Logic Part 2: Translation

This is Part 2. You should see What is Logic? and Logic Part 1: What is Propositional Logic? before reading this.

‘Translation’ refers to the act of converting statements of natural language to statements of a symbolic logical system. In this case I will discuss how to convert statements of English into statements of propositional logic. Translation requires us to know logical connectives used in propositional logic, and ways we use logical connectives in English.

Not all propositions use logical connectives. In order to translate such a proposition, replace it with a capital letter. For example “1+1=2” is a proposition that can be translated as “A.”

What are logical connectives?

Logical connectives are ways of saying when propositions are true. For example “or” is a way of saying at least one of two propositions are true. “The President of the USA is a man or a dog” means that either the proposition “the President of the USA is a man” is true or the proposition “the President of the USA is a dog” is true. We can then replace the propositions with capital letters to get “A or B.” However, the word “or” is also replaced in symbolic logic with the symbol “∨,” so we would write it out as “A ∨ B.”

There are five connectives used in propositional logic: “not” (¬), “and” (∧), “or” (∨), “implies” (→), and “if and only if” (↔):

Negation

The “negation” connective is “¬,” which means “is false.” We also state negation by saying “is not true” or sometimes simply as “not.” For example, “the President of the USA is not a dog” should be taken to mean “it is false that the President of the USA is a dog.” It can be translated into propositional logic as “¬A.” In this case “A” stands for “ the President of the USA is a dog.”

More examples of translating English statements that use negation:

1. It’s false that only two people exist. [“A” stands for “only two people exist.” Translation: “¬A.”]
2. Not all dogs are fish. [This means “it’s false that all dogs are fish.” “B” stands for “all dogs are fish.” Translation: “¬B.”]
3. It is not the case that Samantha went to the store. [This means “it’s false that Samantha went to the store. “C” stands for “Samantha went to the store.” Translation: “¬C.”]
4. Evolution is false. [This will be taken to mean “the central claims of the theory of evolution are not consistent with our observations.” “D” stands for “the central claims of the theory of evolution are consistent with our observations.” Translation: “¬D.”]
5. People are wrong who think that nothing exits. [This means “it’s false that nothing exists.” “E” stands for “nothing exists.” Translation “¬E.”]
6. It’s absurd to think that “2+2=5.” [This implies that “2+2=5” is false. “F” stands for “2+2=5.” Translation: “¬F.”]

Conjunction

The conjunction connective is “∧,” which means “and.” We also state conjunctions by saying “but.” For example, “the President of the USA has an PhD, but dislikes chocolate” should be taken to mean “the President of the USA has a PhD, and the President of the USA dislikes chocolate.” This can be translated as “A ∧ B.” Both “A” and “B” must represent statements that are true or false, so “B” must represent “the President of the USA dislikes chocolate” rather than “dislikes chocolate.”

More examples of translating English statements that use conjunctions are the following:

1. Jill is tall and Jeff is short. [“A” stands for “Jill is tall” and “B” stands for “Jeff is short.” Translation: “A ∧ B.”]
2. Jennifer is a human and a mammal. [“C” stands for “Jennifer is a human” and “D” stands for “Jennifer is a mammal.” Translation: “C ∧ D.”]
3. At least two people exist, but no unicorns exist. [“E” stands for “at least two people exist” and “F” stands for “no unicorns exist.” Translation: “E ∧ F.”]
4. It is true that “1+1=2” and it is true that “2+2=4.” [“G” stands for “1+1=2” and “H” stands for “2+2=4.” Translation: “G ∧ H.”]
5. Evolution is true and we should believe it. [“I” stands for “the central claims of evolution are consistent with our observations” and “J” stands for “we should believe the central claims of evolution.” Translation: “I ∧ J.”]

Disjunction

The disjunction connective is “∨” and it means “or.” We also state this by saying “and/or” or “unless.” It means that one or both of two propositions are true. For example, “Julia went to the store unless she went to school” can be translated as “A ∨ B.” In this case “A” stands for “Julia went to the store” and “B” stands for “Julia went to school.”

Sometimes disjunctions are meant to be exclusive—only one of the two propositions are true. For example, “the President is a man or a woman” could imply that the President is a man or a woman, but not both a man and a woman. In that case we can translate the disjunction as saying “either A or B, but not both-A-and-B.” This can be written in symbolic form as “(A ∨ B) ∧ ¬(A ∧ B). (“A” stands for “the President is a man” and “B” stands for “the President is a woman.”) We use parentheses to group propositions together that are intricately tied. The logical connective that is not in any parentheses is the “main connective.”

More examples of translating English statements that use disjunctions are the following:

1. Either the road leads back to town or the road leads to a farm. [“A” stands for “the road leads back to town” and “B” stands for “the road leads to a farm.” Translation: A ∨ B]
2. Either Jack or Jill is tall. [“C stands for “Jack is tall” and “D” stands for “Jill is tall.” Translation: “C ∨ D.”]
3. Humans are mammals unless they’re reptiles. [“E stands for “humans are mammals” and F” stands for “humans are reptiles.” Translation: “E ∨ F.”]
4. Either evolution is true or creationism is true. [“G” will stand for “the central claims of evolution are consistent with our observations” and “H” will stand for “the central claims of creationism are consistent with our observations.” Translation: “G ∨ H.”]

Material conditional

The material conditional is “→” and means “implies.” We also state material conditionals by saying “if… then” or “only if.” For example, “if all men are mortal, then Socrates is mortal.” We can translate this to be “A → B.” In that case “A” stands for “all men are mortal” and “B” stands for “Socrates is mortal.” The material conditional means that the first part is false or the second part is true.

People often talk about necessary and sufficient conditions, and they can be translated to be propositions using material conditionals. “A is necessary for B” means “B → A.” “A is sufficient for be” means “A → B.” For example, we could say that “the existence of the Sun is sufficient for the existence of light,” or that “the existence of atoms are necessary for the existence of humans.”

Also note that people sometimes say “A if B,” such as “light exists if the Sun exists.” In this case we have to switch the first and second parts of the material conditional to translate it into logical form. “A” can stand for “light exists” and “B” can stand for “the Sun exists.” The statement will then be translated as “B → A.”

More examples of translating English statements that use material conditionals are the following:

1. If Lisa kicked John, then John got hurt. [“A” stands for “Lisa kicked John.” “B” stands for “John got hurt.” Translation: “A → B.”]
2. George likes milk if he likes milkshakes. [“C” stands for “George likes milk.” “D” stands for “George likes milkshakes.” Translation: “D → C.”]
3. Birds are animals only if birds are living organisms. [“E” stands for “all birds are animals” and “F” stands for “birds are living organisms.” Translation: “E → F.”]
4. The fact that all humans are mammals implies that all humans are animals. [“G” stands for “all humans are mammals” and “H” stands for “all humans are animals.” Translation: “G → H.”]
5. Being human is sufficient to be a mammal. [“I” means “something is a human.” “J” means “something is a mammal.” Translation: “I → J.”]
6. Being a mammals is necessary for being a human. [“I” means “something is a human” and “J” means “something is a mammal.” Translation: “J → I.”]

Material equivalence

The material equivalence is “↔” and it means “if and only if.” Sometimes we state it by saying “just in case.” It is used to state that two propositions have the same truth value—they’re both true, or they’re both false. For example, “Socrates is a person if and only if he’s a rational animal” could be translated as “A ↔ B.” In that case “A” stands for “Socrates is a person” and “B” stands for “Socrates is a rational animal.” Either Socrates is a person and a rational animal, or he is neither a person nor a rational animal.

More examples of translating English statements that use material equivalence are the following:

1. Jessica will win the game if and only if Bob loses the game. [“A” stands for “Jessica will win the game.” “B” stands for “Bob loses the game.” Translation: “A ↔ B.”]
2. Water is in the glass just in case H2O is in the glass. [“C” stands for “water is in the glass.” “D” stands for “ H2O is in the glass.” Translation: “C ↔ D.”]
3. Lizards are animals if and only if lizards are creatures. [“E” stands for “lizards are animals.” “F” stands for “lizards are creatures.” Translation: “E ↔ F.”]

Multiple connectives

Many statements use multiple connectives. Propositional logic requires that we keep all of the logical connectives possible. “Neither… nor” is a common example. “Humans are neither plants nor reptiles” means “it is not the case that humans are plants, and it is not the case that humans are reptiles.” This can be translated as “¬A ∧ ¬B.” In this case “A” stands for “all humans are plants” and “B” stands for “all humans are reptiles.”

Another example is “unless.” Although I translated “unless” to mean “or” above, there are other ways people can use the word and the context should be considered. It has been suggested that we should sometimes translate “A unless B” as “¬A ∨ ¬B” or “ B → ¬A.”

When multiple connectives are used, we often need to know which connective is the “main connective.” Identifying the main connective is needed when other connectives are grouped together inside parentheses. For example, “either evolution is consistent with our observations and a species can change over time, or creationism is consistent with our observations.” In this case a disjunction is the main connective and the conjunction is grouped together inside parentheses. “A” stands for “evolution is consistent with our observations, “B” stands for “a species can change over time,” and “C” stands for “creationism is consistent with our observations.” We can then translate the statement as “(A ∧ B) ∨ C.”

It is important to make it clear when two propositions contradict one another. We could need to translate a contradictory statement such as “no humans are reptiles, and some humans are reptiles.” We could translate “no humans are reptiles” as “A,” and “some humans are reptiles” as “B,” but that wouldn’t make it clear that both propositions form a contradiction. For that reason, we should make sure to make it clear that one is the negation of the other. We could simply translate this statement as saying “A ∧ ¬A.” In that case “some humans are reptiles” is taken to be “¬A” (that it’s false that no humans are reptiles).

More examples of translating English statements that use multiple connectives include the following:

1. If Sophia is not a dog, then Sophia is a human. [“A” stands for “Sophia is a dog.” “B” stands for “Sophia is a human.” Translation: “¬A → B.”]
2. Hansel will win the game if and only if Gretel doesn’t win. [“C” stands for “Hansel will win the game.” “D” stands for “Gretel will win the game.” Translation: “C ↔ ¬D”]
3. Neither Democrats nor Republicans will stop corporate welfare. [“E” stands for “Democrats will stop corporate welfare.” “F” stands for “Republicans will stop corporate welfare.” Translation: “¬E ∧ ¬F.”]
4. Either the President of the USA is a reptile, or he’s both a mammal and an animal. [“G” stands for “the President of the USA is a reptile.” “H” stands for “the President of the USA is a mammal.” “I” stands for “the President of the USA is an animal.” Translation: “G ∨ (H ∧ I).”]
5. Either all lizards are mammals or some lizards are not mammals. [“J” stands for “all lizards are mammals.” Translation: “J ∨ ¬J.”]

Scheme of abbreviation

When translating, it’s important to state what each letter stands for. The scheme of abbreviation is what tells us what each letter represents, and multiple statements can be translated using the same scheme of abbreviation. Translation requires that each letter represents the same thing within the scheme of abbreviation.

Consider the following three statements:

1. Jack is tall or Jill is short.
2. Jack is tall if and only if Jill is not short.
3. If Jill is short, then Jack is not tall.

We could use the following scheme of abbreviation for these three statements:

A: Jack is tall.

B: Jill is short.

We could then use this scheme of abbreviation to translate all three statements in the following way:

1. A ∨ B
2. A ↔ ¬B
3. B → ¬A

Arguments

Arguments are a series of two or more statements when any number of statements is meant to be a reason to believe another.  The statement that is being supported is the “conclusion” and the other statements are the “premises.” Arguments are often used for persuasion—people who agree with the premises are likely to agree with the conclusion of a logically valid argument.

Sometimes arguments are stated as a single statement. For example, “Socrates is a man; if Socrates is a man, then he’s mortal; therefore, Socrates is mortal.” When translating arguments, we want to separate each premise and conclusion. We also want to show the maximal number of premises. In this case we want to separate the premises and conclusion into three separate statements:

1. Socrates is a man.
2. If Socrates is a man, then Socrates is mortal.
3. Therefore, Socrates is mortal.

A translation and scheme of abbreviation for this argument is the following:

1. A
2. A → B
3. ∴ B

A: Socrates is a man.

B: Socrates is mortal.

Notice that “∴” is the symbol used to mean “therefore.” It indicates that the final statement is a conclusion. (Quite often, a line is used to separate premises from conclusions as well.)

One reason that translation is important is because we want to know when arguments stated in natural language are logically valid, and we can use propositional logic to know when an argument is logically valid.

The argument given above uses an argument form called “modus ponens.” Every argument with this form is logically valid. You can replace “A” and “B” with any two propositions, and the argument will be logically valid. For example, “A” could stand for “all dogs are mammals” and “B” could stand for “all mammals are animals.” The argument would then be “All dogs are mammals. If all dogs are mammals, then all dogs are animals. Therefore, all dogs are animals.”

It is important that deductive arguments are logically valid—if we know a logically valid deductive argument has true premises, then we should agree with the conclusion. However, not all logically valid arguments are good arguments. One of the most important reasons is that logical validity only concerns the logical form of the argument. A logically valid argument can have false premises.

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Related

• What is Logic?
• Logic Part 1: What is Propositional Logic?