This is part 1 of my series on logic and critical thinking.
 Introductory Logic, Part 1—Introducing both logic in general (such as the difference between a deductive and inductive argument) and propositional logic in particular
 Introductory Logic, Part 2—More propositional logic
 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:
 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).
 Gratuitous evil does exist.
 Therefore, God does not exist.
 If God does not exist, then objective morality does not exist.
 Objective morality does exist.
 Therefore, God exists.
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 easytolearn 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. selfcontradictory, 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:
 If it is raining, then my car is wet.
 It is raining.
 Therefore, my car is wet.
 If P, then Q
 P
 Therefore, Q.
 If P, then Q
 NotQ
 Therefore, notP.
 If it is raining, then my car is wet.
 My car is wet.
 Therefore, it is raining.
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 C_{1}.
 We’ve seen the law of gravity work in conditions C_{2}.
 We’ve seen the law of gravity work in conditions C_{3}.
 …
 We’ve seen the law of gravity work in conditions C_{n}.
 Therefore, (probably) the law of gravity holds true in all physical situations.
 All observed A’s are B’s.
 x is an A.
 Therefore, probably, x is a B.
 We’ve sampled a thousand animals of this species in this forest and threefifths of them have black fur.
 Therefore, probably threefifths of the animals of this species in this forest have black fur.
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.
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:
Connective  Symbolic Logic  English Meaning  Notes 

∧ (conjunction)  p ∧ q  p and q  The p and q parts are called conjuncts. 
∨ (disjunction)  p ∨ q  p or q  The p and q parts are called disjuncts. 
→ (conditional)  p → q  If p, then q  The 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 ↔ q  p, if and only if q  This means the same thing as “p → q and q → p.” 
¬ (negation)  ¬p  Notp  The negation of P is ¬P, and ¬P means “notP” 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:
Operation  Example  Alternate Forms 

conjunction  A ∧ B  A & B, A • B, AB 
implication  A → B  A ⊃ B 
material equivalence  A ↔ B  A ≡ 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  English  Symbolic Logic  When it’s true/false 

Conjunction  p and q  p ∧ q  True if both are true; otherwise false 
Disjunction  p or q  p ∨ q  False if both are false; otherwise true 
Conditional  If p, then q  p → q  False if p is true and q is false; otherwise true 
Biconditional  p, if and only if q  p ↔ q  True if both have the same truthvalue (i.e. both are true or both are false); otherwise false 
Negation  Notp  ¬p  True 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  


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.
English  Symbolic Logic 

p is a sufficient condition for q  p → 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.
English  Symbolic Logic 

p is a necessary and sufficient condition for q  p ↔ 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 truthvalue (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 truthfunctionally equivalent if and only if they yield identical truthvalues in their truthtables, 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 truthtables in stages:
Stage 1  


Stage 2  


Stage 3  


Stage 4  


In every line of the truth tables, the truthvalue is the same for p → q, ¬(p ∧ ¬q), and ¬p ∨ q; thus making all three statements logically equivalent.
Consider also these truthtables:
Stage 1  


Stage 2  


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 truthvalue assignment. A statement is selfcontradictory 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 selfcontradiction is P ∧¬P. We also have “truthfunctional” synonyms for tautologies and contradictions, as well as other terms:
 A statement is truthfunctionally true if and only if every truthvalue assignment makes it true.
 Example: P ∨ ¬P
 A statement is truthfunctionally false if and only if every truthvalue assignment makes it false.
 Example: P ∧ ¬P
 Two statements are truthfunctionally equivalent if and only if every truthvalue assignment gives them the same truth table.
 Example: P → Q and ¬(P ∧ ¬Q)
 Two statements are truthfunctional contradictories if and only if every truthvalue assignment gives them different truthvalues.
 Example: P and ¬P.
 Two statements are truthfunctionally 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 truthfunctionally inconsistent if and only if there is no truthvalue assignment that maks them both true.
 Examples: P ∧ Q and P ∧ ¬Q; also P ∧ ¬P and Q.
 Two statements are truthfunctional contraries if and only if there is no truthvalue assignment that makes both of them true and there is some truthvalue assignment that makes both of them false.
 Example: P ∧ Q and ¬P ∧ Q.
 Two statements are truthfunctional subcontraries if and only if there is no truthvalue assignment that makes both of them false and there is some assignment that makes both of them true.
 Example: P ∨ Q and ¬P ∨Q.
You could prove the following is true via a truth table:
[(P → Q) ∧ P] → QBut it would suck. It would be nice if you could instead do something like this instead:
 P → Q
 P
 Q 1, 2, [rule of logic used to arrive here]
 P → Q
 P
 Q 1, 2, modus ponens
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 Notq Therefore, notp 
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.
 (A ∧ B) → C
 ¬C
 ¬(A ∧ B) 1, 2, modus tollens
 ¬C
 (A ∧ B) → C
 ¬(A ∧ B) 1, 2, modus tollens
Disjunctive Syllogism  

In English  In Symbolic Logic 
p or q Notp Therefore, q 
p ∨ q ¬p ∴ q 
p or q Notq 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:
 A ∨ (B ∧ C)
 ¬A
 B ∧ C 1, 2, disjunctive syllogism
 C 3, simplification
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 14 below:
 A
 A → Z
 Z → B
 C → D
 (B ∨ D) → E
 A → B 2, 3, hypothetical syllogism
 (A → B) ∧ (C → D) 4, 6, conjunction
 A ∨ C 1, addition
 B ∨ D 7, 8, constructive dilemma
 (B ∨ D) ∧ E 5, 9, absorption
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:
 A → (B ∧ C)
 A → B 1, simplification Mistake!
equivalence  name of equivalence 

p ⇔ ¬¬p  double negation 
p → q ⇔ ¬q → ¬p  transposition (also called contraposition) 
p → q ⇔ ¬p ∨ q  material implication 
p ↔ q ⇔ (p → q) ∧ (q → p)  biconditional equivalence 
¬(p ∧ q) ⇔ ¬p ∨ ¬q  De 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 13 below:
 A
 B → ¬A
 ¬(B ∨ B) ↔ (H → C)
 ¬¬A → ¬B 2, transposition
 A → ¬B 3, double negation
 ¬B 1, 4 modus ponens
 ¬B ∧ ¬B 5, indempotence
 ¬(B ∨ B) 6, De Morgan’s laws
 [¬(B ∨ B) → (H → C)] ∧ [(H → C) → ¬(B ∧ B)] 3, biconditional equivalence
 ¬(B ∨ B) → (H → C) 9, simplification
 H → C 7, 9, modus ponens
 ¬H ∨ C 10, material implication
equivalence  name of equivalence 

p ∧ q ⇔ q ∧ p  commutation 
p ∨ q ⇔ q ∨ p  
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ r  association 
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r  
p → (q → r) ⇔ (p ∧ q) → r  exportation 
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:
 A → [B → (C ∧ D)]
 B ∧ A
 (A ∧ B) → (C ∧ D) 1, exportation
 A ∧ B 2, commutation
 C ∧ D 3, 4, modus ponens
 (C ∧ D) ∨ E 5, addition
 E ∨ (C ∧ D) 6, commutation
 (E ∨ C) ∧ (E ∨ D) 7, distribution
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:
 If God exists, then gratuitous evil does not exist.
 Gratuitous evil does exist.
 Therefore, God does not exist.
 G → ¬E
 E
 ¬¬E 2, double negation
 ¬G 1, 3, modus tollens
 If God does not exist, then objective morality does not exist.
 Objective morality does exist.
 Therefore, God exists.
 ¬G → ¬M
 M
 M → G 1, transposition
 G 2, 3, modus ponens
Summarizing Some Logic
Here are the connectives I’ve used:
Type of connective  English  Symbolic Logic  When it’s true/false 

Conjunction  p and q  p ∧ q  True if both are true; otherwise false 
Disjunction  p or q  p ∨ q  False if both are false; otherwise true 
Conditional  If p, then q  p → q  False if p is true and q is false; otherwise true 
Biconditional  p, if and only if q  p ↔ q  True if both have the same truthvalue (i.e. both are true or both are false); otherwise false 
Negation  Notp  ¬p  True 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 Notq Therefore, notp 
p → q ¬q ∴ ¬p 
Disjunctive Syllogism  

In English  In Symbolic Logic 
p or q Notp Therefore, q 
p ∨ q ¬p ∴ q 
p or q Notq 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:
equivalence  name of equivalence 

p ⇔ ¬¬p  double negation 
p → q ⇔ ¬q → ¬p  transposition (also called contraposition) 
p → q ⇔ ¬p ∨ q  material implication 
p ↔ q ⇔ (p → q) ∧ (q → p)  biconditional equivalence 
¬(p ∧ q) ⇔ ¬p ∨ ¬q  De Morgan’s laws 
¬(p ∨ q) ⇔ ¬p ∧ ¬q  
p ⇔ p ∧ p  idempotence 
p ⇔ p ∨ p 
And some more equivalences:
equivalence  name of equivalence 

p ∧ q ⇔ q ∧ p  commutation 
p ∨ q ⇔ q ∨ p  
p ∧ (q ∧ r) ⇔ (p ∧ q) ∧ r  association 
p ∨ (q ∨ r) ⇔ (p ∨ q) ∨ r  
p → (q → r) ⇔ (p ∧ q) → r  exportation 
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:
 If God exists, then gratuitous evil does not exist.
 Gratuitous evil does exist.
 Therefore, God does not exist.
 If God does not exist, then objective morality does not exist.
 Objective morality does exist.
 Therefore, God exists.
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