Classical Logic II – Symbolic Logic

2.1. Symbolic Logic

Now, logic with words seems straightforward enough, but we don’t necessarily want to always talk about specific arguments. Sometimes we wish to examine the generalized form of an argument in order to determine validity (though not soundness: for that we need a specific argument). In order to accomplish this goal, we use symbolic logic.

The general idea for this is to use variables to represent clauses in our sentences, as well as connectives in order to relate our sentences together. This might seem strange, but think of it kind of like algebra: you have a concrete concept (numbers) which you are generalizing using an abstract concept (variables) and then manipulating them together using various operations (plus, minus, square root, etc…). In our case, we are using variables to represent sentences and manipulating them with what are called connectives.

For now, we will represent these clauses using sentence letters. These will be uppercase roman letters (A,B,C,\ldots,P,Q,R,\ldots,X,Y,Z). So for example, we may represent the sentence “It is raining.” with the variable R. There’s nothing special about the letter R, we could easily use I or S. Under no circumstance will we be dealing with arguments that will use more than 26 clauses, however, if a hypothetical logician were ever to need so many variables, he or she can simply add subscripts to the letters (A_1,A_2,\ldots).

So what do we do to make relate different sentences together? We use connectives.

2.2. Logical Connectives

2.2.1. Truth Tables

In order to explain connectives, we’ll first need to go over the concept of a truth table. Basically, a truth table takes a complex sentence and shows how each of the logical connectives modify the truth of the base sentences.

In order to build one, we lay out all of the different truth assignments to the base (or “atomic”) sentences:

A B A and B
T T
T F
F T
F F

We see here that the truth table for the sentence “A and B” will be based on the truth values of A and B and will be laid out underneath the rightmost column. On the left are the truth values for A and B, in each of the four possible combinations of values. The value “T” stands for “true” and the value “F” stands for “false”. In general, for a truth table with n atomic sentences, there will be 2^n possible combinations of truth values, which means 2^n rows.

So let’s check out our very first connective:

2.1.2. Conjunction

As we saw above, one important connective when building complex sentences is “and”. It is usually written as \& or \wedge. We will be using the \wedge notation. More formally, “and” is known as conjunction.

The sentence A\wedge B is read “A and B” and is true if and only if A and B (also called the conjuncts) are both true. Represented as a truth table:

A B A\wedge B
T T T
T F F
F T F
F F F

Thus, perhaps obviously, A\wedge B is true exactly when A is true and B true, and false at all other times.

2.1.3. Disjunction

Well that was easy enough. What about disjunction, also known as an “or” statement. These are statements which indicate multiple options. For example, the sentence “A or B” is written A\vee B. In this example, A and B are known as the “disjuncts”.

Let’s look at the truth table for a disjunction:

A B A\vee B
T T T
T F T
F T T
F F F

Most of this is fairly straightforward. We would expect “true or false” and “false or true” to both evaluate to true, as well as “false or false” to evaluate to false. But what about “true or true”? Should this really evaluate to true? After all, when you go to a restaurant and a meal comes with “soup or salad” you can’t very well take both. Classical logic, however, was designed to deal with more mathematical concepts. Consider the sentence “either two is even or three is odd”. Clearly such a sentence is true, despite having two true disjuncts. If it still doesn’t make sense, just try to roll with it for now, and later we’ll see a version of “or” called “exclusive or” which behaves more like you might expect.

2.1.4. Negation

Unlike conjunction and disjunction which connects two-sentences, negation is a one-place connective (which is an oddly-named concept). The negation of a sentence A is written \sim A or \bar A or \neg A. We will be using the notation \neg A.

A \neg A
T F
F T

Thus, essentially, a negation in classical logic simply flips a sentence from true to false, or vice-versa.

2.1.5. Implication

Implication is really the most interesting of the connectives, and even tends to do most of the work in proofs. It is a translation of the sentence “if A then B” and is written A\rightarrow B. In this example, A is called the antecedent and B is called the consequent.

A B A\rightarrow B
T T T
T F F
F T T
F F T

This one can take a bit of getting used to. The first two lines behave more-or-less as expected. If both the antecedent and the consequent are true, then the the antecedent truly does imply the consequent, making the implication true. If the antecedent is true, but the consequent is false, then the first cannot truly be said to imply the second, making the entire thing false. The last two lines are a big tricky and perhaps counter-intuitive, but they basically say that if the antecedent is false, then there’s no way to really tell whether it implies the consequent. So regardless of whether the consequent is true or not, the entire implication will be true whenever the antecedent is. It’s a bit confusing at first, but when you give it some thought, it starts to make sense.

2.1.6. Biconditional

Our last connective is called the biconditional and is used to indicate that two sentences always have the same truth value. It is written A\leftrightarrow B and read “A if and only if B.

A B A\leftrightarrow B
T T T
T F F
F T F
F F T

Basically this tells us that whenever A is true, then B is true and whenever A is false B is false, and vice-versa.

2.1.7. Complex Sentences

One note before moving on: we aren’t limited to atomic sentences when applying connectives. We can use progressively more complex sentences and string them together to create arbitrary sentences. For example:

\neg(A\wedge B)\leftrightarrow(\neg A\vee\neg B)

is a valid sentence, even though we are connecting together progressively more complicated things.

2.2. Symbolizing Arguments

Let’s try symbolizing our very first argument. If you’ll recall:

  • If it rains, then the grass will be wet.
  • It is raining.
  • Therefore, the grass is wet.

By symbolizing “It is raining.” as R and “The grass is wet.” as W, we arrive at the following:

  • P1) R\rightarrow W
  • P2) R
  • C) W

This type of argument has a formal name modus ponens although that’s not terribly important right now. Basically it says that if we know R then we can infer W (P1). However we do know R (by P2), therefore we can conclude W. One important thing to note is that we don’t care about the tenses of our sentences. R stands equally well for “it is raining” and “it rains”; while W stands for both “the grass will be wet” and “the grass is wet”.

Now, let’s consider some words in natural English and their logical translations.

2.2.1. “But”

In plain English, the word “but” means something different than the word “and”. For example, the sentence “I want to go to a movie, but I’m too tired” has a slightly different meaning than “I want to go to a movie and I’m too tired.” Specifically, “but” tends to be used to imply a slightly different situation than what would be intuited from the first clause. In our example, I want to go to a movie, but when I say I’m too tired, it implies that I might not go. Logically speaking, however, “but” and “and” are identical.

Why is this? Consider the sentence “John went to the mall, but Mary did not.” What is required in order for this sentence to be true? Well, namely it requires first that John went to the mall, and second that Mary didn’t. When all we are considering is the truth values of the atomic sentences (as we do in logic) “but” ends up having the same truth-functional requirements as “and”. As such, the two are logically equivalent and in particular we might translate this sentence as J\wedge\neg M.

2.2.2. “Then”

“Then” can be a tricky concept. In plain English, sometimes it can mean an implication, but sometimes it can also be a conjunction. The key in determining this is typically the word “if”. When you see something like “If it rains, then I will bring an umbrella.” we can hopefully see fairly easily that this will take the logical form of R\rightarrow U. However a sentence such as “I went to the grocery store, then I bought some milk.” is a different creature. In this case, we are not saying that milk will be bought if I go to the grocery store, but rather I did go there, and then I bought the milk. Thus, such a sentence would be translated as G\wedge M. This can be tricky, but a careful analysis of the meaning of the natural English sentence should be enough to determine which situation to apply.

2.2.3. “Nor”

This one can be hard to translate. When you say something like “I will neither give you a bus ticket, nor lend you my car.” the translation is not immediately obvious. What connective is “nor”? But the simplest way to examine this is to look at the word itself: n-or. As you might guess, “nor” is translated as a negated “or” statement. Thus our statement above becomes \neg(B\vee C). And this makes sense. If B\vee C were true, then I must be either taking one of the two options. But if I am doing neither, then the opposite of this sentence will be simply its negation. Later we will see that this sentence is actually equivalent to another version: \neg B\wedge\neg C.

2.2.4. “Only if”

This is easily the hardest translation we’ll be talking about today. If you’ll recall from above, the translation of the sentence “if A, then B” is A\rightarrow B. Similarly, the translation for “B, if A” is also A\rightarrow B. So you might be inclined to think that the antecedent will always be whichever term follows the word “if”. However this would be incorrect.

The sentence “B, only if A” is actually translated as B\rightarrow A. But why is this? Well the simplest thing to do is to make a truth table the two versions of the sentence:

A B A\rightarrow B B\rightarrow A
T T T T
T F F T
F T T F
F F T T

In the last column, we can see that in all rows where B\rightarrow A is true (ie: ignore the third row), B is only true when A is true. On the other hand, if we were to translate “B, only if A as A\rightarrow B then we would run into problems as in the third line (where A\rightarrow B is actually true), B actually holds without A holding.

So while it may seem that this translation violates the temporal ideas of cause coming before effect, it is important to note that, because we are dealing purely with logic, all we care about are the truth values of the component sentences, and that of the final result. With this in mind, one can hopefully see why the correct translation is B\rightarrow A.

2.2.5. “One or the other, but not both”

As mentioned above, in a standard disjunction, if both of the terms are true, then the resulting sentence is true. However this may not always accurately represent what we mean in spoken English. Take, for example, the sentence “John was either born on a Tuesday or a Friday.” Now, clearly we don’t mean that both of these could possibly be true. Should we wish to indicate that in our sentence’s logical form, however, this information will be lost. So what can we do?

Well, with a little bit of thought, it’s easy to see that such a sentence means the same thing as “John was either born on a Tuesday or a Friday, but not both.” With this translation, it becomes much easier to see what we’re looking for. Thus, the correct translation of this sentence would be (T\vee F)\wedge\neg(T\wedge F). This may look complicated, but if you stare at it closely enough, you begin to see how it works: in the first part, we are using our original statement that either John was born on a Tuesday or a Friday; but in the second part we are clarifying that it is not the case that both took place.

This also allows us to work out the problem mentioned about with getting soup or salad: (O\vee A)\wedge\neg (O\wedge A). You can get soup, or you can get salad; but you’re not able to get both soup and salad.

2.3. Questions

You should now be able to answer the following questions.

  1. Translate all the arguments presented in section 1 into their logical forms.
  2. Translate the following sentences into their logical forms:
    1. If John and Michael both ask Louise to the dance, then she will either have a date or will not go.
    2. The statement “two is even” is equivalent to the statement “two plus one is odd”.
    3. If I will neither drink Coke nor Pepsi then I shan’t drink Pepsi and I shan’t drink Coke.
    4. Sharon will go to the cinema, but only if Bill pays for her ticket and buys her some popcorn; but Angela will go either way.
  3. Create truth tables for the following sentences:
    1. (A\rightarrow B)\rightarrow B
    2. \neg(A\wedge B)
    3. A\wedge\neg A
    4. A\vee\neg A
    5. ((A\rightarrow(B\vee C))\rightarrow(A\rightarrow(D\leftrightarrow B)))\vee D
  4. Use truth tables to show that the following sentences are equivalent and explain why a biconditional can be read as “if and only if” or “is equivalent to”:
    • (A\rightarrow B)\wedge(B\rightarrow A)
    • (A\wedge B)\vee(\neg A\wedge\neg B)
    • A\leftrightarrow B
  5. Use a truth table to show that the sentence (A\vee B)\wedge\neg(A\wedge B) can also be translated as \neg(A\leftrightarrow B).
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