.
    • Example: P ∨ ¬P
  • A statement is truth-functionally false if and only if every truth-value assignment makes it false.
    • Example: P ∧ ¬P
  • Two statements are truth-functionally equivalent if and only if every truth-value assignment gives them the same truth table.
    • Example: P → Q and ¬(P ∧ ¬Q)
  • Two statements are truth-functional contradictories if and only if every truth-value assignment gives them different truth-values.
    • Example: P and ¬P.
  • Two statements are truth-functionally consistent if and only if there is some truth value assignment that makes them both true.
    • Example: P ∨ Q and P ∨¬Q.
  • Two statements are truth-functionally inconsistent if and only if there is no truth-value assignment that maks them both true.
    • Examples: P ∧ Q and P ∧ ¬Q; also P ∧ ¬P and Q.
  • Two statements are truth-functional contraries if and only if there is no truth-value assignment that makes both of them true and there is some truth-value assignment that makes both of them false.
    • Example: P ∧ Q and ¬P ∧ Q.
  • Two statements are truth-functional subcontraries if and only if there is no truth-value assignment that makes both of them false and there is some assignment that makes both of them true.
    • Example: P ∨ Q and ¬P ∨Q.
Notice that if two statements are truth-functionally inconsistent, they are either contradictories or contraries. The words contrary and subcontrary are also used in a more general sense. For example, two propositions are contraries if they can’t both be true but they may both be false, e.g. “This ball is green all over” and “This ball is red all over” are contraries. Similarly, two propositions are subcontraries if they can’t both be false but they may both be true, e.g. “I went to the bathroom or I went to the kitchen” and “I didn’t go to the bathroom or I went to the kitchen” are subcontraries: they can’t both be false (I either went to the bathroom or I didn’t) though they can both be true (it’s possible I went to the kitchen). The term contradictory has a normal language definition too, whereby two statements are contradictory if and only if it is impossible for one to be true when the other is false, e.g. the statements “No cats have fur” and “There exists a cat with fur” are contradictories of each other.

You could prove the following is true via a truth table:
[(P → Q) ∧ P] → Q
But it would suck. It would be nice if you could instead do something like this instead:
  1. P → Q
  2. P

  1. Q 1, 2, [rule of logic used to arrive here]
Thankfully, logic has that. Natural deduction refers to types of proof methods that more closely match the way humans naturally think. There are many systems of natural deduction for a variety of logic systems, but for now I’ll concern myself with the system of natural deduction popularly used in propositional logic. For example:
  1. P → Q
  2. P

  1. Q 1, 2, modus ponens
Modus ponens is a famous rule of logic used in natural deduction.

Symbolic Logic: Some Rules of Inference

Some rules of inference I’ve already mentioned:

modus ponens
 
In English In Symbolic Logic
If p then q
p

Therefore, q
p → q
p

∴ q
modus tollens
 
In English In Symbolic Logic
If p then q
Not-q

Therefore, not-p
p → q
¬q

∴ ¬p


In the convention I’m using, the lower case letters p, q, r,...z are placeholders for both simple and compound statements. Thus, below is a valid instance of modus tollens.
  1. (A ∧ B) → C
  2. ¬C

  1. ¬(A ∧ B) 1, 2, modus tollens
It’s worth noting that the order of the premises doesn’t matter when using rules of inference. So below is also a valid use of modus tollens.
  1. ¬C
  2. (A ∧ B) → C

  1. ¬(A ∧ B) 1, 2, modus tollens
Some rules of inference can be used in more than one way. Examples include disjunctive syllogism and simplification.

Disjunctive Syllogism
 
In English In Symbolic Logic
p or q
Not-p

Therefore, q
p ∨ q
¬p

∴ q
p or q
Not-q

Therefore, p
p ∨ q
¬q

∴ p
simplification
 
In English In Symbolic Logic
p and q

Therefore, p
p ∧ q

∴ p
p and q

Therefore, q
p ∧ q

∴ q


Before moving forward, I’ll introduce a quick example of how to use some rules of inference. Suppose we wanted to get C from premises 1 and 2 below:
  1. A ∨ (B ∧ C)
  2. ¬A

  1. B ∧ C 1, 2, disjunctive syllogism
  2. C 3, simplification
Not too hard, right? After learning the above rules of inference, you might even have mentally “seen” that C followed from premises 1 and 2 above. Hopefully you are familiar enough with the symbols by now for me to remove the training wheels of english translation. Some more rules of inference:

conjunction
 
p
q

∴ p ∧ q
constructive dilemma
 
(p → q) ∧ (r → s)
p ∨ r

∴ q ∨ s


hypothetical syllogism
 
p → q
q → r

∴ p → r
addition
 
p

∴ p ∨ q
absorption
 
p → q

∴ p → (p ∧ q)


To illustrate, suppose we wanted to get (B ∨ D) ∧ E from premises 1-4 below:
  1. A
  2. A → Z
  3. Z → B
  4. C → D
  5. (B ∨ D) → E

  1. A → B 2, 3, hypothetical syllogism
  2. (A → B) ∧ (C → D) 4, 6, conjunction
  3. A ∨ C 1, addition
  4. B ∨ D 7, 8, constructive dilemma
  5. (B ∨ D) ∧ E 5, 9, absorption
Symbolic Logic: Some Equivalences

Rules of inference aren’t the only rules of logic. There are also logical equivalences which have the handy property of being replaceable anywhere within a line. In contrast, you can’t use rules of inference anywhere within a line. This for example would be a mistake:
  1. A → (B ∧ C)

  1. A → B 1, simplification Mistake!
Instead rules of inference have to be applied to “whole” lines rather than just anywhere “within” lines. Below are some equivalences:

equivalencename of equivalence
 
p ⇔ ¬¬pdouble negation
 
p → q ⇔ ¬q → ¬ptransposition (also called contraposition)
 
p → q ⇔ ¬p ∨ qmaterial implication
 
p ↔ q ⇔ (p → q) ∧ (q → p) biconditional equivalence
 
¬(p ∧ q) ⇔ ¬p ∨ ¬qDe Morgan’s laws
¬(p ∨ q) ⇔ ¬p ∧ ¬q
 
p ⇔ p ∧ p idempotence
p ⇔ p ∨ p


The contrapositive of a conditional p → q is ¬q → ¬p, e.g. the contrapositive of A → B is ¬B → ¬A, and the contrapositive of ¬C → (A ∧ B) is ¬(A ∧ B) → ¬¬C. Incidentally, the converse of a conditional p → q is q → p. For example, the converse of ¬C → (A ∧ B) is (A ∧ B) → ¬C.

As an example of how to use some equivalences, suppose we want to prove ¬H ∨ C from premises 1-3 below:
  1. A
  2. B → ¬A
  3. ¬(B ∨ B) ↔ (H → C)

  1. ¬¬A → ¬B 2, transposition
  2. A → ¬B 3, double negation
  3. ¬B 1, 4 modus ponens
  4. ¬B ∧ ¬B 5, indempotence
  5. ¬(B ∨ B) 6, De Morgan’s laws
  6. [¬(B ∨ B) → (H → C)] ∧ [(H → C) → ¬(B ∧ B)] 3, biconditional equivalence
  7. ¬(B ∨ B) → (H → C) 9, simplification
  8. H → C 7, 9, modus ponens
  9. ¬H ∨ C 10, material implication
Some more equivalences:

equivalencename of equivalence
 
p ∧ q ⇔ q ∧ pcommutation
p ∨ q ⇔ q ∨ p
 
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ rassociation
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r
 
p → (q → r) ⇔ (p ∧ q) → rexportation
 
p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r)distribution
p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r)


For example, suppose we wanted to get (E ∨ C) ∧ (E ∨ D) from premises 1 and 2:
  1. A → [B → (C ∧ D)]
  2. B ∧ A

  1. (A ∧ B) → (C ∧ D) 1, exportation
  2. A ∧ B 2, commutation
  3. C ∧ D 3, 4, modus ponens
  4. (C ∧ D) ∨ E 5, addition
  5. E ∨ (C ∧ D) 6, commutation
  6. (E ∨ C) ∧ (E ∨ D) 7, distribution
Hopefully, seeing these examples and rules of inference also gives you a feel for how a conclusion can follow logically from the premises. Logic is important if for no other reason than you don’t want to look foolish for calling an argument invalid when it’s really deductively valid. Also, if you want to construct a good deductive argument, remember the rules of logic and try to construct it in such a way that it’s deductively valid.

No, I Won’t Bait and Switch

If you’ve gotten this far, you’ve earned it! I’ll now show how to prove the validity of arguments I talked about at the beginning of this article. First, for the atheist:
  1. If God exists, then gratuitous evil does not exist.
  2. Gratuitous evil does exist.
  3. Therefore, God does not exist.
Let G be “God exists” and E be “Gratuitous evil exists.” Then using symbolic logic:
  1. G → ¬E
  2. E

  1. ¬¬E 2, double negation
  2. ¬G 1, 3, modus tollens
So if a theist is to deny the conclusion that God does not exist, he’ll have to reject a premise. A theist cannot accept the premises and deny the conclusion on pain of irrationality. But the sword of logic cuts both ways. Consider the theist argument below:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
The atheist cannot accept the premises and deny the conclusion on pain of irrationality. Using G to symbolize “God exists” and O as “Objective morality exists” we can prove the validity of the above moral argument as follows:
  1. ¬G → ¬M
  2. M

  1. M → G 1, transposition
  2. G 2, 3, modus ponens
An atheist cannot rationally deny that the conclusion follows logically from the premises on pain of irrationality.

Summarizing Some Logic

Here are the connectives I’ve used:

Type of
connective
EnglishSymbolic
Logic
When it’s true/false
Conjunctionp and qp ∧ qTrue if both are true; otherwise false
Disjunctionp or qp ∨ qFalse if both are false; otherwise true
ConditionalIf p, then qp → qFalse if p is true and q is false; otherwise true
Biconditionalp, if and only if qp ↔ qTrue if both have the same truth-value (i.e. both are true or both are false); otherwise false
NegationNot-p¬pTrue if p is false; false if p is true


Some rules of inference:

modus ponens
 
In English In Symbolic Logic
If p then q
p

Therefore, q
p → q
p

∴ q
modus tollens
 
In English In Symbolic Logic
If p then q
Not-q

Therefore, not-p
p → q
¬q

∴ ¬p


Disjunctive Syllogism
 
In English In Symbolic Logic
p or q
Not-p

Therefore, q
p ∨ q
¬p

∴ q
p or q
Not-q

Therefore, p
p ∨ q
¬q

∴ p
simplification
 
In English In Symbolic Logic
p and q

Therefore, p
p ∧ q

∴ p
p and q

Therefore, q
p ∧ q

∴ q


conjunction
 
p
q

∴ p ∧ q
constructive dilemma
 
(p → q) ∧ (r → s)
p ∨ r

∴ q ∨ s


hypothetical syllogism
 
p → q
q → r

∴ p → r
addition
 
p

∴ p ∨ q
absorption
 
p → q

∴ p → (p ∧ q)


Some equivalences:

equivalencename of equivalence
 
p ⇔ ¬¬pdouble negation
 
p → q ⇔ ¬q → ¬ptransposition (also called contraposition)
 
p → q ⇔ ¬p ∨ qmaterial implication
 
p ↔ q ⇔ (p → q) ∧ (q → p) biconditional equivalence
 
¬(p ∧ q) ⇔ ¬p ∨ ¬qDe Morgan’s laws
¬(p ∨ q) ⇔ ¬p ∧ ¬q
 
p ⇔ p ∧ p idempotence
p ⇔ p ∨ p


And some more equivalences:

equivalencename of equivalence
 
p ∧ q ⇔ q ∧ pcommutation
p ∨ q ⇔ q ∨ p
 
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ rassociation
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r
 
p → (q → r) ⇔ (p ∧ q) → rexportation
 
p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r)distribution
p ∨ (q ∧ r) ⇔ (p ∨ q) ∧ (p ∨ r)


Bait for the Future

Lots more could be said about symbolic logic. As bait for my next entry (Introductory Logic, Part 2), consider this argument from evil:
  1. If God exists, then gratuitous evil does not exist.
  2. Gratuitous evil does exist.
  3. Therefore, God does not exist.
A theist cannot deny both premises on pain of irrationality (it is logically impossible that both premises are false) and I’ll show how one can prove it via symbolic logic in my next entry. Similarly, consider this moral argument:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
An atheist cannot deny both premises on pain of irrationality (it is logically impossible that both premises are false) and I’ll show how one can prove it with symbolic logic in my next entry on symbolic logic.

No comments:

Post a Comment