What are Logic & the Philosophy of Language?:
Logic and the Philosophy of Language are often treated separately, but they are close enough that they are presented together here. Logic is the study of methods of reasoning and argumentation, both proper and improper. The Philosophy of Language involves the study of how our language interacts with our thinking. Atheists and theists don't typically disagree much in these subject areas, but they are nevertheless critical because a proper understanding of them is necessary for the sake of sound reasoning, rational thinking, and reliable communication.
Why is Logic important?:
Logic is perhaps the most important subject which is rarely, if ever, presented in schools. A study of logic can help a person better construct their own arguments and critique the arguments of others. It isn't uncommon for many popular arguments to be filled with errors and fallacies because so many people are simply unaware of how arguments are correctly formulated. A solid grounding in logic and reasoned arguments should be at the heart of any education and it's absence is a primary reason why so many errors in reasoning occur on such a constant basis in all areas of human inquiry.
Why Should Atheists Care about Logic?:
Because so few people ever learn about logic, logical fallacies, and logical arguments, almost everyone makes a lot of basic errors in the arguments they offer and in their critiques of others' arguments. Irreligious atheists who want to do an effective job in critiquing religion, religious beliefs, and theism, absolutely need some grounding in basic logic in order to both avoid those common mistakes and to be able to point them out when reading others' arguments. To put it simply, you cannot argue well, consistently, and reliably unless you know what you are doing and this is the place to start.
What is Modal Logic?:
Modal logic involves the study of the reasoning behind what might be true that is to say, the study of the reasoning behind different modes of reality. Formal equations in modal logic use operators not found in traditional formal logic. For example, the letter L represents "it is necessary that" and the letter M represents "it is possible that." Thus, (Lp) means "it is necessary that p" and (Mp) means "it is possible that p" (where p is an idea or proposition, like "it is raining" or "God exists"). Modal logic is frequently used in higher-level discussions about the existence of gods.
Philosophy of Language in the 20th Century:
No matter what the subject, the one common denominator is the fact that language is being used to clarify and explain it. The Philosophy of Language experienced a resurgence during the 20th century, largely due to the writings of Ludwig Wittgenstein. Some have been studied the ways in which the very categories and formulations of the language we use can constrain and inhibit our ideas and reasoning process. A cross-over between this and the Philosophy of Religion is the nature of Religious Language for example, how it constructs religious beliefs or how it performs actions rather than merely communicates.
Questions asked in Logic and the Philosophy of Language:
What constitutes "good" or "bad" reasoning?
How do we determine whether a given piece of reasoning is good or bad?
What does it mean that a statement has meaning?
What is the difference between literal and figurative meanings?
Important Texts in Logic and the Philosophy of Language:
Principia Mathematica, by Bertrand Russell and Alfred North Whitehead
Prior and Posterior Analytics, by Aristotle
Tractatus Logico-Philosophicus, by Ludwig Wittgenstein
Logical Syntax of Language, by Rudolf Carnap